Study resources
The full reference that goes with the periodic table on the back of your ClipDesk board. Free, works on your phone, and prints cleanly.
Reference only. Values are rounded and may contain mistakes. Check your textbook for work that matters, and only use this during exams if your instructor allows it. The periodic table on your board is the part meant for tests.
Full reference sheetUCSD course guidesPrecalculus and calculus (I, II, III)Probability and statistics (MATH 11, 183, 180A and more)MathPhysicsChemistryAdvanced mathUnit circleAdvanced physicsAdvanced chemistryEngineeringBiology, CS, finance and lab skillsComputer science and theoryMore math for upper-division coursesMore physics for upper-division coursesMore chemistry for upper-division coursesMore engineering for upper-division coursesGraduate engineeringBeyond STEM: other majors and GEsConversions and constantsReference tablesEngineering and science tables
The full reference sheet
The original big board-back reference: periodic table, rulers, formulas, conversions, constants, unit circle, protractor and more. Tap an image to open it full size. Download the PDF to print it at 13 x 9.4 in.
The board itself ships with the simple periodic table back (PDF), so it stays test-friendly.
UCSD course guides
Pick your course to see what it covers and jump straight to the formula cards for it. Course lists come from the UCSD General Catalog. Browse all undergraduate courses or graduate courses. Exams vary by instructor, so check your syllabus.
MathPhysicsChemistryMechanical and AerospaceElectrical and Computer EngineeringComputer ScienceBioengineeringChemical Engineering and NanoEngineeringStructural Engineering
Math
MATH 2
Introduction to College Mathematics
- linear relationships
- polynomials
- rational and radical equations
- exponentials and logs
- geometry and trig
MATH 3B
Foundations of Precalculus
- functions
- polynomial and rational
- exponential and log
- trig
MATH 4C
Precalculus for Science and Engineering
- graphing
- circular functions
- vectors
- conics
- polar coordinates
MATH 10B
Calculus II
- integration
- FTC
- integration methods
- areas and volumes
- separable ODEs
MATH 11
Calculus-Based Intro to Probability and Statistics
- probability
- random variables
- distributions
- CLT
- inference
- regression
MATH 18
Linear Algebra
- matrices
- Gaussian elimination
- determinants
- eigenvalues
- diagonalization
MATH 20B
Calculus for Science and Engineering
- integration methods
- infinite series
- polar coordinates
- complex exponentials
MATH 20C
Calculus and Analytic Geometry for Science and Engineering
- vector functions
- partial derivatives
- extrema
- double integrals
MATH 20D
Introduction to Differential Equations
- first and second order ODEs
- systems
- series solutions
- Laplace transforms
MATH 20E
Vector Calculus
- Jacobians
- line integrals
- Green's, Stokes', Gauss' theorems
MATH 31BH
Honors Multivariable Calculus
- multivariable derivatives
- Jacobians
- extrema
- multiple integrals
MATH 31CH
Honors Vector Calculus
- change of variables
- differential forms
- generalized Stokes
MATH 102
Applied Linear Algebra
- Hermitian matrices
- quadratic forms
- norms
- singular values
MATH 104A
Number Theory I
- unique factorization
- congruences
- reciprocity
- cryptography
MATH 109
Mathematical Reasoning
- proofs
- quantifiers
- induction
- sets
- epsilon-delta
MATH 110
Introduction to Partial Differential Equations
- heat, wave, Laplace equations
- separation of variables
- Fourier series
MATH 111A
Mathematical Modeling I
- epidemic, chemical-reaction and population models
MATH 112A
Introduction to Mathematical Biology I
- population dynamics
- gene regulation
- oscillators
MATH 114
Introduction to Computational Stochastics
- random numbers
- Monte Carlo
- MCMC
MATH 120A
Elements of Complex Analysis
- analytic functions
- Cauchy theorems
- residues
MATH 120B
Applied Complex Analysis
- residues
- conformal maps
- Fourier and Laplace transforms
MATH 130
Differential Equations and Dynamical Systems
- phase portraits
- limit cycles
- bifurcations
MATH 140A
Foundations of Real Analysis I
- real numbers
- topology
- sequences and series
- continuity
MATH 144
Introduction to Fourier Analysis
- Fourier series and transforms
- convergence
MATH 146
Analysis of Ordinary Differential Equations
- linear systems
- matrix exponential
- stability
MATH 148
Analysis of Partial Differential Equations
- characteristics
- wave and heat equations
- Green's functions
MATH 154
Discrete Mathematics and Graph Theory
- trees
- connectivity
- matchings
- coloring
- planar graphs
- flows
MATH 155A
Geometric Computer Graphics
- Bezier curves
- de Casteljau
- barycentric coordinates
MATH 170A
Numerical Analysis: Linear Algebra
- least squares
- orthogonalization
- eigenvalue computation
MATH 170B
Numerical Analysis: Approximation and Nonlinear Equations
- root finding
- interpolation
- approximation
MATH 170C
Numerical Analysis: ODEs
- numerical differentiation and integration
- ODE solvers
MATH 171A
Numerical Optimization: Linear Programming
- simplex
- duality
- network flows
MATH 171B
Numerical Optimization: Nonlinear Programming
- Newton methods
- constrained optimization
- KKT
MATH 173A
Optimization Methods for Data Science I
- convexity
- LP and QP
- SVMs
MATH 174
Numerical Methods for Physical Modeling
- floating point
- solvers
- ODE and BVP methods
MATH 175
Numerical Methods for PDEs
- finite difference and finite element methods
MATH 180A
Introduction to Probability
- probability spaces
- random variables
- expectation
- CLT
MATH 180B
Introduction to Stochastic Processes I
- multivariate normal
- random walks
- Poisson process
MATH 180C
Introduction to Stochastic Processes II
- Markov chains
- queues
- branching
MATH 181A
Mathematical Statistics I
- estimation
- MLE
- confidence intervals
- hypothesis tests
MATH 181D
Statistical Learning
- dimension reduction
- classification
- boosting
- SVMs
MATH 183
Statistical Methods
- distributions
- CLT
- intervals
- tests
- curve fitting
MATH 184
Enumerative Combinatorics
- permutations
- partitions
- inclusion-exclusion
- generating functions
MATH 187A
Introduction to Cryptography
- classical ciphers
- DES
- public-key systems
MATH 189
Exploratory Data Analysis and Inference
- R
- sampling
- bootstrap
- regression diagnostics
MATH 194
The Mathematics of Finance
- martingales
- arbitrage pricing
- options
Physics
PHYS 1A
Mechanics
- Newtonian motion
- forces
- gravity
- energy
- momentum
- rotation
- fluids
PHYS 1B
Electricity and Magnetism
- electric and magnetic fields
- DC and AC circuits
PHYS 1C
Waves, Optics, and Modern Physics
- oscillations
- waves
- sound
- light
- quantum basics
PHYS 2A
Physics: Mechanics
- vectors
- kinematics
- Newton's laws
- energy
- momentum
- rotation
- gravitation
PHYS 2B
Physics: Electricity and Magnetism
- electric fields
- Gauss's law
- circuits
- magnetism
- induction
- Maxwell
PHYS 2C
Physics: Fluids, Waves, Thermodynamics, Optics
- fluids
- waves
- heat
- entropy
- EM waves
- optics
PHYS 2D
Physics: Relativity and Quantum Physics
- relativity
- Bohr model
- wave-particle duality
- Schrodinger equation
- radioactivity
PHYS 4A
Physics for Majors: Mechanics
- kinematics
- dynamics
- conservation laws
- rotation
- oscillations
- central forces
PHYS 4B
Physics for Majors: Fluids, Waves, Stat and Thermal
- damped oscillations
- fluids
- waves
- thermodynamics
- kinetic theory
PHYS 4C
Physics for Majors: Electricity and Magnetism
- Coulomb
- fields
- Gauss's law
- capacitors
- circuits
- Faraday
PHYS 4D
Physics for Majors: EM Waves, Relativity, Optics
- Maxwell
- EM waves
- special relativity
- interference
- diffraction
PHYS 4E
Physics for Majors: Quantum Physics
- Schrodinger equation
- spin
- identical particles
- quantum statistics
PHYS 41
Scientific Computing with Python
- Python
- data analysis
- visualization
- numerical methods
- simulation
PHYS 100A
Electromagnetism I
- electrostatics
- conductors and dielectrics
- steady currents
PHYS 100C
Electromagnetism III
- EM waves
- radiation
- charged particles in fields
PHYS 105A
Mathematical and Computational Physics I
- Fourier methods
- special functions
- BVPs
- Green's functions
- PDEs
PHYS 110A
Mechanics I
- phase flows
- oscillations
- Lagrangian dynamics
- central forces
PHYS 110B
Mechanics II
- noninertial frames
- rigid bodies
- Hamiltonian dynamics
- chaos
PHYS 116
Fluid Dynamics for Physicists
- Navier-Stokes
- instabilities
- boundary layers
- turbulence
- shocks
PHYS 120
Circuits and Electronics
- analog design
- amplifiers
- filters
- sensors
- digitization
PHYS 130B
Quantum Physics II
- matrix mechanics
- angular momentum
- spin
- approximation methods
PHYS 140A
Statistical and Thermal Physics I
- thermodynamics
- partition function
- free energy
- quantum gases
PHYS 140B
Statistical and Thermal Physics II
- third law
- kinetic theory
- transport
- critical phenomena
PHYS 152A
Condensed Matter Physics
- crystal structure
- diffraction
- phonons
- band theory
- semiconductors
PHYS 154
Elementary Particle Physics
- quarks
- leptons
- interactions
- symmetries
- electroweak
PHYS 160
Stars and Compact Objects
- stellar structure and evolution
- white dwarfs
- neutron stars
PHYS 170
Medical Instruments
- spectroscopy
- microscopy
- ultrasound
- X-ray
- MRI
- lasers
Chemistry
CHEM 6A
General Chemistry I
- atomic theory
- bonding
- molecular geometry
- stoichiometry
- reaction types
CHEM 6B
General Chemistry II
- gases
- liquids
- solids
- thermochemistry
- equilibria
- solubility
CHEM 6C
General Chemistry III
- kinetics
- acid-base equilibria
- electrochemistry
- coordination
- nuclear
CHEM 7L
General Chemistry Laboratory
- analytical, inorganic, physical and synthetic lab techniques
CHEM 40A
Organic Chemistry for Life Sciences I
- bonding
- stereochemistry
- spectroscopy
- substitution
CHEM 40B
Organic Chemistry for Life Sciences II
- carbonyl reactivity
- enzymatic catalysis
- electrophilic reactions
CHEM 40C
Organic Chemistry III
- carboxylic acids
- carbohydrates
- proteins
- fatty acids
- natural products
CHEM 41A
Organic Chemistry I: Structure and Reactivity
- bonding
- isomerism
- stereochemistry
- conformations
CHEM 41B
Organic Chemistry II: Reactivity and Synthesis
- alcohols
- ethers
- aromatics
- organometallics
- carbonyls
CHEM 41C
Organic Chemistry III: Synthesis and Macromolecules
- compound classes
- multi-step synthesis
CHEM 43A
Organic Chemistry Laboratory
- separation
- purification
- spectroscopy
- product analysis
CHEM 100A
Analytical Chemistry Laboratory
- quantitative analysis
- gravimetric methods
- intro to instrumental analysis
CHEM 100B
Instrumental Chemistry Laboratory
- GC
- MS
- HPLC
- ion chromatography
- AAS
- fluorescence
- IR
CHEM 105A
Physical Chemistry Laboratory
- experimental physical chemistry
CHEM 114A
Biochemical Structure and Function
- nucleic acids
- proteins
- enzymes
- carbohydrates
- lipids
CHEM 114B
Biochemical Energetics and Metabolism
- glycolysis
- Krebs cycle
- oxidative phosphorylation
CHEM 114C
Biosynthesis of Macromolecules
- protein and nucleic acid biosynthesis
- cellular control
CHEM 120A
Inorganic Chemistry I
- main-group structure, bonding, reactivity
CHEM 120B
Inorganic Chemistry II
- transition metals
- coordination chemistry
- organometallics
CHEM 126A
Physical Biochemistry I
- thermodynamics and kinetics of biomolecules
- enzyme kinetics
CHEM 126B
Physical Biochemistry II
- quantum and statistical mechanics of biomolecules
- spectroscopy
- NMR
CHEM 130
Chemical Physics: Quantum Mechanics
- quantum mechanics
- atomic and molecular spectroscopy
CHEM 131
Chemical Physics: Stat Thermo I
- thermodynamics
- chemical and phase equilibria
- solutions
CHEM 132
Chemical Physics: Stat Thermo II
- statistics
- kinetic theory
- reaction kinetics
CHEM 143B
Organic Chemistry Laboratory
- synthetic organic methods
CHEM 146
Kinetics and Mechanism of Organic Reactions
- rate laws
- transition state theory
- isotope and solvent effects
CHEM 154
Mechanisms of Organic Reactions
- substitutions
- eliminations
- rearrangements
- radicals
- photochemistry
CHEM 156
Structure and Properties of Organic Molecules
- catalytic rate laws
- free-energy relationships
- pKa
CHEM 158
Applied Spectroscopy
- spectroscopic structure determination
CHEM 167
Medicinal Chemistry
- receptor-protein interactions
- drug design
- pharmacokinetics
CHEM 171
Environmental Chemistry I
- air pollution
- ozone depletion
- climate change
- energy
CHEM 173
Atmospheric Chemistry
- photochemistry
- radical reactions
- aerosols
- ozone
CHEM 190
Mathematics for Physical Chemistry
- linear algebra
- group theory
- transforms
- differential equations
Mechanical and Aerospace (MAE)
MAE 02
Introduction to Aerospace Engineering
- aerodynamics
- propulsion
- flight mechanics
- orbital mechanics
MAE 03
Introduction to Mechanical Design
- design process
- CAD
- prototyping
- shop skills
MAE 08
MATLAB Programming for Engineering Analysis
- arrays
- plotting
- symbolic math
- equation solving
MAE 11
Thermodynamics
- energy
- heat
- work
- first and second laws
- power and refrigeration cycles
- combustion
MAE 20
Elements of Materials Science
- atomic structure
- defects
- phase diagrams
- mechanical and electrical properties
- diffusion
MAE 21
Aerospace Materials Science
- structure
- defects
- phase diagrams
- testing
- processing
MAE 30A
Statics and Introduction to Dynamics
- rigid-body statics
- centroids
- particle kinematics and kinetics
- friction
MAE 30B
Dynamics and Vibrations
- energy methods
- Lagrange
- impact
- 3-D rigid body
- 1-DOF vibrations
MAE 40
Linear Circuits
- RLC circuits
- Kirchhoff
- node and mesh analysis
- op-amps
- motors
MAE 101A
Introductory Fluid Mechanics
- statics
- kinematics
- conservation laws
- Bernoulli
- potential flow
- similitude
MAE 101B
Advanced Fluid Mechanics
- laminar and turbulent flow
- pipe friction
- boundary layers
- drag and lift
- compressible flow
- shocks
MAE 101D
Intermediate Heat Transfer
- numerical conduction
- boiling
- condensation
- natural convection
- heat exchangers
MAE 104
Aerodynamics
- subsonic and supersonic flow over wings and bodies
- thin-wing theory
MAE 105
Introduction to Mathematical Physics
- Fourier series
- Sturm-Liouville
- PDEs
- integral transforms
MAE 107
Computational Methods in Engineering
- linear and nonlinear solvers
- interpolation
- least squares
- ODE numerics
MAE 108
Probability and Statistical Methods for Engineering
- probability
- random variables
- CLT
- reliability
- estimation
- hypothesis tests
MAE 110
Thermodynamic Systems
- IC, diesel, gas turbine cycles
- mixtures
- chemical and phase equilibrium
MAE 113
Fundamentals of Propulsion
- compressible flow
- combustion
- gas turbines
- rockets
MAE 114
Space Propulsion
- liquid and solid rockets
- electric propulsion
- launch
MAE 118
Introduction to Energy and Environment
- energy sources
- heat engines
- pollution
- climate change
MAE 119
Introduction to Renewable Energy: Solar and Wind
- solar radiation
- PV
- solar thermal
- wind
- Betz's law
MAE 120
Introduction to Nuclear Energy
- fission
- fusion
- reactors
- waste
MAE 121
Air Pollution Transport and Dispersion Modeling
- atmospheric stability
- plume dispersion
- air quality models
MAE 125
Building Energy Efficiency
- building thermodynamics
- envelope heat transfer
- HVAC
- codes
MAE 130
Advanced Vibrations
- forced vibration
- multi-DOF
- continuous systems
- Hamilton's equations
MAE 131A
Solid Mechanics I
- stress and strain
- Hooke's law
- torsion
- beam bending
- combined loading
- buckling
MAE 131B
Solid Mechanics II
- 3-D stress
- elasticity
- stress concentration
- failure criteria
- energy methods
MAE 133
Finite Element Methods in MAE
- stiffness and mass matrices
- static and dynamic FE analysis
MAE 142
Dynamics and Control of Aerospace Vehicles
- vehicle dynamics
- stability
- aircraft modes
- state-space control
MAE 143A
Signals and Systems
- state-space modeling
- linearization
- Laplace
- transfer functions
- z-transform
- Fourier
MAE 143B
Linear Control
- frequency-domain feedback
- PID
- Routh-Hurwitz
- root locus
- Bode and Nyquist
MAE 144
Embedded Control and Robotics
- unstable robot
- embedded Linux
- continuous and discrete control
MAE 145
Introduction to Robotic Planning and Estimation
- planning
- A*
- configuration space
- SLAM
MAE 146
Introduction to Machine Learning Algorithms
- supervised learning
- regression
- neural networks
- k-means
MAE 148
Introduction to Autonomous Vehicles
- 1/8-scale autonomous cars
- computer vision
- navigation
- sensor fusion
MAE 149
Sensor Networks
- sensors
- signal processing
- wireless networking
MAE 150
Computational Methods for Design
- tolerances
- Monte Carlo
- linkages and cams
- optimization
- FEA
MAE 153
Design of Machine Components
- failure theories
- static and fatigue design
- shafts
- gears
- bearings
- fasteners
MAE 154
Product Design and Entrepreneurship
- user needs
- concepts
- prototyping
- business plans
MAE 155A
Aerospace Engineering Design I
- conceptual, preliminary and detailed vehicle design
MAE 156A
Fundamental Principles of Mechanical Design I
- design process
- component analysis
- team project
MAE 160
Mechanical Behavior of Materials
- elasticity
- plasticity
- dislocations
- creep
- strengthening
- fracture
- fatigue
- composites
MAE 165
Fatigue and Failure Analysis of Engineering Components
- crack growth
- fracture toughness
- fatigue testing
- fractography
MAE 170
Experimental Techniques
- measurement and control
- error and noise
- filtering
- DAQ
MAE 171A
Mechanical Engineering Laboratory I
- fluids, solids and control experiments
- error analysis
- reports
MAE 175A
Aerospace Engineering Laboratory I
- wind tunnel
- gyroscope control
- vibration table
MAE 180
Orbital Mechanics
- orbital motion
- perturbations
- orbit determination
- maneuvers
- interplanetary
MAE 181
Space Mission Analysis and Design
- mission architecture
- orbit design
- subsystems
- cost and risk
MAE 182
Spacecraft Guidance and Navigation
- guidance
- tracking
- navigation
- link budgets
MAE 184
Flight Simulation Techniques
- 6-DOF equations of motion
- numerical integration
- linearization
MAE 185
Computational Fluid Mechanics
- finite difference and volume solutions of Navier-Stokes
Electrical and Computer Engineering (ECE)
ECE 5
Intro to Electrical and Computer Engineering
- circuits
- embedded systems
- signals
- DSP
ECE 15
Engineering Computation: C
- control and data structures
- pointers
- memory management
ECE 16
Rapid Hardware and Software Design for Interfacing
- EMG controller
- sampling
- signal processing
- microcontrollers
ECE 17
Object-Oriented Programming: C++
- OO design
- modern C++
- design patterns
- STL
ECE 18
Programming in Python and OO Principles
- Python
- common modules
- OO design
ECE 25
Introduction to Digital Design
- Boolean algebra
- combinational and sequential logic
ECE 30
Introduction to Computer Engineering
- information representation
- computer organization
- assembly
ECE 35
Introduction to Analog Design
- Kirchhoff
- Thevenin and Norton
- nodal analysis
- transients
- sinusoidal steady state
ECE 45
Circuits and Systems
- steady state
- first and second order systems
- Fourier and Laplace
- filters
ECE 65
Components and Circuits Laboratory
- diodes
- BJTs and FETs
- small-signal circuits
ECE 100
Linear Electronic Systems
- frequency response
- Laplace
- op-amp filters
- stability
ECE 101
Linear Systems Fundamentals
- complex variables
- Fourier, Laplace, z-transforms
- sampling
- stability
ECE 102
Introduction to Active Circuit Design
- nonlinear devices
- linearization
- small-signal circuits
ECE 103
Fundamentals of Devices and Materials
- semiconductor physics
- doping
- p-n junctions
- BJTs
- MOS
ECE 107
Electromagnetism
- Maxwell
- plane waves
- transmission lines
- Smith chart
- waveguides
ECE 108
Digital Circuits
- CMOS logic
- noise margins
- delays
- power
- memory
ECE 109
Engineering Probability and Statistics
- probability
- random variables
- CLT
- estimation
- hypothesis tests
ECE 111
Advanced Digital Design Project
- FSM synthesis
- hazards
- pipelining
- design automation
ECE 118
Computer Interfacing
- buses
- interrupts
- DMA
- serial and parallel communication
- A/D and D/A
ECE 121A
Power Systems Analysis and Fundamentals
- generation and distribution
- steady-state analysis
- economic operation
ECE 121B
Energy Conversion
- electromechanical conversion
- three-phase
- transformers
- machines
ECE 123
Antenna Systems Engineering
- impedance
- patterns
- gain
- polarization
- arrays
- link budgets
ECE 124
Motor Drives
- DC and induction motor drives
- vector control
ECE 125A
Introduction to Power Electronics I
- power semiconductors
- rectifiers
- DC-DC converters
- thermal design
ECE 125B
Introduction to Power Electronics II
- converter design
- PWM rectifiers
- inverters
ECE 135A
Semiconductor Physics
- crystal structure
- band structure
- carrier transport
- junctions
ECE 135B
Electronic Devices
- BJT, JFET and MOS structure and operation
- DC and AC characteristics
ECE 141A
Software Foundations I
- C++ data structures
- algorithms
- design patterns
ECE 143
Programming for Data Analysis
- Python
- NumPy
- Pandas
- Matplotlib
ECE 148
Introduction to Autonomous Vehicles
- scale autonomous cars
- ROS
- computer vision
- deep learning
ECE 153
Probability and Random Processes for Engineers
- stationary processes
- power spectral density
- Gaussian processes
- noise
ECE 155
Digital Communications Theory
- digital modulation
- ISI
- fading
- optimal detection
ECE 157A
Communications Systems Laboratory I
- modulation and demodulation
- signal impairments
ECE 158A
Data Networks I
- layered architecture
- data link
- multiple access
- flow control
- routing
ECE 159
Introduction to Data Processing and Information Theory
- entropy
- mutual information
- channel capacity
- codes
ECE 161A
Introduction to Digital Signal Processing
- DTFT
- FFT
- FIR and IIR filter design
ECE 161B
Digital Signal Processing
- sampling and quantization
- oversampling
- multirate processing
ECE 163
Electronic Circuits and Systems
- feedback
- stability
- active filters
- switched-capacitor
- PLLs
- data conversion
ECE 164
Analog Integrated Circuit Design
- op-amps
- regulators
- oscillators
- noise
ECE 165
Digital Integrated Circuit Design
- VLSI
- logic styles
- clocking
- ALUs
- memory
ECE 166
Microwave Systems and Circuits
- distributed circuits
- S-parameters
- matching
- amplifiers
ECE 171A
Linear Control System Theory
- frequency-domain stability
- root locus
- Bode/Nyquist
- compensators
ECE 171B
Linear Control System Theory (state space)
- state space
- controllability and observability
- pole placement
ECE 174
Linear and Nonlinear Optimization
- least squares
- constrained optimization
- nonlinear methods
ECE 175A
Elements of Machine Intelligence
- statistical classifiers
- regression
- clustering
ECE 176
Introduction to Deep Learning and Applications
- neural networks
- training
- PyTorch
ECE 181
Physical Optics and Fourier Optics
- ray and wave optics
- diffraction
- Fourier imaging
ECE 182
Electromagnetic Optics and Fiber Optics
- polarization
- waveguide modes
- fiber dispersion
ECE 183
Optical Electronics
- lasers
- photodetection
- electro-optics
- fiber systems
ECE 187
Introduction to Biomedical Imaging and Sensing
- radiography
- CT
- MRI
- ultrasound
- microscopy
Computer Science (CSE)
CSE 5
Principles and Practice of Artificial Intelligence
- AI foundations
- ML basics
- ethics
CSE 8A
Intro to Programming and Computational Problem-Solving I
- variables
- conditionals
- loops
- functions
- lists
- dictionaries
CSE 11
Intro to Programming: Accelerated
- OOP
- Java
- recursion
- file I/O
CSE 12
Basic Data Structures and Object-Oriented Design
- linked lists
- stacks
- queues
- trees
- hash tables
CSE 20
Discrete Mathematics
- logic
- sets
- relations
- functions
- proofs
- induction
CSE 21
Mathematics for Algorithms and Systems
- counting
- complexity
- recurrences
- graphs
- probability
CSE 29
Systems Programming and Software Tools
- C
- program execution
- gdb
- valgrind
- make
- UNIX
CSE 30
Computer Organization
- assembly
- digital logic
- processor design
- memory hierarchy
CSE 42
Building and Programming Electronic Devices
- Arduino team devices
CSE 100
Advanced Data Structures
- balanced trees
- graphs
- priority queues
- hash tables
- amortized analysis
CSE 101
Design and Analysis of Algorithms
- sorting
- searching
- graph algorithms
- NP-completeness
CSE 103
Practical Introduction to Probability and Statistics
- distributions
- hypothesis testing
- Bayes and MLE
- entropy
CSE 105
Theory of Computability
- automata
- formal languages
- Turing machines
- undecidability
CSE 106
Discrete and Continuous Optimization
- shortest paths
- flows
- LP, integer and convex programming
- gradient methods
CSE 107
Introduction to Modern Cryptography
- block ciphers
- public-key
- protocols
- zero-knowledge
CSE 110
Software Engineering
- specification
- design
- testing
- agile
- version control
CSE 120
Operating Systems Principles
- processes
- synchronization
- scheduling
- virtual memory
- file systems
CSE 124
Networked Services
- data centers
- TCP
- RPC
- load balancing
- secure communication
CSE 127
Introduction to Computer Security
- crypto basics
- threat analysis
- access control
- exploits
CSE 130
Programming Languages: Principles and Paradigms
- language design and comparison
CSE 131
Compiler Construction
- parsing
- type checking
- code generation
- optimization
CSE 140
Components and Design Techniques for Digital Systems
- Boolean logic
- combinational and sequential design
- state machines
CSE 141
Introduction to Computer Architecture
- processor and control design
- memory systems
CSE 142
Computer Architecture: Software Perspective
- pipelining
- caches
- SIMD
- multicore
- x86
CSE 143
Microelectronic System Design
- VLSI
- logic design
- CAD
- layout
CSE 147
Introduction to Embedded Systems
- processors
- sensors
- actuators
- real-time scheduling
CSE 150A
Intro to AI: Probabilistic Reasoning
- Bayesian networks
- MDPs
- inference
CSE 150B
Intro to AI: Search and Reasoning
- A*
- adversarial search
- MCTS
- reinforcement learning
CSE 151A
Machine Learning: Learning Algorithms
- kNN
- decision trees
- boosting
- perceptrons
- k-means
CSE 151B
Deep Learning
- neural networks
- backprop
- CNNs
- RNNs
- transformers
CSE 152A
Introduction to Computer Vision I
- image formation
- filtering
- features
- stereo
CSE 156
Statistical Natural Language Processing
- language models
- classification
- sequence models
- parsing
CSE 158
Recommender Systems and Web Mining
- predictive analytics
- recommendation methods
CSE 160
Introduction to Parallel Computing
- parallel architectures
- MPI
- speedup
- Amdahl's law
CSE 167
Computer Graphics
- transformations
- curves
- surfaces
- shading
- graphics APIs
CSE 168
Computer Graphics II: Rendering
- ray tracing
- acceleration structures
- sampling
- global illumination
CSE 170
Interaction Design
- user-centered design
- prototyping
- evaluation
CSE 180
Biology Meets Computing
- bioinformatics with Python
CSE 181
Molecular Sequence Analysis
- alignment
- database search
- phylogenetics
Bioengineering (BENG)
BENG 1
Introduction to Bioengineering
- design projects
- prototyping
- testing
- ethics
BENG 2
Introductory Computer Programming and MATLAB
- MATLAB basics
- plotting
- curve fitting
BENG 100
Probability and Statistics for Bioengineers
- probability
- distributions
- CLT
- hypothesis testing
BENG 102
Molecular Components of Living Systems
- proteins
- nucleic acids
- metabolites
- cellular design
BENG 103B
Bioengineering Mass Transfer
- diffusion
- convection
- reaction-diffusion
- active transport
BENG 110
Musculoskeletal Biomechanics
- statics
- dynamics
- kinematics
- tissue material properties
BENG 112A
Soft Tissue Biomechanics
- stress and strain
- constitutive laws
- viscoelasticity
- finite elasticity
BENG 112B
Fluid and Cell Biomechanics
- hydrostatics
- circulation
- biopolymers
- cell mechanics
BENG 120
Organic Chemistry Structural and Design Principles
- carbon compounds
- stereochemistry
- biomolecules
BENG 123
Dynamic Simulation in Bioengineering
- reaction networks
- kinetics
- simulation
BENG 125
Modeling and Computation in Bioengineering
- ODE, SDE and PDE models
- excitable cells
- time series
BENG 130
Biotechnology Thermodynamics and Kinetics
- thermodynamics
- molecular reactions
- kinetic models
BENG 133
Numerical Analysis and Computational Engineering
- number representation
- linear systems
- ODE solvers
BENG 134
Measurements, Statistics, and Probability
- estimation
- hypothesis tests
- regression
BENG 135
Biomedical Signals and Systems
- convolution
- Fourier
- sampling
- Laplace
BENG 140A
Bioengineering Physiology
- cell function
- control systems
- muscle
- neural
- circulatory
BENG 140B
Bioengineering Physiology II
- digestive
- respiratory
- renal systems
- metabolism
BENG 141
Biomedical Optics and Imaging
- light in tissue
- OCT
- microscopy
- spectroscopy
BENG 152
Biosystems Engineering Laboratory
- instrumentation
- biosignal acquisition
- microcontrollers
BENG 160
Chemical and Molecular Bioengineering Techniques
- protein and nucleic acid assays
- separations
- purification
BENG 161A
Bioreactor Engineering
- enzyme kinetics
- mass transfer
- fermentation design and scale-up
BENG 161B
Biochemical Engineering
- recombinant production
- bioseparations
- bioprocess economics
BENG 166A
Cell and Tissue Engineering
- cell behavior
- microenvironment
- tissue substitutes
BENG 168
Biomolecular Engineering
- recombinant DNA
- biomolecule design
- diagnostics
BENG 186A
Principles of Biomaterials Design
- polymers
- biocompatibility
- devices
- regulation
BENG 186B
Principles of Bioinstrumentation Design
- transducers
- biopotentials
- sensors
- electrical safety
BENG 189
Physiological Systems Engineering
- models of muscle, kidney transport and neural circuits
Chemical Engineering and NanoEngineering (CENG, NANO)
CENG 100
Material and Energy Balances
- steady and unsteady balances
- degrees of freedom
- reaction kinetics
- phase equilibrium
CENG 101A
Introductory Fluid Mechanics
- hydrostatics
- Bernoulli
- viscous and pipe flow
- boundary layers
- drag
CENG 101B
Heat Transfer
- conduction
- convection
- radiation
- exchanger design
CENG 101C
Mass Transfer
- diffusive and convective mass transfer
- coefficients
CENG 102
Chemical Engineering Thermodynamics
- properties of substances and mixtures
- phase and chemical equilibria
- cycles
CENG 113
Chemical Reaction Engineering
- rate equations
- batch and continuous reactors
- selectivity
- residence time
CENG 114
Probability and Statistical Methods for Engineers
- probability
- CLT
- regression
- hypothesis tests
CENG 120
Chemical Process Dynamics and Control
- linearized models
- stability
- PID
- Bode and Nyquist
- root locus
CENG 122
Separation Processes
- distillation
- extraction
- absorption
- membranes
CENG 124A
Chemical Plant and Process Design I
- flow diagrams
- costing
- simulation
- recycle
- safety
CENG 157
Process Technology in the Semiconductor Industry
- crystal growth
- thin films
- oxidation
- CVD and PVD
- doping
- etching
CENG 160
Introduction to Biochemical Engineering
- biomacromolecules
- enzyme kinetics
- metabolism
- bioreactors
CENG 170
Experimental Methods for Chemical Engineers
- measurement
- error analysis
- DAQ
- technical writing
NANO 11
Introduction to NanoEngineering
- scaling laws
- nanomaterials synthesis and properties
- nanomedicine
NANO 15
Engineering Computation Using MATLAB
- MATLAB for nano, chemical and materials problems
NANO 20L
Introduction to Nanomaterials Synthesis Lab
- chemical synthesis of nanomaterials
- characterization
NANO 100L
Physical Properties of Materials Lab
- thermal, optical, acoustic and magnetic measurements
NANO 102
Foundations in NanoEngineering: Chemical Principles
- thermodynamics of small systems
- diffusion
- colloids
- catalysis
- surfaces
NANO 103
Foundations in NanoEngineering: Biochemical Principles
- biomolecules
- interactions
- signaling
- detection
NANO 104
Foundations in NanoEngineering: Physical Principles
- quantum mechanics
- Schrodinger
- band theory
- nanoelectronics
NANO 106
Crystallography of Materials
- crystal symmetry
- tensors
- X-ray, neutron and electron diffraction
NANO 107
Electronic Devices and Circuits for NanoEngineers
- CMOS
- scaling
- amplifiers
- filters
- digital logic
NANO 108
Materials Science and Engineering
- atomic structure
- defects
- phase diagrams
- mechanical, electrical, magnetic properties
NANO 110
Molecular Modeling of Nanoscale Systems
- molecular mechanics
- MD
- Monte Carlo
NANO 111
Characterization of NanoEngineering Systems
- optical and electron microscopy
- scanning probes
- spectroscopy
NANO 114
Probability and Statistical Methods for Engineers
- probability
- regression
- intervals
- tests
NANO 117
Multiscale Transport
- conduction
- convection
- laminar flow
- diffusion
NANO 141A
Engineering Mechanics I: Equilibrium
- force and moment vectors
- free body diagrams
- friction
NANO 141B
Engineering Mechanics II: Motion
- particle and rigid-body dynamics
- energy
- Lagrange
- vibration
NANO 148
Thermodynamics of Materials
- laws
- statistical thermodynamics
- multiphase equilibria
- electrochemistry
NANO 150
Mechanics of Nanomaterials
- continuum and atomistic models
- elasticity
- plasticity
- fracture
NANO 158
Phase Transformations and Kinetics
- diffusion
- transformations
- microstructure
NANO 159
Electrochemistry: Fundamentals and Applications
- electrode reactions
- techniques
- energy applications
NANO 161
Material Selection in Engineering
- requirements
- property-based selection
- optimization
NANO 164
Micro- and Nano-materials for Energy Storage and Conversion
- fuel cells
- batteries
- PV
- thermoelectrics
NANO 168
Electrical, Dielectric, and Magnetic Properties of Materials
- semiconductors
- piezo and ferroelectrics
- magnetic materials
NANO 174
Mechanical Behavior of Materials
- plasticity
- strengthening
- failure modes
NANO 181
Data Science in Materials Science
- ML
- regression
- dimensionality reduction
- classification
Structural Engineering (SE)
SE 1
Introduction to Structures and Design
- structural behavior
- design process
- failures
- ethics
SE 9
Algorithms and Programming for Structural Engineering
- MATLAB
- functions
- loops
- OOP
- physics models
SE 101C
Mechanics III: Vibrations
- damped 1-DOF and MDOF
- normal modes
- Lagrange
- modal superposition
SE 102
Introduction to Computing for Engineers
- interpolation
- integration
- ODEs
- linear and nonlinear systems
SE 103
Conceptual Structural Design
- materials
- loads and load paths
- code design
- analysis methods
SE 105
Structural Materials
- stress and strain
- strength
- fatigue
- creep
- concrete, wood, steel, aluminum, composites
SE 110A
Solid Mechanics I
- stress and strain
- torsion
- bending
- shear
- deflections
- indeterminate problems
SE 110B
Solid Mechanics II
- unsymmetric bending
- curved beams
- shear center
- column stability
- elasticity
SE 115
Fluid Mechanics for Structural Engineering
- statics
- conservation
- Bernoulli
- pipe and open-channel flow
SE 125
Statistics, Probability and Reliability
- probability
- inference
- distributions
- structural reliability
SE 130A
Structural Analysis I
- indeterminate analysis
- matrix methods
- direct stiffness
SE 131A
Finite Element Analysis
- Galerkin models
- heat conduction
- stress analysis
- error analysis
SE 132
Machine Learning for Structural Engineering
- supervised and unsupervised learning
- neural networks
- Python
SE 140A
Design of Civil Structures I
- team design, build and test
- codes
- sustainability
SE 142
Mechanics and Design of Composite Structures
- micromechanics
- stiffness
- ply stress
- failure
SE 143A
Aerospace Structural Design I
- aircraft and spacecraft design
- minimum-weight design
SE 150A
Design of Steel Structures I
- LRFD
- steel properties
- tension, compression, beams, connections
SE 151A
Design of Reinforced Concrete
- concrete and reinforcement
- limit state design
- detailing
SE 152
Seismic Design of Structures
- seismic philosophy
- ductility
- lateral systems
- nonlinear behavior
SE 154
Design of Timber Structures
- wood
- beams
- columns
- diaphragms
- shear walls
- connections
SE 160A
Aerospace Structural Mechanics I
- flight loads
- 3D stress and strain
- failure theories
- thin-walled multicell
SE 160B
Aerospace Structural Mechanics II
- energy methods
- FEA
- plate and shell bending
- buckling
- flutter
SE 163
Nondestructive Evaluation
- signal processing
- ultrasonic testing
- impact-echo
- acoustic emission
SE 164
Sensors and Data Acquisition for Structural Engineering
- sensing
- uncertainty
- signal conditioning
- DAQ
SE 165
Structural Health Monitoring
- DAQ
- feature extraction
- statistical modeling
SE 167
Signal Processing and Spectral Analysis for Structural Engineering
- Fourier transform
- digital filtering
- feature extraction
SE 180
Earthquake Engineering
- seismic hazard
- response spectra
- modal and time-history analysis
- codes
SE 181
Geotechnical Engineering
- soil properties
- compaction
- effective stress
- seepage
- consolidation
- shear strength
SE 182
Foundation Engineering
- soil exploration
- shallow and deep foundations
- settlement
- bearing capacity
SE 184
Ground Improvement
- mechanical, hydraulic, chemical and inclusion methods
Precalculus and calculus (I, II, III)
Precalculus: functions and algebra
\(y=a\,f(b(x-h))+k:\ \text{vertical stretch }a,\ \text{horizontal stretch }1/b,\ \text{shift right }h,\ \text{up }k;\quad (f\circ g)(x)=f(g(x)),\quad f^{-1}:\ \text{swap }x,y\)
\(a^{x}=e^{x\ln a},\quad \log_{a}x=\dfrac{\ln x}{\ln a},\quad \ln(ab)=\ln a+\ln b,\quad \ln a^{n}=n\ln a,\quad e^{\ln x}=x\)
\(\text{Remainder thm: }f(c)=\text{remainder of }f(x)\div(x-c);\quad \text{rational roots: }\pm\dfrac{p}{q},\ p\mid a_{0},\ q\mid a_{n}\)
\(a_{n}=a_{1}+(n-1)d,\ S_{n}=\dfrac{n(a_{1}+a_{n})}2;\quad a_{n}=a_{1}r^{n-1},\ S_{n}=a_{1}\dfrac{1-r^{n}}{1-r},\ S_{\infty}=\dfrac{a_{1}}{1-r}\ (|r|<1)\)
\(\text{Circle: }(x-h)^{2}+(y-k)^{2}=r^{2};\ \text{ellipse: }\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\ c^{2}=a^{2}-b^{2};\ \text{hyperbola: }\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1,\ c^{2}=a^{2}+b^{2};\ \text{parabola: }(x-h)^{2}=4p(y-k)\)
\(x=r\cos\theta,\ y=r\sin\theta,\ r^{2}=x^{2}+y^{2},\ \tan\theta=\dfrac yx\)
Trig identities (complete)
\(\sin^{2}\theta+\cos^{2}\theta=1,\quad 1+\tan^{2}\theta=\sec^{2}\theta,\quad 1+\cot^{2}\theta=\csc^{2}\theta\)
\(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B,\quad \cos(A\pm B)=\cos A\cos B\mp\sin A\sin B,\quad \tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}\)
\(\sin2\theta=2\sin\theta\cos\theta,\quad \cos2\theta=\cos^{2}\theta-\sin^{2}\theta=2\cos^{2}\theta-1=1-2\sin^{2}\theta,\quad \tan2\theta=\dfrac{2\tan\theta}{1-\tan^{2}\theta}\)
\(\sin^{2}\theta=\dfrac{1-\cos2\theta}2,\quad \cos^{2}\theta=\dfrac{1+\cos2\theta}2,\quad \sin\dfrac\theta2=\pm\sqrt{\dfrac{1-\cos\theta}2},\quad \cos\dfrac\theta2=\pm\sqrt{\dfrac{1+\cos\theta}2}\)
\(\sin A\cos B=\tfrac12[\sin(A+B)+\sin(A-B)],\quad \cos A\cos B=\tfrac12[\cos(A-B)+\cos(A+B)],\quad \sin A\sin B=\tfrac12[\cos(A-B)-\cos(A+B)]\)
\(\sin(-\theta)=-\sin\theta,\ \cos(-\theta)=\cos\theta;\quad \sin\left(\tfrac\pi2-\theta\right)=\cos\theta;\quad \text{periods: }\sin,\cos,\sec,\csc\ 2\pi;\ \tan,\cot\ \pi\)
\(\arcsin:\ [-1,1]\to[-\tfrac\pi2,\tfrac\pi2];\quad \arccos:\ [-1,1]\to[0,\pi];\quad \arctan:\ \mathbb R\to(-\tfrac\pi2,\tfrac\pi2)\)
Calculus I: limits and continuity
\(\lim_{x\to a}f(x)=L\ \text{exists iff left and right limits both equal }L;\quad f\ \text{continuous at }a\iff\lim_{x\to a}f=f(a)\)
\(\lim_{x\to0}\dfrac{\sin x}x=1,\quad \lim_{x\to0}\dfrac{1-\cos x}x=0,\quad \lim_{x\to0}\dfrac{e^{x}-1}x=1,\quad \lim_{x\to0}\dfrac{\ln(1+x)}x=1,\quad \lim_{n\to\infty}\left(1+\dfrac1n\right)^{n}=e\)
\(\text{Squeeze: }g\le f\le h,\ \lim g=\lim h=L\Rightarrow\lim f=L;\quad \text{IVT: }f\text{ continuous on }[a,b]\Rightarrow f\text{ takes every value between }f(a),f(b)\)
\(\text{L'Hopital for }\dfrac00,\ \dfrac\infty\infty:\ \lim\dfrac fg=\lim\dfrac{f'}{g'};\quad 0\cdot\infty\to\text{rewrite as a quotient};\quad 0^{0},1^{\infty},\infty^{0}\to\text{take }\ln\)
\(\text{Limits at infinity of rational functions: compare leading terms (degree top < bottom gives 0; equal gives ratio of leading coefficients; top > bottom diverges)}\)
Calculus I: derivative rules (complete)
\(f'(x)=\lim_{h\to0}\dfrac{f(x+h)-f(x)}h,\quad (x^{n})'=nx^{n-1},\quad (a^{x})'=a^{x}\ln a,\quad (\log_{a}x)'=\dfrac1{x\ln a}\)
\((\sin x)'=\cos x,\ (\cos x)'=-\sin x,\ (\tan x)'=\sec^{2}x,\ (\cot x)'=-\csc^{2}x,\ (\sec x)'=\sec x\tan x,\ (\csc x)'=-\csc x\cot x\)
\((\arcsin x)'=\dfrac1{\sqrt{1-x^{2}}},\ (\arccos x)'=-\dfrac1{\sqrt{1-x^{2}}},\ (\arctan x)'=\dfrac1{1+x^{2}},\ (\sinh x)'=\cosh x,\ (\cosh x)'=\sinh x\)
\((fg)'=f'g+fg',\quad \left(\dfrac fg\right)'=\dfrac{f'g-fg'}{g^{2}},\quad [f(g(x))]'=f'(g(x))g'(x),\quad (f^{-1})'(y)=\dfrac1{f'(x)}\)
\(\text{Implicit: differentiate both sides, solve for }y';\quad \text{log differentiation: }y=f^{g}\Rightarrow\ln y=g\ln f\)
Calculus I: applications of derivatives
\(\text{Critical points: }f'=0\text{ or undefined;}\quad f''>0\Rightarrow\text{concave up (min at a critical point)};\quad f''<0\Rightarrow\text{concave down (max)};\ \text{inflection where }f''\text{ changes sign}\)
\(\text{MVT: }f'(c)=\dfrac{f(b)-f(a)}{b-a};\quad \text{Rolle: if }f(a)=f(b)\text{ then }f'(c)=0;\quad L(x)=f(a)+f'(a)(x-a);\quad dy=f'(x)\,dx\)
Related rates: draw a picture, name variables, write an equation linking them, differentiate with respect to time, plug in numbers last.
Optimization: write the quantity to optimize, use a constraint to reduce to one variable, find critical points, check endpoints and the second derivative.
\(s'=v,\quad v'=a;\quad \text{curve sketching: domain, intercepts, asymptotes, }f',\ f''\text{, end behavior}\)
Calculus II: integration techniques
\(\text{Substitution: }\int f(g(x))g'(x)dx=\int f(u)du;\quad \text{by parts: }\int u\,dv=uv-\int v\,du\ \text{(LIATE)};\quad \int\ln x\,dx=x\ln x-x+C\)
\(\int\dfrac{dx}{a^{2}+x^{2}}=\dfrac1a\arctan\dfrac xa+C,\quad \int\dfrac{dx}{\sqrt{a^{2}-x^{2}}}=\arcsin\dfrac xa+C,\quad \int\sec^{3}x\,dx=\tfrac12(\sec x\tan x+\ln|\sec x+\tan x|)+C\)
\(\int\sin^{m}x\cos^{n}x\,dx:\ \text{odd power: save one factor and use }\sin^{2}+\cos^{2}=1;\ \text{both even: half-angle identities}\)
\(\text{Trig substitution: }\sqrt{a^{2}-x^{2}}\Rightarrow x=a\sin\theta;\ \sqrt{a^{2}+x^{2}}\Rightarrow x=a\tan\theta;\ \sqrt{x^{2}-a^{2}}\Rightarrow x=a\sec\theta\)
\(\text{Partial fractions: }\dfrac{A}{x-a},\ \ \dfrac{A}{x-a}+\dfrac{B}{(x-a)^{2}},\ \ \dfrac{Ax+B}{x^{2}+bx+c}\ (\text{irreducible});\ \text{divide first if degree top}\ge\text{bottom}\)
\(\text{Improper: }\int_{1}^{\infty}\dfrac{dx}{x^{p}}\text{ converges iff }p>1;\quad \int_{0}^{1}\dfrac{dx}{x^{p}}\text{ converges iff }p<1;\quad \text{use comparison to decide}\)
Calculus II: applications of integration
\(\text{FTC: }\dfrac d{dx}\int_{a}^{x}f(t)dt=f(x),\quad \int_{a}^{b}f\,dx=F(b)-F(a),\quad f_{avg}=\dfrac1{b-a}\int_{a}^{b}f\,dx\)
\(\text{Area between curves: }\int(\text{top}-\text{bottom})dx;\quad \text{disk: }\pi\int R^{2};\ \text{washer: }\pi\int(R^{2}-r^{2});\ \text{shells: }2\pi\int(\text{radius})(\text{height})dx\)
\(\text{Arc length: }\int\sqrt{1+f'(x)^{2}}\,dx;\quad \text{surface of revolution: }2\pi\int f\sqrt{1+f'^{2}}\,dx;\quad W=\int F\,dx,\ F_{\text{spring}}=kx\)
\(\bar x=\dfrac{\int x\rho\,dx}{\int\rho\,dx}\ \text{(center of mass)},\quad \text{separable ODE: }\int\dfrac{dy}{g(y)}=\int f(x)dx\)
Calculus II: sequences and series tests
\(\text{n-th term: if }a_{n}\not\to0\text{ then }\sum a_{n}\text{ diverges};\quad \text{geometric }\sum ar^{n}\text{ converges iff }|r|<1;\quad p\text{-series }\sum\dfrac1{n^{p}}\text{ converges iff }p>1\)
\(\text{Integral test; comparison; limit comparison (}\lim a_{n}/b_{n}\in(0,\infty));\quad \text{alternating series: }b_{n}\downarrow0\Rightarrow\text{converges}\)
\(\text{Ratio: }L=\lim\left|\dfrac{a_{n+1}}{a_{n}}\right|;\ L<1\text{ converges absolutely},\ L>1\text{ diverges},\ L=1\text{ inconclusive};\quad \text{root test: }\lim|a_{n}|^{1/n}\)
\(\text{Radius of convergence }R=\lim\left|\dfrac{a_{n}}{a_{n+1}}\right|;\ \text{test endpoints separately};\quad f(x)=\sum\dfrac{f^{(n)}(a)}{n!}(x-a)^{n}\)
\(e^{x}=\sum\dfrac{x^{n}}{n!},\quad \sin x=\sum\dfrac{(-1)^{n}x^{2n+1}}{(2n+1)!},\quad \cos x=\sum\dfrac{(-1)^{n}x^{2n}}{(2n)!},\quad \dfrac1{1-x}=\sum x^{n},\quad \ln(1+x)=\sum\dfrac{(-1)^{n+1}x^{n}}n\)
\(\arctan x=\sum\dfrac{(-1)^{n}x^{2n+1}}{2n+1},\quad (1+x)^{k}=\sum\dbinom kn x^{n},\quad |R_{n}(x)|\le\dfrac{M|x-a|^{n+1}}{(n+1)!}\ \text{(Taylor remainder)}\)
Calculus II: parametric and polar
\(\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt},\quad L=\int\sqrt{x'^{2}+y'^{2}}\,dt,\quad A=\int y\,dx=\int y(t)x'(t)\,dt\)
\(\text{Polar area: }A=\tfrac12\int r^{2}d\theta;\quad L=\int\sqrt{r^{2}+(dr/d\theta)^{2}}\,d\theta;\quad \text{conic: }r=\dfrac{ed}{1\pm e\cos\theta}\)
Polar curves: circle r = a; cardioid r = a(1 +/- cos theta); rose r = a cos(n theta) (n petals if odd, 2n if even); lemniscate r^2 = a^2 cos(2 theta); spiral r = a theta.
Calculus III: vectors, lines, planes, surfaces
\(\mathbf a\cdot\mathbf b=|a||b|\cos\theta,\quad |\mathbf a\times\mathbf b|=|a||b|\sin\theta=\text{parallelogram area},\quad |\mathbf a\cdot(\mathbf b\times\mathbf c)|=\text{parallelepiped volume}\)
\(\text{proj}_{\mathbf b}\mathbf a=\dfrac{\mathbf a\cdot\mathbf b}{\mathbf b\cdot\mathbf b}\mathbf b;\quad \text{line: }\mathbf r=\mathbf r_{0}+t\mathbf v;\quad \text{plane: }\mathbf n\cdot(\mathbf r-\mathbf r_{0})=0\iff ax+by+cz=d\)
\(\text{dist(point, plane)}=\dfrac{|ax_{0}+by_{0}+cz_{0}-d|}{\sqrt{a^{2}+b^{2}+c^{2}}},\quad \text{dist(point, line)}=\dfrac{|\overrightarrow{PQ}\times\mathbf v|}{|\mathbf v|}\)
\(\text{Cylindrical: }x=r\cos\theta,\ y=r\sin\theta,\ z=z;\quad \text{spherical: }x=\rho\sin\phi\cos\theta,\ y=\rho\sin\phi\sin\theta,\ z=\rho\cos\phi\)
Quadric surfaces: ellipsoid x^2/a^2 + y^2/b^2 + z^2/c^2 = 1; hyperboloid of one sheet (one minus sign); two sheets (two minus signs); elliptic paraboloid z = x^2/a^2 + y^2/b^2; hyperbolic paraboloid z = x^2/a^2 - y^2/b^2 (saddle); cone z^2 = x^2/a^2 + y^2/b^2.
Calculus III: curves in space
\(\mathbf r(t),\ \mathbf v=\mathbf r',\ \mathbf a=\mathbf r'',\ \text{speed}=|\mathbf r'|,\quad s=\int|\mathbf r'|\,dt,\quad \mathbf T=\dfrac{\mathbf r'}{|\mathbf r'|},\quad \mathbf N=\dfrac{\mathbf T'}{|\mathbf T'|},\quad \mathbf B=\mathbf T\times\mathbf N\)
\(\kappa=\dfrac{|\mathbf T'|}{|\mathbf r'|}=\dfrac{|\mathbf r'\times\mathbf r''|}{|\mathbf r'|^{3}},\quad a_{T}=\dfrac{\mathbf r'\cdot\mathbf r''}{|\mathbf r'|},\quad a_{N}=\dfrac{|\mathbf r'\times\mathbf r''|}{|\mathbf r'|},\quad \text{projectile: }\mathbf r=\mathbf r_{0}+\mathbf v_{0}t+\tfrac12\mathbf gt^{2}\)
Calculus III: partial derivatives and optimization
\(\nabla f=(f_{x},f_{y},f_{z}),\quad D_{\mathbf u}f=\nabla f\cdot\mathbf u,\quad \text{max rate of change}=|\nabla f|\ \text{in the direction of }\nabla f;\ \nabla f\perp\text{level curves}\)
\(\text{Tangent plane: }z=f(a,b)+f_{x}(a,b)(x-a)+f_{y}(a,b)(y-b);\quad \text{chain rule: }\dfrac{dz}{dt}=f_{x}x'+f_{y}y';\quad \dfrac{dy}{dx}=-\dfrac{F_{x}}{F_{y}}\)
\(\text{Second derivative test: }D=f_{xx}f_{yy}-f_{xy}^{2};\ D>0,f_{xx}>0\Rightarrow\min;\ D>0,f_{xx}<0\Rightarrow\max;\ D<0\Rightarrow\text{saddle}\)
\(\text{Lagrange: }\nabla f=\lambda\nabla g,\ g=c;\quad \text{absolute extrema on a closed region: check interior critical points and the boundary}\)
Calculus III: multiple integrals
\(\iint_{D}f\,dA\ (\text{type I: }a\le x\le b,\ g_{1}(x)\le y\le g_{2}(x));\quad \text{polar: }dA=r\,dr\,d\theta;\quad \text{swap order by redrawing the region}\)
\(\text{Cylindrical: }dV=r\,dz\,dr\,d\theta;\quad \text{spherical: }dV=\rho^{2}\sin\phi\,d\rho\,d\phi\,d\theta;\quad \text{Jacobian: }dx\,dy=\left|\dfrac{\partial(x,y)}{\partial(u,v)}\right|du\,dv\)
\(\text{Area}=\iint dA,\ \text{volume}=\iint z\,dA,\ \text{mass}=\iint\rho\,dA,\ \bar x=\dfrac{\iint x\rho\,dA}{m},\ I_{z}=\iint(x^{2}+y^{2})\rho\,dA,\ \text{surface area}=\iint\sqrt{1+f_{x}^{2}+f_{y}^{2}}\,dA\)
Calculus III: line and surface integrals
\(\int_{C}f\,ds=\int f(\mathbf r(t))|\mathbf r'(t)|\,dt,\quad W=\int_{C}\mathbf F\cdot d\mathbf r=\int\mathbf F(\mathbf r(t))\cdot\mathbf r'(t)\,dt\)
\(\mathbf F=\nabla f\ (\text{conservative})\Rightarrow\int_{C}\mathbf F\cdot d\mathbf r=f(B)-f(A);\quad \operatorname{curl}\mathbf F=0\text{ on a simply connected region}\Rightarrow\text{conservative}\)
\(\text{Green: }\oint P\,dx+Q\,dy=\iint(Q_{x}-P_{y})dA;\quad \text{Stokes: }\oint\mathbf F\cdot d\mathbf r=\iint(\nabla\times\mathbf F)\cdot d\mathbf S;\quad \text{Divergence: }\iint_{S}\mathbf F\cdot d\mathbf S=\iiint\nabla\cdot\mathbf F\,dV\)
\(\iint_{S}\mathbf F\cdot d\mathbf S=\iint\mathbf F\cdot(\mathbf r_{u}\times\mathbf r_{v})\,dA,\quad \nabla\cdot(\nabla\times\mathbf F)=0,\quad \nabla\times(\nabla f)=0\)
Probability and statistics (MATH 11, 183, 180A and more)
Probability rules and counting
\(P(A\cup B)=P(A)+P(B)-P(A\cap B),\quad P(A|B)=\dfrac{P(A\cap B)}{P(B)},\quad \text{independent iff }P(A\cap B)=P(A)P(B),\quad P(A^{c})=1-P(A)\)
\(\text{Total probability: }P(B)=\sum P(B|A_{i})P(A_{i});\quad \text{Bayes: }P(A_{i}|B)=\dfrac{P(B|A_{i})P(A_{i})}{\sum P(B|A_{j})P(A_{j})}\)
\(n!,\quad P(n,r)=\dfrac{n!}{(n-r)!},\quad \dbinom nr=\dfrac{n!}{r!(n-r)!},\quad \text{with repetition: }n^{r};\quad \text{multinomial: }\dfrac{n!}{n_{1}!\cdots n_{k}!}\)
\(E[aX+b]=aE[X]+b,\quad \operatorname{Var}(aX+b)=a^{2}\operatorname{Var}(X),\quad E[X+Y]=E[X]+E[Y],\quad \operatorname{Var}(X+Y)=\operatorname{Var}X+\operatorname{Var}Y\ (\text{if independent})\)
\(\operatorname{Cov}(X,Y)=E[XY]-E[X]E[Y],\quad \rho=\dfrac{\operatorname{Cov}(X,Y)}{\sigma_{X}\sigma_{Y}},\quad f_{X}(x)=\int f(x,y)\,dy\ \text{(marginal)}\)
Distributions cheat sheet
\(\text{Binomial}(n,p):\ P(k)=\dbinom nk p^{k}(1-p)^{n-k},\ \mu=np,\ \sigma^{2}=np(1-p);\quad \text{Geometric}(p):\ P(k)=(1-p)^{k-1}p,\ \mu=\dfrac1p\)
\(\text{Poisson}(\lambda):\ P(k)=\dfrac{\lambda^{k}e^{-\lambda}}{k!},\ \mu=\sigma^{2}=\lambda;\quad \text{Uniform}(a,b):\ \mu=\dfrac{a+b}2,\ \sigma^{2}=\dfrac{(b-a)^{2}}{12}\)
\(\text{Exponential}(\lambda):\ f=\lambda e^{-\lambda x},\ F=1-e^{-\lambda x},\ \mu=\dfrac1\lambda,\ \sigma^{2}=\dfrac1{\lambda^{2}},\ \text{memoryless}\)
\(\text{Normal}(\mu,\sigma^{2}):\ z=\dfrac{x-\mu}\sigma;\ 68\%\text{-}95\%\text{-}99.7\%\ \text{within }1,2,3\sigma;\quad z_{0.90}=1.282,\ z_{0.95}=1.645,\ z_{0.975}=1.960,\ z_{0.995}=2.576\)
\(\text{Normal approx. to binomial if }np\ge10\text{ and }n(1-p)\ge10;\quad \text{sampling distribution of }\bar x:\ \mu,\ \dfrac\sigma{\sqrt n};\quad \text{CLT for large }n\)
Descriptive statistics and inference
\(\bar x=\dfrac{\sum x}n,\quad s^{2}=\dfrac{\sum(x-\bar x)^{2}}{n-1},\quad IQR=Q_{3}-Q_{1},\quad \text{outlier if beyond }Q_{1}-1.5\,IQR\text{ or }Q_{3}+1.5\,IQR\)
\(\text{CI for mean: }\bar x\pm t^{*}\dfrac s{\sqrt n};\quad \text{CI for proportion: }\hat p\pm z^{*}\sqrt{\dfrac{\hat p(1-\hat p)}n};\quad \text{sample size: }n=\left(\dfrac{z^{*}\sigma}E\right)^{2}\)
\(\text{One-sample t: }t=\dfrac{\bar x-\mu_{0}}{s/\sqrt n},\ df=n-1;\quad \text{two-sample: }t=\dfrac{\bar x_{1}-\bar x_{2}}{\sqrt{s_{1}^{2}/n_{1}+s_{2}^{2}/n_{2}}};\quad \text{proportion: }z=\dfrac{\hat p-p_{0}}{\sqrt{p_{0}(1-p_{0})/n}}\)
Hypothesis test steps: state H0 and Ha, pick alpha, compute the test statistic, find the p-value, compare to alpha, conclude in context. p-value is the probability of data this extreme if H0 is true, not the probability H0 is true.
\(\chi^{2}=\sum\dfrac{(O-E)^{2}}E;\ \text{goodness of fit }df=k-1;\ \text{independence }df=(r-1)(c-1);\quad \text{ANOVA: }F=\dfrac{MSB}{MSW}\)
\(\text{Regression: }\hat y=a+bx,\ b=r\dfrac{s_{y}}{s_{x}},\ a=\bar y-b\bar x;\quad R^{2}=r^{2};\quad t=\dfrac b{SE_{b}},\ df=n-2;\quad \text{residual}=y-\hat y\)
Math
Algebra
\(x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)
\(a^{m}a^{n}=a^{m+n},\quad (a^{m})^{n}=a^{mn},\quad a^{-n}=\dfrac{1}{a^{n}}\)
\(\log(xy)=\log x+\log y,\quad \log\dfrac{x}{y}=\log x-\log y,\quad \log x^{n}=n\log x\)
\(\log_{b}x=\dfrac{\ln x}{\ln b}\)
\(y=mx+b,\quad m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Trigonometry
\(\sin^{2}\theta+\cos^{2}\theta=1,\quad \tan\theta=\dfrac{\sin\theta}{\cos\theta}\)
\(\sin 2\theta=2\sin\theta\cos\theta,\quad \cos 2\theta=\cos^{2}\theta-\sin^{2}\theta\)
\(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\)
\(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\)
\(c^{2}=a^{2}+b^{2}-2ab\cos C\)
\(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\)
Derivatives
\(\dfrac{d}{dx}x^{n}=nx^{n-1},\quad \dfrac{d}{dx}e^{x}=e^{x},\quad \dfrac{d}{dx}\ln x=\dfrac1x\)
\(\dfrac{d}{dx}\sin x=\cos x,\quad \dfrac{d}{dx}\cos x=-\sin x,\quad \dfrac{d}{dx}\tan x=\sec^{2}x\)
\(\dfrac{d}{dx}\arcsin x=\dfrac{1}{\sqrt{1-x^{2}}},\quad \dfrac{d}{dx}\arctan x=\dfrac{1}{1+x^{2}}\)
\((fg)'=f'g+fg',\quad \left(\dfrac fg\right)'=\dfrac{f'g-fg'}{g^{2}}\)
\(\dfrac{d}{dx}f(g(x))=f'(g(x))\,g'(x)\)
\(\lim\dfrac{f}{g}=\lim\dfrac{f'}{g'}\ \text{(L'Hôpital, for }0/0\text{ or }\infty/\infty)\)
Integrals
\(\int x^{n}dx=\dfrac{x^{n+1}}{n+1}+C\ (n\neq-1),\quad \int\dfrac{dx}{x}=\ln|x|+C,\quad \int e^{x}dx=e^{x}+C\)
\(\int\sin x\,dx=-\cos x+C,\quad \int\cos x\,dx=\sin x+C,\quad \int\sec^{2}x\,dx=\tan x+C\)
\(\int\tan x\,dx=-\ln|\cos x|+C,\quad \int\sec x\,dx=\ln|\sec x+\tan x|+C\)
\(\int u\,dv=uv-\int v\,du\ \text{(choose }u\text{ by LIATE)}\)
Series
\(e^{x}=\sum_{n=0}^{\infty}\dfrac{x^{n}}{n!}\)
\(\sin x=x-\dfrac{x^{3}}{3!}+\dfrac{x^{5}}{5!}-\cdots\)
\(\cos x=1-\dfrac{x^{2}}{2!}+\dfrac{x^{4}}{4!}-\cdots\)
\(\dfrac{1}{1-x}=\sum_{n=0}^{\infty}x^{n}\ (|x|<1)\)
Geometry
\(\text{Circle: }A=\pi r^{2},\ C=2\pi r;\quad \text{Triangle: }A=\tfrac12bh\)
\(\text{Sphere: }V=\tfrac43\pi r^{3},\ A=4\pi r^{2}\)
\(\text{Cylinder: }V=\pi r^{2}h;\quad \text{Cone: }V=\tfrac13\pi r^{2}h\)
Probability and statistics
\(\bar x=\dfrac{\sum x}{n},\quad s=\sqrt{\dfrac{\sum(x-\bar x)^{2}}{n-1}},\quad z=\dfrac{x-\mu}{\sigma}\)
\(P(k)=\dbinom nk p^{k}(1-p)^{n-k},\quad \mu=np,\quad \sigma=\sqrt{np(1-p)}\)
\({}_{n}P_{r}=\dfrac{n!}{(n-r)!},\quad {}_{n}C_{r}=\dfrac{n!}{r!(n-r)!}\)
\(68\%\,/\,95\%\,/\,99.7\%\ \text{within }1\sigma\,/\,2\sigma\,/\,3\sigma\)
\(z_{90\%,95\%,99\%}=1.645,\ 1.960,\ 2.576\)
Vectors, matrices, complex
\(\vec A\cdot\vec B=AB\cos\theta=A_{x}B_{x}+A_{y}B_{y},\quad |\vec A\times\vec B|=AB\sin\theta\)
\(\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc,\quad A^{-1}=\dfrac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\)
\(e^{i\theta}=\cos\theta+i\sin\theta\)
Physics
Motion
\(v=v_{0}+at,\quad \Delta x=v_{0}t+\tfrac12at^{2}\)
\(v^{2}=v_{0}^{2}+2a\Delta x,\quad \Delta x=\tfrac12(v_{0}+v)t\)
\(x=v_{0}\cos\theta\,t,\quad y=v_{0}\sin\theta\,t-\tfrac12gt^{2}\)
\(a_{c}=\dfrac{v^{2}}{r},\quad v=\omega r,\quad \omega=\dfrac{2\pi}{T}\)
Forces and momentum
\(\Sigma F=ma,\quad W=mg,\quad f=\mu N,\quad F_{\text{spring}}=-kx\)
\(F_{g}=\dfrac{Gm_{1}m_{2}}{r^{2}},\quad \tau=rF\sin\theta\)
\(p=mv,\quad J=F\Delta t=\Delta p\)
Rotation
\(\tau=I\alpha,\quad L=I\omega,\quad KE_{\text{rot}}=\tfrac12I\omega^{2}\)
\(I_{\text{sphere}}=\tfrac25MR^{2},\ I_{\text{disk}}=\tfrac12MR^{2},\ I_{\text{hoop}}=MR^{2},\ I_{\text{rod}}=\tfrac1{12}ML^{2}\)
Work and energy
\(W=Fd\cos\theta,\quad P=\dfrac Wt,\quad KE=\tfrac12mv^{2}\)
\(PE_{g}=mgh,\quad PE_{\text{spring}}=\tfrac12kx^{2},\quad E=KE+PE\)
Waves and optics
\(v=f\lambda,\quad f=\dfrac1T,\quad n=\dfrac cv\)
\(n_{1}\sin\theta_{1}=n_{2}\sin\theta_{2},\quad \dfrac1f=\dfrac1{d_{o}}+\dfrac1{d_{i}}\)
Electricity
\(V=IR,\quad P=IV=I^{2}R=\dfrac{V^{2}}{R}\)
\(R_{s}=R_{1}+R_{2},\quad \dfrac1{R_{p}}=\dfrac1{R_{1}}+\dfrac1{R_{2}}\)
\(\Sigma I_{\text{in}}=\Sigma I_{\text{out}},\quad \Sigma V=0\)
\(F=\dfrac{kq_{1}q_{2}}{r^{2}},\quad k=8.99\times10^{9}\ \text{N·m}^{2}/\text{C}^{2}\)
\(E=\dfrac Fq,\quad C=\dfrac QV,\quad \tau=RC\)
Capacitors and magnetism
\(\dfrac1{C_{s}}=\dfrac1{C_{1}}+\dfrac1{C_{2}},\quad C_{p}=C_{1}+C_{2},\quad U=\tfrac12CV^{2}\)
\(F=qvB\sin\theta,\quad F=BIL\sin\theta,\quad \mathcal E=-\dfrac{d\Phi}{dt}\)
Heat and fluids
\(Q=mc\Delta T,\quad Q=mL,\quad \Delta U=Q-W\)
\(\rho=\dfrac mV,\quad P=\dfrac FA,\quad P=P_{0}+\rho gh\)
\(e=\dfrac{W}{Q_{H}},\quad e_{\text{Carnot}}=1-\dfrac{T_{C}}{T_{H}}\)
\(A_{1}v_{1}=A_{2}v_{2},\quad P+\tfrac12\rho v^{2}+\rho gy=\text{const},\quad F_{b}=\rho Vg\)
Modern physics and electronics
\(E=hf,\quad \lambda=\dfrac hp,\quad E=mc^{2},\quad K_{\max}=hf-\phi\)
\(\text{dB}=20\log\dfrac{V_{2}}{V_{1}}=10\log\dfrac{P_{2}}{P_{1}},\quad f_{\text{res}}=\dfrac1{2\pi\sqrt{LC}}\)
Chemistry
Gases and moles
\(PV=nRT,\quad n=\dfrac mM,\quad N=nN_{A}\)
\(\%\text{ yield}=\dfrac{\text{actual}}{\text{theoretical}}\times100\)
\(M=\dfrac{\text{mol solute}}{\text{L solution}},\quad M_{1}V_{1}=M_{2}V_{2}\)
Acids and bases
\(\text{pH}=-\log[\text{H}^{+}],\quad \text{pOH}=-\log[\text{OH}^{-}],\quad \text{pH}+\text{pOH}=14\)
\(K_{w}=[\text{H}^{+}][\text{OH}^{-}]=1.0\times10^{-14},\quad \text{p}K_{a}=-\log K_{a}\)
\(\text{pH}=\text{p}K_{a}+\log\dfrac{[\text{A}^{-}]}{[\text{HA}]}\)
\(\text{Strong acids: HCl, HBr, HI, HNO}_{3}\text{, H}_{2}\text{SO}_{4}\text{, HClO}_{4}\)
\(\text{Strong bases: group 1 hydroxides, Ca(OH)}_{2}\text{, Sr(OH)}_{2}\text{, Ba(OH)}_{2}\)
Kinetics
\(\text{rate}=k[A]^{m}[B]^{n}\)
\(\text{0th: }[A]=[A]_{0}-kt,\quad \text{1st: }\ln[A]=\ln[A]_{0}-kt,\quad \text{2nd: }\dfrac1{[A]}=\dfrac1{[A]_{0}}+kt\)
\(t_{1/2}=\dfrac{0.693}{k}\ \text{(1st order)},\quad \ln k=\ln A-\dfrac{E_{a}}{RT}\)
Energy and equilibrium
\(q=mc\Delta T\quad (c_{\text{water}}=4.184\ \text{J/g·°C})\)
\(\Delta G=\Delta H-T\Delta S=-RT\ln K\)
\(K=\dfrac{[C]^{c}[D]^{d}}{[A]^{a}[B]^{b}}\ \text{for }aA+bB\rightleftharpoons cC+dD\)
\(E^{\circ}_{\text{cell}}=E^{\circ}_{\text{cathode}}-E^{\circ}_{\text{anode}},\quad \Delta G^{\circ}=-nFE^{\circ}\)
\(E=E^{\circ}-\dfrac{0.0592}{n}\log Q\ \ (25\ {}^{\circ}\text{C})\)
Solutions and atoms
\(\Delta T=iKm,\quad \pi=iMRT,\quad E=h\nu,\quad c=\lambda\nu\)
\(E_{n}=-\dfrac{2.178\times10^{-18}\ \text{J}}{n^{2}}\)
Molecular shapes (VSEPR)
\(2:\text{ linear }180^{\circ},\quad 3:\text{ trigonal planar }120^{\circ},\quad 4:\text{ tetrahedral }109.5^{\circ}\)
\(5:\text{ trigonal bipyramidal }90^{\circ}/120^{\circ},\quad 6:\text{ octahedral }90^{\circ}\)
Solubility rules
\(\text{Soluble: group 1, NH}_{4}^{+}\text{, NO}_{3}^{-}\text{, C}_{2}\text{H}_{3}\text{O}_{2}^{-}\)
\(\text{Cl}^{-}\text{, Br}^{-}\text{, I}^{-}\text{ soluble except Ag}^{+}\text{, Pb}^{2+}\text{, Hg}_{2}^{2+}\)
\(\text{SO}_{4}^{2-}\text{ soluble except Ba}^{2+}\text{, Pb}^{2+}\text{, Sr}^{2+}\text{, Ca}^{2+}\)
\(\text{OH}^{-}\text{, S}^{2-}\text{, CO}_{3}^{2-}\text{, PO}_{4}^{3-}\text{ insoluble except group 1 and NH}_{4}^{+}\)
Common polyatomic ions
\(\text{NH}_{4}^{+}\text{ ammonium},\ \text{NO}_{3}^{-}\text{ nitrate},\ \text{SO}_{4}^{2-}\text{ sulfate},\ \text{CO}_{3}^{2-}\text{ carbonate}\)
\(\text{PO}_{4}^{3-}\text{ phosphate},\ \text{OH}^{-}\text{ hydroxide},\ \text{HCO}_{3}^{-}\text{ bicarbonate}\)
\(\text{C}_{2}\text{H}_{3}\text{O}_{2}^{-}\text{ acetate},\ \text{MnO}_{4}^{-}\text{ permanganate},\ \text{CN}^{-}\text{ cyanide}\)
Organic prefixes and groups
\(\text{1 meth, 2 eth, 3 prop, 4 but, 5 pent, 6 hex, 7 hept, 8 oct, 9 non, 10 dec}\)
\(\text{alcohol }-\text{OH},\ \text{ether R–O–R},\ \text{aldehyde }-\text{CHO},\ \text{ketone C=O},\ \text{acid }-\text{COOH},\ \text{ester }-\text{COO}-\)
\(\text{amine }-\text{NH}_{2},\ \text{amide }-\text{CONH}_{2},\ \text{alkene C=C},\ \text{alkyne C≡C}\)
Advanced math
Differential equations: first order
\(\text{Separable: }\dfrac{dy}{dx}=f(x)g(y)\Rightarrow\int\dfrac{dy}{g(y)}=\int f(x)\,dx\)
\(\text{Linear: }y'+P(x)y=Q(x),\quad \mu=e^{\int P\,dx},\quad y=\dfrac1\mu\int\mu Q\,dx\)
\(\text{Exact: }M\,dx+N\,dy=0\text{ with }M_{y}=N_{x}\Rightarrow\exists\,F:\ F_{x}=M,\ F_{y}=N\)
\(\text{Bernoulli: }y'+Py=Qy^{n}\ \text{use }v=y^{1-n}\)
\(\text{Growth/decay: }y=y_{0}e^{kt};\quad \text{Newton cooling: }T=T_{s}+(T_{0}-T_{s})e^{-kt}\)
Differential equations: second order
\(ay''+by'+cy=0:\ ar^{2}+br+c=0\)
\(\text{Real distinct: }y=C_{1}e^{r_{1}x}+C_{2}e^{r_{2}x};\quad \text{repeated: }y=(C_{1}+C_{2}x)e^{rx}\)
\(\text{Complex }r=\alpha\pm\beta i:\ y=e^{\alpha x}(C_{1}\cos\beta x+C_{2}\sin\beta x)\)
\(\text{Wronskian: }W=y_{1}y_{2}'-y_{2}y_{1}',\quad y_{p}=-y_{1}\!\int\!\dfrac{y_{2}g}{W}dx+y_{2}\!\int\!\dfrac{y_{1}g}{W}dx\)
\(\text{Undetermined coefficients: guess the form of }g(x);\ \text{multiply by }x\text{ if it solves the homogeneous eq.}\)
\(\text{Cauchy-Euler: }x^{2}y''+axy'+by=0,\ y=x^{r}\)
Laplace transforms
\(\mathcal L\{1\}=\dfrac1s,\quad \mathcal L\{t^{n}\}=\dfrac{n!}{s^{n+1}},\quad \mathcal L\{e^{at}\}=\dfrac1{s-a}\)
\(\mathcal L\{\sin bt\}=\dfrac{b}{s^{2}+b^{2}},\quad \mathcal L\{\cos bt\}=\dfrac{s}{s^{2}+b^{2}}\)
\(\mathcal L\{\sinh bt\}=\dfrac{b}{s^{2}-b^{2}},\quad \mathcal L\{\cosh bt\}=\dfrac{s}{s^{2}-b^{2}}\)
\(\mathcal L\{e^{at}\sin bt\}=\dfrac{b}{(s-a)^{2}+b^{2}},\quad \mathcal L\{e^{at}\cos bt\}=\dfrac{s-a}{(s-a)^{2}+b^{2}}\)
\(\mathcal L\{te^{at}\}=\dfrac1{(s-a)^{2}},\quad \mathcal L\{u_{c}(t)\}=\dfrac{e^{-cs}}{s},\quad \mathcal L\{\delta(t-c)\}=e^{-cs}\)
\(\mathcal L\{f'\}=sF-f(0),\quad \mathcal L\{f''\}=s^{2}F-sf(0)-f'(0)\)
\(\mathcal L\{e^{ct}f\}=F(s-c),\quad \mathcal L\{tf\}=-F'(s),\quad \mathcal L\{f*g\}=F(s)G(s)\)
\(\mathcal L\{u_{c}(t)f(t-c)\}=e^{-cs}F(s)\)
Fourier series and transform
\(f(x)=\dfrac{a_{0}}{2}+\sum_{n=1}^{\infty}\left(a_{n}\cos\dfrac{n\pi x}{L}+b_{n}\sin\dfrac{n\pi x}{L}\right)\)
\(a_{n}=\dfrac1L\int_{-L}^{L}f\cos\dfrac{n\pi x}{L}dx,\quad b_{n}=\dfrac1L\int_{-L}^{L}f\sin\dfrac{n\pi x}{L}dx\)
\(F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt,\quad f(t)=\dfrac1{2\pi}\int_{-\infty}^{\infty}F(\omega)e^{j\omega t}d\omega\)
\(\delta(t)\leftrightarrow1,\quad e^{-at}u(t)\leftrightarrow\dfrac1{a+j\omega},\quad f(t-t_{0})\leftrightarrow e^{-j\omega t_{0}}F(\omega)\)
\(f*g\leftrightarrow FG,\quad \text{Parseval: }\int|f|^{2}dt=\dfrac1{2\pi}\int|F|^{2}d\omega\)
Linear algebra
\(Ax=b;\quad (AB)^{T}=B^{T}A^{T},\quad (AB)^{-1}=B^{-1}A^{-1}\)
\(\det(A-\lambda I)=0,\quad Av=\lambda v,\quad \operatorname{tr}A=\sum\lambda_{i},\quad \det A=\prod\lambda_{i}\)
\(\text{rank}(A)+\text{nullity}(A)=n,\quad A=PDP^{-1}\text{ (diagonalization)}\)
\(\text{proj}_{u}v=\dfrac{v\cdot u}{u\cdot u}u,\quad \cos\theta=\dfrac{u\cdot v}{\|u\|\|v\|}\)
\(\text{Least squares: }A^{T}A\hat x=A^{T}b;\quad \text{SVD: }A=U\Sigma V^{T}\)
\(\text{Cramer: }x_{i}=\dfrac{\det A_{i}}{\det A};\quad \text{orthogonal: }Q^{T}Q=I\)
Multivariable and vector calculus
\(\nabla f=\left(f_{x},f_{y},f_{z}\right),\quad D_{u}f=\nabla f\cdot u,\quad \nabla\cdot F=\dfrac{\partial F_{x}}{\partial x}+\dfrac{\partial F_{y}}{\partial y}+\dfrac{\partial F_{z}}{\partial z}\)
\(\nabla\times F=\begin{vmatrix}\hat i&\hat j&\hat k\\\partial_{x}&\partial_{y}&\partial_{z}\\F_{x}&F_{y}&F_{z}\end{vmatrix}\)
\(\text{Polar: }dA=r\,dr\,d\theta;\quad \text{cylindrical: }dV=r\,dr\,d\theta\,dz;\quad \text{spherical: }dV=\rho^{2}\sin\phi\,d\rho\,d\phi\,d\theta\)
\(\text{Green: }\oint_{C}P\,dx+Q\,dy=\iint\left(Q_{x}-P_{y}\right)dA\)
\(\text{Stokes: }\oint F\cdot dr=\iint(\nabla\times F)\cdot dS;\quad \text{Divergence: }\oint F\cdot dS=\iiint\nabla\cdot F\,dV\)
\(\text{Lagrange multipliers: }\nabla f=\lambda\nabla g;\quad \text{Jacobian: }dx\,dy=\left|\dfrac{\partial(x,y)}{\partial(u,v)}\right|du\,dv\)
Complex numbers and analysis
\(z=a+bi=re^{i\theta},\quad r=\sqrt{a^{2}+b^{2}},\quad \theta=\arg z\)
\(\text{De Moivre: }z^{n}=r^{n}(\cos n\theta+i\sin n\theta),\quad \text{roots: }z^{1/n}=r^{1/n}e^{i(\theta+2\pi k)/n}\)
\(\text{Cauchy-Riemann: }u_{x}=v_{y},\ u_{y}=-v_{x};\quad \oint_{C}f(z)\,dz=2\pi i\sum\operatorname{Res}\)
\(\sin\theta=\dfrac{e^{i\theta}-e^{-i\theta}}{2i},\quad \cos\theta=\dfrac{e^{i\theta}+e^{-i\theta}}{2},\quad e^{i\pi}+1=0\)
Numerical methods
\(\text{Newton: }x_{n+1}=x_{n}-\dfrac{f(x_{n})}{f'(x_{n})}\)
\(\text{Euler: }y_{n+1}=y_{n}+hf(t_{n},y_{n})\)
\(\text{RK4: }y_{n+1}=y_{n}+\tfrac h6(k_{1}+2k_{2}+2k_{3}+k_{4})\)
\(\text{Trapezoid: }\int_{a}^{b}f\approx\dfrac h2\left[f_{0}+2f_{1}+\cdots+2f_{n-1}+f_{n}\right]\)
\(\text{Simpson: }\int_{a}^{b}f\approx\dfrac h3\left[f_{0}+4f_{1}+2f_{2}+\cdots+4f_{n-1}+f_{n}\right]\)
\(\text{Taylor: }f(x)=\sum\dfrac{f^{(n)}(a)}{n!}(x-a)^{n}\)
Discrete math and number theory
\(\sum_{k=1}^{n}k=\dfrac{n(n+1)}2,\quad \sum k^{2}=\dfrac{n(n+1)(2n+1)}6,\quad \sum_{k=0}^{n}r^{k}=\dfrac{1-r^{n+1}}{1-r}\)
\((x+y)^{n}=\sum_{k=0}^{n}\dbinom nk x^{n-k}y^{k},\quad |A\cup B|=|A|+|B|-|A\cap B|\)
\(a\equiv b\pmod m\iff m\mid(a-b);\quad \gcd(a,b)=\gcd(b,a\bmod b)\)
\(\text{Fermat: }a^{p-1}\equiv1\pmod p;\quad \text{Euler: }a^{\varphi(n)}\equiv1\pmod n\)
Statistics: inference
\(\text{CI for mean: }\bar x\pm z^{*}\dfrac{\sigma}{\sqrt n}\ \ \text{or}\ \ \bar x\pm t^{*}\dfrac{s}{\sqrt n}\)
\(t=\dfrac{\bar x-\mu_{0}}{s/\sqrt n},\quad \chi^{2}=\sum\dfrac{(O-E)^{2}}{E}\)
\(\hat y=a+bx,\quad b=r\dfrac{s_{y}}{s_{x}},\quad a=\bar y-b\bar x,\quad r^{2}=\text{explained variance}\)
\(P(A|B)=\dfrac{P(B|A)P(A)}{P(B)},\quad P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
\(\text{Poisson: }P(k)=\dfrac{\lambda^{k}e^{-\lambda}}{k!};\quad \text{Normal: }f(x)=\dfrac1{\sigma\sqrt{2\pi}}e^{-(x-\mu)^{2}/2\sigma^{2}}\)
\(E[X]=\sum xP(x),\quad \operatorname{Var}(X)=E[X^{2}]-E[X]^{2},\quad \text{standard error}=\dfrac{s}{\sqrt n}\)
Unit circle
Each point is \((\cos\theta,\ \sin\theta)\). Quadrants: All Students Take Calculus.
Special angles
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| radians | 0 | \(\pi/6\) | \(\pi/4\) | \(\pi/3\) | \(\pi/2\) |
| sin | 0 | \(1/2\) | \(\sqrt2/2\) | \(\sqrt3/2\) | 1 |
| cos | 1 | \(\sqrt3/2\) | \(\sqrt2/2\) | \(1/2\) | 0 |
| tan | 0 | \(\sqrt3/3\) | 1 | \(\sqrt3\) | undefined |
Memory aids
SOH CAH TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj
\(\text{radians}=\text{degrees}\times\pi/180\)
OIL RIG: oxidation is loss, reduction is gain (of electrons)
Advanced physics
Electromagnetism
\(\oint E\cdot dA=\dfrac{Q}{\varepsilon_{0}},\quad \oint B\cdot dA=0\)
\(\oint E\cdot dl=-\dfrac{d\Phi_{B}}{dt},\quad \oint B\cdot dl=\mu_{0}\left(I+\varepsilon_{0}\dfrac{d\Phi_{E}}{dt}\right)\)
\(F=q(E+v\times B),\quad dB=\dfrac{\mu_{0}}{4\pi}\dfrac{I\,dl\times\hat r}{r^{2}},\quad B_{\text{wire}}=\dfrac{\mu_{0}I}{2\pi r}\)
\(c=\dfrac1{\sqrt{\mu_{0}\varepsilon_{0}}},\quad S=\dfrac{E\times B}{\mu_{0}},\quad u=\tfrac12\varepsilon_{0}E^{2}+\dfrac{B^{2}}{2\mu_{0}}\)
\(V=\dfrac{kq}{r},\quad E=-\nabla V,\quad \text{solenoid: }B=\mu_{0}nI,\quad L=\dfrac{\mu_{0}N^{2}A}{\ell}\)
Quantum mechanics
\(i\hbar\dfrac{\partial\psi}{\partial t}=\hat H\psi,\quad \hat H\psi=E\psi,\quad \hat H=-\dfrac{\hbar^{2}}{2m}\nabla^{2}+V\)
\(\Delta x\,\Delta p\ge\dfrac\hbar2,\quad [x,p]=i\hbar,\quad \lambda=\dfrac hp,\quad E=hf=\hbar\omega\)
\(\text{Particle in a box: }E_{n}=\dfrac{n^{2}\pi^{2}\hbar^{2}}{2mL^{2}},\quad \psi_{n}=\sqrt{\tfrac2L}\sin\dfrac{n\pi x}{L}\)
\(\text{Harmonic oscillator: }E_{n}=\hbar\omega\left(n+\tfrac12\right);\quad \text{hydrogen: }E_{n}=-\dfrac{13.6\ \text{eV}}{n^{2}}\)
\(\langle A\rangle=\int\psi^{*}\hat A\psi\,dx,\quad \int|\psi|^{2}dx=1,\quad a_{0}=0.529\ \text{Å}\)
Special relativity
\(\gamma=\dfrac1{\sqrt{1-v^{2}/c^{2}}},\quad \Delta t=\gamma\Delta t_{0},\quad L=\dfrac{L_{0}}{\gamma}\)
\(E=\gamma mc^{2},\quad E^{2}=(pc)^{2}+(mc^{2})^{2},\quad p=\gamma mv\)
\(u'=\dfrac{u-v}{1-uv/c^{2}},\quad f_{\text{obs}}=f\sqrt{\dfrac{1+\beta}{1-\beta}}\)
Thermal and statistical physics
\(S=k_{B}\ln\Omega,\quad \dfrac1T=\dfrac{\partial S}{\partial U},\quad Z=\sum e^{-E_{i}/k_{B}T},\quad P_{i}=\dfrac{e^{-E_{i}/k_{B}T}}{Z}\)
\(v_{\text{rms}}=\sqrt{\dfrac{3k_{B}T}{m}},\quad \tfrac12k_{B}T\text{ per quadratic degree of freedom}\)
\(dU=T\,dS-P\,dV,\quad F=U-TS,\quad G=H-TS,\quad C_{P}-C_{V}=nR\ \text{(ideal gas)}\)
\(\text{Blackbody: }P=\sigma AT^{4},\ \sigma=5.670\times10^{-8};\quad \lambda_{\max}T=2.898\times10^{-3}\ \text{m·K}\)
Classical mechanics (advanced)
\(L=T-V,\quad \dfrac{d}{dt}\dfrac{\partial L}{\partial\dot q}=\dfrac{\partial L}{\partial q},\quad H=\sum p\dot q-L\)
\(T^{2}=\dfrac{4\pi^{2}a^{3}}{GM},\quad v^{2}=GM\left(\dfrac2r-\dfrac1a\right),\quad v_{\text{esc}}=\sqrt{\dfrac{2GM}{r}}\)
\(\ddot x+2\zeta\omega_{n}\dot x+\omega_{n}^{2}x=0,\quad \omega_{n}=\sqrt{k/m},\quad \omega_{d}=\omega_{n}\sqrt{1-\zeta^{2}}\)
\(a_{r}=\ddot r-r\dot\theta^{2},\quad a_{\theta}=r\ddot\theta+2\dot r\dot\theta,\quad L=r\times p\)
Optics and waves (advanced)
\(d\sin\theta=m\lambda\ \text{(slits/gratings)},\quad a\sin\theta=m\lambda\ \text{(single-slit minima)}\)
\(\theta_{\min}=1.22\dfrac\lambda D,\quad \dfrac1f=\dfrac1{d_{o}}+\dfrac1{d_{i}},\quad M=-\dfrac{d_{i}}{d_{o}}\)
\(\tan\theta_{B}=\dfrac{n_{2}}{n_{1}},\quad \sin\theta_{c}=\dfrac{n_{2}}{n_{1}},\quad I=I_{0}\cos^{2}\theta\ \text{(Malus)}\)
\(f_{\text{obs}}=f\dfrac{v\pm v_{o}}{v\mp v_{s}},\quad I=\dfrac{P}{4\pi r^{2}},\quad \beta=10\log\dfrac I{10^{-12}}\ \text{dB}\)
Advanced chemistry
Thermochemistry
\(\Delta H^{\circ}_{rxn}=\sum\Delta H^{\circ}_{f}(\text{products})-\sum\Delta H^{\circ}_{f}(\text{reactants})\)
\(\Delta S^{\circ}_{rxn}=\sum S^{\circ}(\text{products})-\sum S^{\circ}(\text{reactants})\)
\(\Delta G=\Delta G^{\circ}+RT\ln Q,\quad \Delta G^{\circ}=-RT\ln K=-nFE^{\circ}\)
\(\Delta U=q+w,\quad w=-P\Delta V,\quad \Delta H=\Delta U+\Delta nRT\)
\(\ln\dfrac{K_{2}}{K_{1}}=-\dfrac{\Delta H^{\circ}}{R}\left(\dfrac1{T_{2}}-\dfrac1{T_{1}}\right)\ \text{(van 't Hoff)}\)
\(\ln\dfrac{P_{2}}{P_{1}}=-\dfrac{\Delta H_{vap}}{R}\left(\dfrac1{T_{2}}-\dfrac1{T_{1}}\right)\ \text{(Clausius-Clapeyron)}\)
Gases, solutions and phase behavior
\(\left(P+\dfrac{an^{2}}{V^{2}}\right)(V-nb)=nRT\ \text{(van der Waals)}\)
\(\dfrac{r_{1}}{r_{2}}=\sqrt{\dfrac{M_{2}}{M_{1}}}\ \text{(Graham)},\quad v_{\text{rms}}=\sqrt{\dfrac{3RT}{M}}\)
\(P_{i}=x_{i}P_{i}^{*}\ \text{(Raoult)},\quad P_{i}=\chi_{i}P_{\text{tot}}\ \text{(Dalton)}\)
\(\Delta T_{f}=iK_{f}m,\quad \Delta T_{b}=iK_{b}m,\quad \Pi=iMRT,\quad S_{\text{gas}}=k_{H}P\)
\(F=C-P+2\ \text{(Gibbs phase rule)}\)
Kinetics and mechanisms
\(\text{Arrhenius: }k=Ae^{-E_{a}/RT},\quad \ln\dfrac{k_{2}}{k_{1}}=\dfrac{E_{a}}{R}\left(\dfrac1{T_{1}}-\dfrac1{T_{2}}\right)\)
\(\text{Eyring: }k=\dfrac{k_{B}T}{h}e^{-\Delta G^{\ddagger}/RT},\quad \text{steady state: }\dfrac{d[I]}{dt}\approx0\)
\(\text{Half-lives: 0th }\dfrac{[A]_{0}}{2k},\ \text{1st }\dfrac{\ln2}{k},\ \text{2nd }\dfrac1{k[A]_{0}}\)
\(\text{Michaelis-Menten: }v=\dfrac{V_{\max}[S]}{K_{m}+[S]},\quad \dfrac1v=\dfrac{K_{m}}{V_{\max}}\dfrac1{[S]}+\dfrac1{V_{\max}}\)
Quantum chemistry and spectroscopy
\(\dfrac1\lambda=R_{H}\left(\dfrac1{n_{1}^{2}}-\dfrac1{n_{2}^{2}}\right),\quad R_{H}=1.097\times10^{7}\ \text{m}^{-1}\)
\(E_{J}=BJ(J+1)\ \text{(rotation)},\quad E_{v}=h\nu\left(v+\tfrac12\right)\ \text{(vibration)}\)
\(\tilde\nu=\dfrac1{2\pi c}\sqrt{\dfrac k\mu},\quad \mu=\dfrac{m_{1}m_{2}}{m_{1}+m_{2}}\)
\(\dfrac{N_{j}}{N_{i}}=e^{-\Delta E/k_{B}T},\quad \text{bond order}=\tfrac12(n_{\text{bonding}}-n_{\text{antibonding}})\)
\(\text{Formal charge}=V-N_{\text{nonbonding}}-\tfrac12N_{\text{bonding}}\)
Analytical chemistry
\(A=\varepsilon bc=-\log T,\quad T=10^{-A}\)
\(\text{pH}=\text{p}K_{a}+\log\dfrac{[\text{base}]}{[\text{acid}]},\quad K_{sp}=[\text{M}]^{m}[\text{A}]^{n}\)
\(I=\tfrac12\sum c_{i}z_{i}^{2},\quad \text{LOD}\approx\dfrac{3\sigma_{\text{blank}}}{m}\)
\(R_{s}=\dfrac{2(t_{2}-t_{1})}{w_{1}+w_{2}},\quad N=16\left(\dfrac{t_{R}}{w}\right)^{2},\quad k'=\dfrac{t_{R}-t_{M}}{t_{M}}\)
\(\text{Error propagation: add/subtract }\sigma=\sqrt{\sigma_{a}^{2}+\sigma_{b}^{2}};\ \text{multiply/divide }\dfrac{\sigma}{x}=\sqrt{\left(\dfrac{\sigma_{a}}{a}\right)^{2}+\left(\dfrac{\sigma_{b}}{b}\right)^{2}}\)
\(m=\dfrac{QM}{nF}\ \text{(electrolysis)},\quad Q=It\)
Organic: reaction cheat sheet
SN2: rate = k[RX][Nu]; one step; backside attack, inversion; methyl > 1° > 2° (3° does not react); strong nucleophile, polar aprotic solvent.
SN1: rate = k[RX]; carbocation intermediate, racemization; 3° > 2°; weak nucleophile, polar protic solvent.
E2: rate = k[RX][base]; anti-periplanar; strong base; Zaitsev (more substituted alkene) unless bulky base (Hofmann).
E1: carbocation, competes with SN1, favored by heat; Zaitsev product.
Alkene + HX: Markovnikov. HBr + peroxides: anti-Markovnikov. H2O/H2SO4: Markovnikov alcohol. BH3 then H2O2/NaOH: anti-Markovnikov, syn addition.
Alkene + X2: anti addition (halonium). Alkene + H2/Pd: syn addition. mCPBA: epoxide. OsO4: syn diol. O3 then Me2S: cleavage to carbonyls.
NaBH4 reduces aldehydes and ketones. LiAlH4 also reduces acids, esters, amides. PCC: 1° alcohol to aldehyde. Jones/KMnO4: 1° alcohol to carboxylic acid.
Grignard (RMgX) adds to carbonyl: aldehyde gives 2° alcohol, ketone gives 3° alcohol, ester gives 3° alcohol (2 equivalents), CO2 gives acid. Needs dry ether.
Acyl substitution reactivity: acyl chloride > anhydride > ester ~ acid > amide. Fischer esterification: acid + alcohol, H+ catalyst, equilibrium.
Aldol: enolate + carbonyl, then dehydration with heat. Diels-Alder: diene + dienophile gives cyclohexene, concerted, endo favored.
Electrophilic aromatic substitution: nitration (HNO3/H2SO4), halogenation (X2/FeX3), sulfonation, Friedel-Crafts alkylation/acylation (AlCl3). Activators (OH, OR, NH2, alkyl) are ortho/para; deactivators (NO2, CN, C=O, CF3) are meta; halogens deactivate but are ortho/para.
Organic: acidity (approximate pKa)
Strong acids (HCl, H2SO4) -7 to -3; carboxylic acid 4-5; phenol 10; thiol 10; water 15.7; alcohol 16-18; ketone alpha-H 19-20; ester alpha-H 25; terminal alkyne 25; amine N-H 35-38; alkene C-H 44; alkane C-H 50.
Acid strength trends: stronger with more stable conjugate base (electronegativity across a period, size down a group, resonance, induction, hybridization sp > sp2 > sp3).
Organic: IR (cm-1)
O-H alcohol 3200-3600 broad; O-H acid 2500-3300 very broad; N-H 3300-3500; sp C-H 3300; sp2 C-H 3000-3100; sp3 C-H 2850-3000; aldehyde C-H 2700-2850 (two peaks).
C≡N 2210-2260; C≡C 2100-2260; C=O: ketone ~1715, aldehyde ~1725, acid 1700-1725, ester 1735-1750, amide 1650-1690, acyl chloride ~1800; C=C 1620-1680; aromatic C=C 1450-1600; C-O 1000-1300.
Organic: NMR and mass spec
1H NMR (ppm): alkyl 0.9-1.7; allylic 1.6-2.6; alpha to C=O 2.0-2.7; alkyne H 2-3; H-C-X 2.5-4; H-C-O 3.3-4.5; vinyl 4.5-6.5; aromatic 6.5-8.5; aldehyde 9-10; carboxylic acid 10-12; OH and NH variable 1-5.
13C NMR (ppm): alkyl 0-50; C-O / C-N 50-90; alkene and aromatic 100-150; C=O 160-220. n+1 rule for splitting. Integration gives relative hydrogens.
\(\text{Degrees of unsaturation}=\dfrac{2C+2+N-H-X}{2}\)
Mass spec: M+ is the molecular ion. Chlorine gives M and M+2 about 3:1; bromine gives M and M+2 about 1:1. Odd molecular mass means odd number of nitrogens.
Organic: nomenclature and functional groups
Priority (high to low): carboxylic acid > ester > amide > nitrile > aldehyde > ketone > alcohol > amine > alkene > alkyne > alkane/halide/ether.
Prefixes: meth 1, eth 2, prop 3, but 4, pent 5, hex 6, hept 7, oct 8, non 9, dec 10. Suffixes: -ol alcohol, -al aldehyde, -one ketone, -oic acid, -oate ester, -amide, -amine, -nitrile, -ene, -yne.
Stereochemistry: R/S by CIP priorities (lowest priority away: clockwise = R). Enantiomers are nonsuperimposable mirror images; diastereomers are not mirror images; meso compounds have stereocenters but are achiral. E/Z for alkenes. 2^n maximum stereoisomers.
Biochemistry
\(\Delta G^{\circ\prime}=-RT\ln K'_{eq},\quad \text{ATP hydrolysis }\Delta G^{\circ\prime}\approx-30.5\ \text{kJ/mol}\)
\(\text{pI}=\dfrac{\text{p}K_{a1}+\text{p}K_{a2}}2\ \text{(amino acid with neutral side chain)}\)
Amino acids (3-letter / 1-letter): Ala A, Arg R, Asn N, Asp D, Cys C, Gln Q, Glu E, Gly G, His H, Ile I, Leu L, Lys K, Met M, Phe F, Pro P, Ser S, Thr T, Trp W, Tyr Y, Val V.
Central dogma: DNA to RNA (transcription) to protein (translation). Start codon AUG; stop codons UAA, UAG, UGA. Base pairs: A-T (A-U in RNA), G-C.
Metabolism: glucose + 6 O2 gives 6 CO2 + 6 H2O + energy; photosynthesis is the reverse in light.
Engineering
Statics
\(\Sigma F_{x}=0,\ \Sigma F_{y}=0,\ \Sigma M=0,\quad M=r\times F,\quad M=Fd\)
\(\text{Friction: }F\le\mu_{s}N,\quad F_{k}=\mu_{k}N,\quad \tan\theta_{\text{slip}}=\mu_{s}\)
\(\bar x=\dfrac{\int x\,dA}{A},\quad I_{\text{rect}}=\dfrac{bh^{3}}{12},\quad I_{\text{circle}}=\dfrac{\pi d^{4}}{64},\quad I=\bar I+Ad^{2}\)
\(\text{Truss: }2j=m+r\ \text{(determinate)};\ \text{method of joints, method of sections}\)
\(\text{Distributed load: equivalent force }=\text{area under load curve, at its centroid}\)
\(\dfrac{dV}{dx}=-w,\quad \dfrac{dM}{dx}=V\)
Mechanics of materials
\(\sigma=\dfrac FA,\quad \varepsilon=\dfrac{\Delta L}L,\quad \sigma=E\varepsilon,\quad \nu=-\dfrac{\varepsilon_{\text{lat}}}{\varepsilon_{\text{axial}}},\quad G=\dfrac{E}{2(1+\nu)}\)
\(\delta=\dfrac{PL}{AE},\quad \delta_{T}=\alpha L\Delta T\)
\(\tau=\dfrac{T\rho}J,\quad J_{\text{solid}}=\dfrac{\pi d^{4}}{32},\quad \phi=\dfrac{TL}{GJ}\)
\(\sigma=\dfrac{My}I,\quad \tau=\dfrac{VQ}{Ib},\quad \dfrac1\rho=\dfrac MEI\)
\(\sigma_{\text{hoop}}=\dfrac{pr}t,\quad \sigma_{\text{long}}=\dfrac{pr}{2t}\ \text{(thin-wall cylinder)},\quad \sigma_{\text{sphere}}=\dfrac{pr}{2t}\)
\(P_{cr}=\dfrac{\pi^{2}EI}{(KL)^{2}}\ \text{(Euler buckling)}\)
\(\sigma_{\text{avg}}=\dfrac{\sigma_{x}+\sigma_{y}}2,\quad R=\sqrt{\left(\dfrac{\sigma_{x}-\sigma_{y}}2\right)^{2}+\tau_{xy}^{2}},\quad \sigma_{1,2}=\sigma_{\text{avg}}\pm R,\quad \tau_{\max}=R\)
\(\sigma_{vm}=\sqrt{\sigma_{1}^{2}-\sigma_{1}\sigma_{2}+\sigma_{2}^{2}},\quad FS=\dfrac{\sigma_{y}}{\sigma_{\text{allow}}}\)
Beam deflection (max)
\(\text{Cantilever, end load: }\delta=\dfrac{PL^{3}}{3EI}\)
\(\text{Cantilever, uniform load: }\delta=\dfrac{wL^{4}}{8EI}\)
\(\text{Simply supported, center load: }\delta=\dfrac{PL^{3}}{48EI}\)
\(\text{Simply supported, uniform load: }\delta=\dfrac{5wL^{4}}{384EI}\)
\(\text{Simply supported, center load: }M_{\max}=\dfrac{PL}{4};\quad \text{uniform: }M_{\max}=\dfrac{wL^{2}}8\)
\(\text{Fixed-fixed, center load: }\delta=\dfrac{PL^{3}}{192EI}\)
Dynamics
\(\Sigma F=ma,\quad \Sigma M_{G}=I_{G}\alpha,\quad v_{B}=v_{A}+\omega\times r_{B/A}\)
\(T_{1}+U_{1\to2}=T_{2},\quad T=\tfrac12mv^{2}+\tfrac12I\omega^{2}\)
\(m v_{1}+\int F\,dt=m v_{2},\quad e=\dfrac{v_{B}'-v_{A}'}{v_{A}-v_{B}}\ \text{(restitution)}\)
\(c_{c}=2\sqrt{km},\quad \zeta=\dfrac c{c_{c}},\quad \text{log decrement }\delta=\ln\dfrac{x_{1}}{x_{2}}\)
Engineering thermodynamics
\(\text{Closed: }Q-W=\Delta U;\quad \text{open (steady): }\dot Q-\dot W=\dot m\left[\Delta h+\dfrac{\Delta V^{2}}{2}+g\Delta z\right]\)
\(Pv=RT,\quad c_{p}-c_{v}=R,\quad k=\dfrac{c_{p}}{c_{v}},\quad \Delta s\ge\int\dfrac{\delta Q}{T}\)
\(\text{Isentropic ideal gas: }Pv^{k}=\text{const},\quad \dfrac{T_{2}}{T_{1}}=\left(\dfrac{P_{2}}{P_{1}}\right)^{(k-1)/k}\)
\(\eta_{\text{Carnot}}=1-\dfrac{T_{C}}{T_{H}},\quad COP_{R}=\dfrac{Q_{L}}{W}=\dfrac{T_{L}}{T_{H}-T_{L}},\quad COP_{HP}=\dfrac{Q_{H}}{W}\)
\(\eta_{\text{Otto}}=1-\dfrac1{r^{k-1}},\quad \eta_{\text{Brayton}}=1-\dfrac1{r_{p}^{(k-1)/k}},\quad \eta_{\text{Rankine}}=\dfrac{W_{\text{net}}}{Q_{\text{in}}}\)
\(\eta_{\text{turbine}}=\dfrac{h_{1}-h_{2}}{h_{1}-h_{2s}},\quad \text{polytropic: }Pv^{n}=\text{const}\)
Fluid mechanics
\(\dot m=\rho AV,\quad A_{1}V_{1}=A_{2}V_{2},\quad P+\tfrac12\rho V^{2}+\rho gz=\text{const}\)
\(Re=\dfrac{\rho VD}\mu,\quad \text{laminar pipe flow }Re<2300,\quad \text{turbulent }Re>4000\)
\(h_{f}=f\dfrac LD\dfrac{V^{2}}{2g}\ \text{(Darcy-Weisbach)},\quad f_{\text{laminar}}=\dfrac{64}{Re}\)
\(\Delta P=\dfrac{128\mu LQ}{\pi D^{4}}\ \text{(Hagen-Poiseuille)},\quad F_{D}=\tfrac12\rho V^{2}C_{D}A,\quad F_{L}=\tfrac12\rho V^{2}C_{L}A\)
\(F_{\text{hydro}}=\rho gh_{c}A,\quad F_{b}=\rho_{f}Vg,\quad P_{\text{pump}}=\dfrac{\rho gQH}\eta\)
\(V=\dfrac1nR_{h}^{2/3}S^{1/2}\ \text{(Manning, SI)},\quad Ma=\dfrac Vc,\quad Fr=\dfrac V{\sqrt{gL}}\)
Heat transfer
\(q=-kA\dfrac{dT}{dx},\quad R_{\text{wall}}=\dfrac L{kA},\quad Q_{\text{cyl}}=\dfrac{2\pi kL(T_{1}-T_{2})}{\ln(r_{2}/r_{1})}\)
\(q=hA(T_{s}-T_{\infty}),\quad q=\varepsilon\sigma A\left(T_{s}^{4}-T_{\text{sur}}^{4}\right),\quad R_{\text{conv}}=\dfrac1{hA}\)
\(Nu=\dfrac{hL}k,\quad Pr=\dfrac{c_{p}\mu}k,\quad Bi=\dfrac{hL_{c}}k,\quad Nu=0.023Re^{0.8}Pr^{n}\ \ (n=0.4\text{ heating},\,0.3\text{ cooling})\)
\(\dfrac{T-T_{\infty}}{T_{i}-T_{\infty}}=e^{-t/\tau},\quad \tau=\dfrac{\rho Vc}{hA}\ \ (Bi<0.1)\)
\(Q=UA\,\Delta T_{lm},\quad \Delta T_{lm}=\dfrac{\Delta T_{1}-\Delta T_{2}}{\ln(\Delta T_{1}/\Delta T_{2})},\quad \varepsilon=\dfrac{Q}{Q_{\max}}\ \text{(effectiveness-NTU)}\)
Electric circuits
\(\text{KCL: }\Sigma I=0,\quad \text{KVL: }\Sigma V=0,\quad V=IR,\quad P=VI\)
\(\text{Thevenin: }V_{th}=V_{oc},\quad R_{th}=\dfrac{V_{oc}}{I_{sc}},\quad P_{\max}\text{ when }R_{L}=R_{th}\)
\(i_{C}=C\dfrac{dv}{dt},\quad v_{L}=L\dfrac{di}{dt},\quad E_{C}=\tfrac12CV^{2},\quad E_{L}=\tfrac12LI^{2}\)
\(v(t)=v_{f}+(v_{0}-v_{f})e^{-t/\tau},\quad \tau_{RC}=RC,\quad \tau_{RL}=\dfrac LR\)
\(Z_{R}=R,\quad Z_{L}=j\omega L,\quad Z_{C}=\dfrac1{j\omega C},\quad \omega_{0}=\dfrac1{\sqrt{LC}},\quad Q=\dfrac{\omega_{0}L}R\)
\(V_{\text{rms}}=\dfrac{V_{p}}{\sqrt2},\quad P=V_{\text{rms}}I_{\text{rms}}\cos\phi,\quad S=VI,\quad PF=\cos\phi,\quad P_{3\phi}=\sqrt3V_{L}I_{L}\cos\phi\)
\(\dfrac{V_{1}}{V_{2}}=\dfrac{N_{1}}{N_{2}}=\dfrac{I_{2}}{I_{1}}\ \text{(ideal transformer)},\quad f_{c}=\dfrac1{2\pi RC}\)
Electronics: op-amps and transistors
\(\text{Ideal op-amp: }V_{+}=V_{-},\ I_{\text{in}}=0\)
\(\text{Inverting: }\dfrac{V_{o}}{V_{i}}=-\dfrac{R_{f}}{R_{in}},\quad \text{non-inverting: }1+\dfrac{R_{f}}{R_{in}},\quad \text{integrator: }V_{o}=-\dfrac1{RC}\int V_{i}\,dt\)
\(I_{C}=\beta I_{B},\quad I_{E}=(\beta+1)I_{B},\quad V_{BE}\approx0.7\ \text{V (silicon diode drop)}\)
\(\text{MOSFET saturation: }I_{D}=\tfrac12k(V_{GS}-V_{th})^{2};\quad \text{triode: }I_{D}=k\left[(V_{GS}-V_{th})V_{DS}-\tfrac12V_{DS}^{2}\right]\)
\(A_{dB}=20\log\dfrac{V_{o}}{V_{i}},\quad f_{3dB}=\dfrac1{2\pi RC},\quad \text{GBW}=A\cdot f=\text{const}\)
Signals and systems
\(y(t)=x(t)*h(t)=\int x(\tau)h(t-\tau)\,d\tau,\quad Y(s)=H(s)X(s)\)
\(f_{s}\ge2f_{\max}\ \text{(Nyquist)},\quad \text{LSB}=\dfrac{V_{\text{ref}}}{2^{n}}\ \text{(ADC)},\quad \text{SNR}\approx6.02n+1.76\ \text{dB}\)
\(X(z)=\sum x[n]z^{-n},\quad H(\omega)=\sum h[n]e^{-j\omega n},\quad \text{DFT: }X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi kn/N}\)
\(\text{Stable (continuous) if all poles have }\operatorname{Re}(s)<0;\ \text{(discrete) if all poles inside }|z|=1\)
Control systems
\(G(s)=\dfrac{Y(s)}{U(s)},\quad T(s)=\dfrac{G}{1+GH},\quad \text{characteristic eq.: }1+GH=0\)
\(G(s)=\dfrac{\omega_{n}^{2}}{s^{2}+2\zeta\omega_{n}s+\omega_{n}^{2}},\quad \%OS=100\,e^{-\pi\zeta/\sqrt{1-\zeta^{2}}},\quad t_{s}\approx\dfrac4{\zeta\omega_{n}},\quad t_{r}\approx\dfrac{1.8}{\omega_{n}}\)
\(u(t)=K_{p}e+K_{i}\int e\,dt+K_{d}\dfrac{de}{dt},\quad C(s)=K_{p}+\dfrac{K_{i}}s+K_{d}s\)
\(e_{ss}=\lim_{s\to0}\dfrac{sR(s)}{1+G(s)},\quad \text{Routh: no sign changes in first column}\Rightarrow\text{stable}\)
\(\text{Bode: }20\log|G(j\omega)|;\ \text{phase margin}=180^{\circ}+\angle G\text{ at gain crossover}\)
Digital logic and computer engineering
\(A+AB=A,\quad A+\bar AB=A+B,\quad \overline{AB}=\bar A+\bar B,\quad \overline{A+B}=\bar A\bar B\)
\(\text{NAND and NOR are universal; XOR: }A\oplus B=A\bar B+\bar AB\)
\(n\text{-bit two's complement range: }-2^{n-1}\ \text{to}\ 2^{n-1}-1;\quad \text{negate: invert bits, add 1}\)
Flip-flops: D (Q follows D at clock edge), T (toggles when T=1), JK (J=K=1 toggles). Setup and hold times must be met. Gray code changes one bit at a time.
K-map: group 1s in powers of 2 (1, 2, 4, 8), edges wrap, larger groups give simpler terms.
Materials science
\(\text{Atoms per cell: SC 1, BCC 2, FCC 4;}\quad \text{APF: SC }0.52,\ \text{BCC }0.68,\ \text{FCC }0.74\)
\(a_{SC}=2R,\quad a_{BCC}=\dfrac{4R}{\sqrt3},\quad a_{FCC}=2\sqrt2R,\quad \rho=\dfrac{nA}{V_{c}N_{A}}\)
\(n\lambda=2d\sin\theta\ \text{(Bragg)},\quad d_{hkl}=\dfrac a{\sqrt{h^{2}+k^{2}+l^{2}}}\ \text{(cubic)}\)
\(J=-D\dfrac{dC}{dx}\ \text{(Fick I)},\quad D=D_{0}e^{-Q/RT},\quad \dfrac{\partial C}{\partial t}=D\dfrac{\partial^{2}C}{\partial x^{2}}\ \text{(Fick II)}\)
\(\sigma_{y}=\sigma_{0}+k_{y}d^{-1/2}\ \text{(Hall-Petch)},\quad K_{Ic}=Y\sigma\sqrt{\pi a}\)
\(W_{\alpha}=\dfrac{C_{\beta}-C_{0}}{C_{\beta}-C_{\alpha}}\ \text{(lever rule)},\quad \text{Miller indices: reciprocals of intercepts, cleared of fractions}\)
Chemical engineering
\(\text{Accumulation}=\text{in}-\text{out}+\text{generation}-\text{consumption}\)
\(X=\dfrac{F_{A0}-F_{A}}{F_{A0}},\quad \text{CSTR: }V=\dfrac{F_{A0}X}{-r_{A}},\quad \text{PFR: }V=F_{A0}\int_{0}^{X}\dfrac{dX}{-r_{A}}\)
\(\log_{10}P^{\text{sat}}=A-\dfrac B{C+T}\ \text{(Antoine)},\quad y_{i}P=x_{i}P_{i}^{\text{sat}}\ \text{(Raoult, VLE)}\)
\(N_{A}=k_{c}\Delta C_{A},\quad \text{Sc}=\dfrac\mu{\rho D},\quad \text{Sh}=\dfrac{k_{c}L}D\)
Civil and structural
\(Q=KiA\ \text{(Darcy, seepage)},\quad \sigma'=\sigma-u\ \text{(effective stress)}\)
\(\text{Concrete: }f_{c}'\approx20\text{-}40\ \text{MPa; steel yield }\approx250\text{-}500\ \text{MPa}\)
\(\text{Rational method: }Q=CiA,\quad \text{Manning: see Fluid mechanics}\)
Engineering economics and project math
\(F=P(1+i)^{n},\quad P=\dfrac F{(1+i)^{n}},\quad P=A\dfrac{(1+i)^{n}-1}{i(1+i)^{n}},\quad A=P\dfrac{i(1+i)^{n}}{(1+i)^{n}-1}\)
\(NPV=\sum_{t=0}^{n}\dfrac{C_{t}}{(1+r)^{t}},\quad \text{break-even }Q=\dfrac{F}{p-v},\quad \text{rule of 72: }t\approx\dfrac{72}{r\%}\)
\(\text{Dimensional analysis: write every quantity with units and cancel; check both sides of an equation have the same dimensions.}\)
Biology, CS, finance and lab skills
Biology and ecology
\(p+q=1,\quad p^{2}+2pq+q^{2}=1\ \text{(Hardy-Weinberg)}\)
\(\dfrac{dN}{dt}=rN,\quad \dfrac{dN}{dt}=rN\left(1-\dfrac NK\right),\quad Q_{10}=\left(\dfrac{R_{2}}{R_{1}}\right)^{10/(T_{2}-T_{1})}\)
\(E_{ion}=\dfrac{RT}{zF}\ln\dfrac{[\text{out}]}{[\text{in}]}\ \text{(Nernst, membrane potential)},\quad 6\text{CO}_{2}+6\text{H}_{2}\text{O}\xrightarrow{\text{light}}\text{C}_{6}\text{H}_{12}\text{O}_{6}+6\text{O}_{2}\)
Computer science
\(T(n)=aT(n/b)+f(n)\ \text{(master theorem)};\quad \text{binary search }O(\log n),\ \text{merge sort }O(n\log n),\ \text{quicksort avg }O(n\log n),\ \text{worst }O(n^{2})\)
\(\text{BFS/DFS }O(V+E),\quad \text{Dijkstra }O((V+E)\log V),\quad \text{hash table avg }O(1)\)
\(\text{Bitwise: AND }\&,\ \text{OR }|,\ \text{XOR }\wedge,\ \text{NOT }\sim,\ \text{shift }x\ll k=x\cdot2^{k}\)
\(\text{Information: }H=-\sum p_{i}\log_{2}p_{i},\quad \text{bits for }N\text{ values}=\lceil\log_{2}N\rceil\)
Finance and economics
\(A=P\left(1+\dfrac rn\right)^{nt},\quad A=Pe^{rt},\quad \text{CAGR}=\left(\dfrac{V_{f}}{V_{i}}\right)^{1/t}-1\)
\(E_{d}=\dfrac{\%\Delta Q}{\%\Delta P},\quad GDP=C+I+G+(X-M),\quad \text{MPC}+\text{MPS}=1,\quad \text{multiplier}=\dfrac1{1-\text{MPC}}\)
\(\text{Profit}=TR-TC,\quad MR=MC\ \text{(profit max)},\quad \text{markup}=\dfrac{p-c}{c}\)
Lab skills
Significant figures: nonzero digits count; zeros between digits count; leading zeros do not; trailing zeros count only with a decimal point. Multiply/divide: fewest sig figs. Add/subtract: fewest decimal places.
\(\%\text{ error}=\dfrac{|\text{measured}-\text{accepted}|}{\text{accepted}}\times100,\quad \text{precision}\ne\text{accuracy}\)
Graphing: linearize (plot y vs x^2, ln y vs x, 1/y vs x) so the slope and intercept carry the physics. Always label axes with units. Use a ruler for best-fit lines; read slope from two far-apart points on the line, not data points.
Dilution: M1V1 = M2V2. Serial dilution factor = (aliquot) / (aliquot + diluent). Always add acid to water.
Computer science and theory
Logic, proofs and sets
Truth tables: P→Q is false only when P is true and Q is false. Contrapositive (¬Q→¬P) is equivalent; converse and inverse are not.
\(\neg(\forall x\,P)\equiv\exists x\,\neg P,\quad \neg(\exists x\,P)\equiv\forall x\,\neg P,\quad A\cap(B\cup C)=(A\cap B)\cup(A\cap C)\)
Induction: prove the base case, assume P(k), prove P(k+1). Strong induction assumes P(1)...P(k). Contradiction: assume the negation and derive a falsehood.
\(|A\times B|=|A||B|,\quad |\mathcal P(A)|=2^{|A|},\quad \text{equivalence relation}=\text{reflexive, symmetric, transitive}\)
Counting and recurrences
\(P(n,r)=\dfrac{n!}{(n-r)!},\quad \dbinom nr=\dfrac{n!}{r!(n-r)!},\quad \text{stars and bars: }\dbinom{n+k-1}{k-1}\ \text{(n items, k bins)}\)
\(T(n)=2T(n/2)+n\Rightarrow\Theta(n\log n),\quad T(n)=T(n/2)+1\Rightarrow\Theta(\log n),\quad T(n)=T(n-1)+n\Rightarrow\Theta(n^{2})\)
\(\text{Master theorem: compare }f(n)\text{ to }n^{\log_{b}a}:\ \text{smaller}\Rightarrow\Theta(n^{\log_{b}a});\ \text{equal}\Rightarrow\Theta(n^{\log_{b}a}\log n);\ \text{larger}\Rightarrow\Theta(f)\)
\(E[X]=\sum E[X_{i}]\ \text{(linearity of expectation, always)},\quad \text{pigeonhole: }\lceil n/k\rceil\)
Data structures: time complexity
Array: access O(1), search O(n), insert/delete O(n). Linked list: access/search O(n), insert/delete at known node O(1). Stack and queue: push/pop/enqueue/dequeue O(1).
Hash table: average O(1), worst O(n). Binary search tree: average O(log n), worst O(n); balanced (AVL, red-black) O(log n). Heap: insert and extract-min O(log n), build-heap O(n), peek O(1).
Sorting: insertion/bubble/selection O(n^2); merge O(n log n) (stable, extra space); quicksort average O(n log n), worst O(n^2); heapsort O(n log n); comparison sorts need at least n log n; counting/radix sort O(n + k).
Amortized analysis: dynamic array doubling gives O(1) amortized append. Union-find with path compression and union by rank is nearly O(1) per operation.
Graph algorithms
\(\text{BFS/DFS: }O(V+E);\quad \text{Dijkstra (heap): }O((V+E)\log V)\ \text{(no negative edges)};\quad \text{Bellman-Ford: }O(VE);\quad \text{Floyd-Warshall: }O(V^{3})\)
\(\text{MST: Kruskal }O(E\log E)\ \text{(sort edges, union-find)},\ \text{Prim }O(E\log V);\quad \text{topological sort: DAGs only, }O(V+E)\)
Max-flow min-cut: max flow equals min cut capacity (Ford-Fulkerson, Edmonds-Karp O(VE^2)). Bipartite matching reduces to max flow.
NP-complete examples: SAT, 3-SAT, clique, vertex cover, Hamiltonian cycle, subset sum, graph coloring. To show NP-complete: show in NP, then reduce a known NP-complete problem to it. P is contained in NP; P vs NP is open.
Automata and computability
Regular languages: recognized by DFA, NFA, regex (all equivalent). Closed under union, intersection, complement, concatenation, star. Pumping lemma shows languages like {0^n 1^n} are not regular.
Context-free languages: recognized by PDAs and generated by CFGs. Not closed under intersection or complement. Turing machines recognize recursively enumerable languages.
Halting problem is undecidable (diagonalization). Rice's theorem: any nontrivial property of the language a TM recognizes is undecidable. Reduction direction: A reduces to B means B is at least as hard as A.
Operating systems
Process states: new, ready, running, waiting, terminated. Context switch saves and restores registers. Thread shares address space; process does not.
\(\text{Scheduling: FCFS, SJF (optimal average wait for known bursts), Round Robin (quantum }q\text{), priority (starvation fixed by aging)};\quad \text{turnaround}=\text{finish}-\text{arrival},\ \text{wait}=\text{turnaround}-\text{burst}\)
Deadlock needs all four: mutual exclusion, hold and wait, no preemption, circular wait. Semaphore: wait(P) decrements, signal(V) increments. Mutex gives mutual exclusion; condition variables wait for events.
\(\text{Virtual memory: page number}=\left\lfloor\dfrac{\text{addr}}{\text{page size}}\right\rfloor,\ \text{offset}=\text{addr}\bmod\text{page size};\quad \text{EAT}=h\,t_{m}+(1-h)(t_{m}+t_{\text{miss}})\)
Page replacement: FIFO (Belady's anomaly possible), LRU, optimal. Thrashing: too little memory for the working set.
Computer networks
Layers: application (HTTP, DNS), transport (TCP, UDP), network (IP), link (Ethernet, Wi-Fi), physical. TCP: reliable, ordered, congestion control (slow start, AIMD). UDP: connectionless, no guarantees.
\(\text{Transmission delay}=\dfrac LR,\quad \text{propagation delay}=\dfrac dS,\quad \text{total}=d_{\text{proc}}+d_{\text{queue}}+d_{\text{trans}}+d_{\text{prop}}\)
\(\text{Subnetting: /}n\text{ gives }2^{32-n}\text{ addresses},\ 2^{32-n}-2\text{ usable hosts; e.g. /24 = 255.255.255.0}\)
\(\text{Throughput}\le\min(R_{\text{links}}),\quad \text{TCP window: throughput}\approx\dfrac{\text{window}}{\text{RTT}},\quad \text{Shannon: }C=B\log_{2}(1+\text{SNR})\)
Computer architecture
\(\text{CPU time}=\text{IC}\times\text{CPI}\times T_{\text{clk}},\quad \text{speedup}_{\text{Amdahl}}=\dfrac1{(1-f)+f/s}\)
\(\text{AMAT}=t_{\text{hit}}+\text{miss rate}\times t_{\text{penalty}},\quad \text{cache index bits}=\log_{2}(\text{sets}),\ \text{offset bits}=\log_{2}(\text{block size})\)
5-stage pipeline: fetch, decode, execute, memory, write-back. Hazards: structural, data (forwarding, stalls), control (branch prediction). Ideal CPI is 1.
Memory hierarchy: registers, L1, L2, L3, DRAM, disk; locality (temporal and spatial). Write-through vs write-back; direct-mapped, set-associative, fully associative.
Machine learning and deep learning
\(\text{MSE}=\dfrac1n\sum(y_{i}-\hat y_{i})^{2},\quad \text{cross-entropy}=-\sum y_{i}\log\hat y_{i},\quad \sigma(z)=\dfrac1{1+e^{-z}},\quad \text{softmax}_{i}=\dfrac{e^{z_{i}}}{\sum e^{z_{j}}}\)
\(\theta\leftarrow\theta-\eta\nabla_{\theta}J,\quad \text{L2 penalty: }\lambda\|w\|^{2},\quad \text{bias-variance: }\text{error}=\text{bias}^{2}+\text{variance}+\text{noise}\)
\(\text{Precision}=\dfrac{TP}{TP+FP},\quad \text{recall}=\dfrac{TP}{TP+FN},\quad F_{1}=\dfrac{2PR}{P+R},\quad \text{accuracy}=\dfrac{TP+TN}{\text{all}}\)
Backprop applies the chain rule layer by layer. Overfitting: low train error, high test error; fix with regularization, dropout, more data, early stopping. CNNs share weights across space; transformers use attention softmax(QK^T/sqrt(d))V.
\(\text{Bayes: }P(y|x)\propto P(x|y)P(y);\quad \text{k-means: assign to nearest centroid, recompute means}\)
Databases and SQL
SELECT cols FROM t WHERE cond GROUP BY col HAVING cond ORDER BY col; JOIN types: INNER, LEFT, RIGHT, FULL. Aggregates: COUNT, SUM, AVG, MIN, MAX. WHERE filters rows, HAVING filters groups.
Normal forms: 1NF atomic values; 2NF no partial dependency on a composite key; 3NF no transitive dependency; BCNF every determinant is a key.
ACID: atomicity, consistency, isolation, durability. B+ trees give O(log n) index lookups. Relational algebra: selection σ, projection π, join ⋈, union, difference.
Cryptography and security
\(\text{RSA: }n=pq,\ \varphi=(p-1)(q-1),\ ed\equiv1\pmod\varphi,\quad c=m^{e}\bmod n,\ m=c^{d}\bmod n\)
\(\text{Diffie-Hellman: public }g^{a},g^{b};\ \text{shared key }g^{ab}\bmod p;\quad \text{birthday bound: collisions after }\sim2^{n/2}\text{ hashes}\)
Symmetric (AES) is fast; asymmetric (RSA, ECC) for key exchange and signatures. Hash (SHA-256) is one-way. MAC authenticates; signature adds non-repudiation. Never reuse a one-time pad key or an AES-GCM nonce.
Common attacks: buffer overflow, SQL injection (use parameterized queries), XSS, CSRF, man-in-the-middle, replay. Principles: least privilege, defense in depth, never roll your own crypto.
Parallel computing and graphics
\(\text{Speedup}=\dfrac{T_{1}}{T_{p}},\quad \text{efficiency}=\dfrac{S}{p},\quad \text{Amdahl: }S\le\dfrac1{(1-f)+f/p},\quad \text{Gustafson: }S=p-\alpha(p-1)\)
\(\text{Homogeneous transform: }\begin{pmatrix}R&t\\0&1\end{pmatrix};\quad R_{z}(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}\)
\(\text{Phong: }I=k_{a}+k_{d}(n\cdot l)+k_{s}(r\cdot v)^{\alpha};\quad \text{ray-sphere: solve }|o+td-c|^{2}=r^{2}\ (\text{quadratic in }t)\)
\(\text{Bezier cubic: }B(t)=(1-t)^{3}P_{0}+3(1-t)^{2}tP_{1}+3(1-t)t^{2}P_{2}+t^{3}P_{3}\)
More math for upper-division courses
Real analysis
\(\lim_{x\to a}f=L\iff\forall\varepsilon>0\ \exists\delta>0:\ 0<|x-a|<\delta\Rightarrow|f(x)-L|<\varepsilon\)
Sequence converges if Cauchy; monotone bounded sequences converge. Compact sets in R^n are closed and bounded (Heine-Borel). Continuous image of a compact set is compact.
Series tests: divergence, comparison, limit comparison, ratio (L<1 converges), root, integral, alternating series (terms decrease to 0). Uniform convergence preserves continuity and allows swapping limit and integral.
Mean value theorem: f(b)-f(a) = f'(c)(b-a). Fundamental theorem of calculus. Inverse and implicit function theorems need an invertible Jacobian.
Abstract algebra
Group axioms: closure, associativity, identity, inverses. Lagrange: the order of a subgroup divides the order of the group. Cyclic groups are abelian; every subgroup of a cyclic group is cyclic.
\(|G|=|H|[G:H],\quad |\operatorname{Sym}(n)|=n!,\quad |\mathbb Z_{n}^{\times}|=\varphi(n),\quad G/\ker\phi\cong\operatorname{im}\phi\ \text{(first isomorphism theorem)}\)
Rings: abelian under addition, associative multiplication, distributive. Field: commutative ring where every nonzero element is invertible. Z_p is a field iff p is prime. Ideals are kernels of ring homomorphisms.
Probability and stochastic processes
\(\text{Markov chain: }\pi=\pi P,\ \sum\pi_{i}=1;\quad P^{(n)}=P^{n};\quad \text{expected hitting times solve a linear system}\)
\(\text{Poisson process: }P(N_{t}=k)=\dfrac{(\lambda t)^{k}e^{-\lambda t}}{k!},\quad \text{interarrival}\sim\text{Exp}(\lambda),\ E=\dfrac1\lambda\)
\(\operatorname{Var}(aX+bY)=a^{2}\operatorname{Var}X+b^{2}\operatorname{Var}Y+2ab\operatorname{Cov}(X,Y),\quad \operatorname{Cov}=E[XY]-E[X]E[Y]\)
\(\text{CLT: }\dfrac{\bar X-\mu}{\sigma/\sqrt n}\to N(0,1);\quad \text{MLE: maximize }\ell(\theta)=\sum\ln f(x_{i};\theta)\)
\(\text{Normal: }E=\mu,\ \operatorname{Var}=\sigma^{2};\ \text{exponential: }\operatorname{Var}=1/\lambda^{2};\ \text{binomial: }\operatorname{Var}=np(1-p);\ \text{Poisson: }E=\operatorname{Var}=\lambda\)
PDEs: heat, wave, Laplace
\(\text{Heat: }u_{t}=\alpha u_{xx};\quad \text{wave: }u_{tt}=c^{2}u_{xx};\quad \text{Laplace: }u_{xx}+u_{yy}=0\)
\(\text{Separation: }u=X(x)T(t)\Rightarrow X''=-\lambda X;\quad X_{n}=\sin\dfrac{n\pi x}{L}\ (\text{Dirichlet}),\ \lambda_{n}=\left(\dfrac{n\pi}{L}\right)^{2}\)
\(\text{Heat solution: }u=\sum b_{n}e^{-\alpha\lambda_{n}t}\sin\dfrac{n\pi x}{L};\quad \text{d'Alembert: }u=\tfrac12[f(x+ct)+f(x-ct)]+\dfrac1{2c}\int_{x-ct}^{x+ct}g\)
Classify second-order PDEs by B^2-4AC: negative elliptic, zero parabolic, positive hyperbolic. Check boundary conditions: Dirichlet (value), Neumann (derivative), Robin (mixed).
Numerical analysis and optimization
\(\text{Condition number }\kappa(A)=\|A\|\|A^{-1}\|;\quad \text{relative error in }x\lesssim\kappa\cdot\text{relative error in }b\)
Error orders: forward Euler O(h) global, trapezoid O(h^2), Simpson O(h^4), RK4 O(h^4). Stability of explicit methods limits the step size. Interpolation of degree n through n+1 points is unique.
\(\text{KKT (min }f\text{ s.t. }g\le0\text{): }\nabla f+\sum\mu_{i}\nabla g_{i}=0,\ \mu_{i}\ge0,\ \mu_{i}g_{i}=0;\quad \text{convex: local min}=\text{global min}\)
Simplex moves along vertices of the feasible polytope; strong duality: primal and dual optimal values are equal. Gradient descent step size must be small enough; Newton converges quadratically near a minimum.
More physics for upper-division courses
Nuclear and particle physics
\(N=N_{0}e^{-\lambda t},\quad t_{1/2}=\dfrac{\ln2}{\lambda},\quad A=\lambda N,\quad B=\left[Zm_{p}+Nm_{n}-m\right]c^{2}\)
Alpha decay: Z-2, A-4. Beta-minus: n to p, Z+1. Beta-plus/EC: Z-1. Gamma: no change in Z or A. Fission of U-235 releases about 200 MeV; binding energy per nucleon peaks near Fe-56.
Standard model: 6 quarks (u d c s t b), 6 leptons (e, mu, tau and neutrinos), gauge bosons (photon, W, Z, gluon), Higgs. Conserved: charge, baryon number, lepton number, energy-momentum.
Astrophysics and cosmology
\(L=4\pi R^{2}\sigma T^{4},\quad R_{s}=\dfrac{2GM}{c^{2}},\quad v=H_{0}d,\quad H_{0}\approx70\ \text{km/s/Mpc},\quad 1+z=\dfrac{\lambda_{\text{obs}}}{\lambda_{\text{emit}}}\)
\(m-M=5\log_{10}\dfrac d{10\ \text{pc}},\quad \text{Chandrasekhar limit}\approx1.4M_{\odot},\quad \text{Friedmann: }H^{2}=\dfrac{8\pi G}{3}\rho-\dfrac k{a^{2}}\)
Stellar evolution: main sequence, red giant, then white dwarf (low mass) or supernova to neutron star or black hole (high mass). CMB temperature is 2.725 K.
Condensed matter, semiconductors, plasma
\(E_{F}=\dfrac{\hbar^{2}}{2m}(3\pi^{2}n)^{2/3},\quad f(E)=\dfrac1{e^{(E-E_{F})/k_{B}T}+1},\quad n_{i}\propto e^{-E_{g}/2k_{B}T}\)
\(\text{Bragg/Laue: }2d\sin\theta=n\lambda;\quad \text{Debye: }C_{V}\propto T^{3}\ (\text{low }T);\quad \text{Hall: }V_{H}=\dfrac{IB}{nqt}\)
\(\text{Plasma frequency }\omega_{p}=\sqrt{\dfrac{ne^{2}}{\varepsilon_{0}m_{e}}},\quad \text{Debye length }\lambda_{D}=\sqrt{\dfrac{\varepsilon_{0}k_{B}T}{ne^{2}}},\quad \text{gyroradius }r=\dfrac{mv_{\perp}}{qB}\)
More chemistry for upper-division courses
Inorganic and coordination chemistry
Crystal field theory: octahedral splitting gives t2g (lower) and eg (higher), splitting Δo. Strong-field ligands (CN-, CO) give low spin; weak-field (I-, Br-) high spin. Spectrochemical series: I- < Br- < Cl- < F- < OH- < H2O < NH3 < en < CN- < CO.
18-electron rule: metal d electrons + ligand electrons = 18 for stable organometallic complexes. Hard and soft acids and bases: hard-hard and soft-soft prefer to bind.
\(\text{Bond order}=\tfrac12(n_{b}-n_{a});\quad \text{CFSE (octahedral)}=-0.4\Delta_{o}\,n_{t_{2g}}+0.6\Delta_{o}\,n_{e_{g}}\)
Polymers
\(M_{n}=\dfrac{\sum N_{i}M_{i}}{\sum N_{i}},\quad M_{w}=\dfrac{\sum N_{i}M_{i}^{2}}{\sum N_{i}M_{i}},\quad \text{PDI}=\dfrac{M_{w}}{M_{n}}\ge1\)
\(\text{Carothers: }X_{n}=\dfrac1{1-p}\ \text{(step growth, conversion }p)\)
Chain growth: initiation, propagation, termination. Step growth needs high conversion for high molecular weight. Tg: glassy below, rubbery above. Thermoplastics melt; thermosets are crosslinked.
Mass spectrometry, chromatography, spectroscopy
GC separates volatile compounds; HPLC for non-volatile. Ionization: EI (hard, fragments), ESI (soft, multiple charges), MALDI. Analyzers: quadrupole, TOF, ion trap, Orbitrap.
\(\dfrac mz\ \text{(mass-to-charge)};\quad \text{ESI charge states: }m/z=\dfrac{M+nH}{n};\quad \text{resolving power}=\dfrac m{\Delta m}\)
\(\text{Fluorescence: quantum yield }\Phi=\dfrac{\text{photons emitted}}{\text{photons absorbed}};\quad \text{selection rules: IR needs a changing dipole, Raman a changing polarizability}\)
Environmental and atmospheric chemistry
Ozone: O2 + hv gives 2 O; O + O2 gives O3. Catalytic destruction: Cl + O3 gives ClO + O2; ClO + O gives Cl + O2. Tropospheric ozone and smog form from NOx + VOCs + sunlight.
\(\text{Greenhouse forcing }\Delta F\approx5.35\ln\dfrac{C}{C_{0}}\ \text{W/m}^{2}\ (\text{CO}_{2});\quad \text{acid rain: pH}<5.6\ \text{(natural rain)}\)
\(\text{Henry's law: }[\text{gas}]=k_{H}P;\quad \text{alkalinity}\approx[\text{HCO}_{3}^{-}]+2[\text{CO}_{3}^{2-}]+[\text{OH}^{-}]-[\text{H}^{+}]\)
Medicinal chemistry and pharmacology
Lipinski rule of five (oral drugs): MW below 500, logP below 5, H-bond donors at most 5, acceptors at most 10.
\(\text{Therapeutic index}=\dfrac{TD_{50}}{ED_{50}},\quad \text{Hill: }\theta=\dfrac{[L]^{n}}{K_{d}^{n}+[L]^{n}},\quad K_{i}=\dfrac{IC_{50}}{1+[S]/K_{m}}\ \text{(Cheng-Prusoff)}\)
\(\text{First-order elimination: }C=C_{0}e^{-kt},\ t_{1/2}=\dfrac{0.693}{k},\ \text{clearance}=kV_{d},\ \text{bioavailability }F=\dfrac{AUC_{po}}{AUC_{iv}}\)
Metabolism and nuclear chemistry
Glycolysis: glucose to 2 pyruvate, net 2 ATP + 2 NADH. Pyruvate to acetyl-CoA; citric acid cycle per glucose: 6 NADH, 2 FADH2, 2 ATP (GTP). Oxidative phosphorylation gives about 30-32 ATP per glucose in total.
\(\text{Nuclear stability: neutron/proton ratio near 1 for light elements, rises to about 1.5; magic numbers 2, 8, 20, 28, 50, 82, 126};\quad E=\Delta m\,c^{2}\)
\(\text{Dating: }t=\dfrac{t_{1/2}}{\ln2}\ln\dfrac{N_{0}}{N}\)
More engineering for upper-division courses
Communications and information theory
\(C=B\log_{2}\left(1+\dfrac SN\right),\quad H=-\sum p_{i}\log_{2}p_{i},\quad I(X;Y)=H(X)-H(X|Y)\)
\(\text{Q-function: }P_{e,\text{BPSK}}=Q\!\left(\sqrt{\dfrac{2E_{b}}{N_{0}}}\right),\quad \text{AM: }s(t)=[1+m\,x(t)]\cos\omega_{c}t,\quad \text{FM bandwidth (Carson)}\approx2(\Delta f+f_{m})\)
\(\text{Hamming distance }d_{\min}:\ \text{detects }d_{\min}-1\text{ errors, corrects }\left\lfloor\dfrac{d_{\min}-1}2\right\rfloor;\quad \text{code rate }R=\dfrac kn\)
Digital signal processing
\(X(e^{j\omega})=\sum x[n]e^{-j\omega n},\quad \text{FFT: }O(N\log N)\ \text{vs DFT }O(N^{2}),\quad \Delta f=\dfrac{f_{s}}{N}\)
\(\text{FIR: }y[n]=\sum_{k}b_{k}x[n-k]\ (\text{always stable, linear phase possible});\quad \text{IIR: feedback, poles inside unit circle}\)
\(\text{Aliasing if }f_{s}<2f_{\max};\quad \text{quantization noise}\ \sigma^{2}=\dfrac{\Delta^{2}}{12};\quad \text{upsample by }L:\ \text{insert }L-1\text{ zeros, then low-pass}\)
Electromagnetics: transmission lines and waves
\(\Gamma=\dfrac{Z_{L}-Z_{0}}{Z_{L}+Z_{0}},\quad \text{VSWR}=\dfrac{1+|\Gamma|}{1-|\Gamma|},\quad Z_{in}=Z_{0}\dfrac{Z_{L}+jZ_{0}\tan\beta\ell}{Z_{0}+jZ_{L}\tan\beta\ell}\)
\(v_{p}=\dfrac1{\sqrt{\mu\varepsilon}},\quad \lambda=\dfrac{v_{p}}f,\quad \eta=\sqrt{\dfrac\mu\varepsilon}\ (\eta_{0}\approx377\ \Omega),\quad \delta=\sqrt{\dfrac{2}{\omega\mu\sigma}}\ \text{(skin depth)}\)
\(\text{Quarter-wave transformer: }Z_{T}=\sqrt{Z_{0}Z_{L}};\quad \text{rectangular waveguide cutoff: }f_{c}=\dfrac c2\sqrt{\left(\dfrac ma\right)^{2}+\left(\dfrac nb\right)^{2}}\)
\(\text{Friis: }P_{r}=P_{t}G_{t}G_{r}\left(\dfrac\lambda{4\pi R}\right)^{2},\quad \text{dipole directivity}\approx1.64\)
Semiconductor devices
\(n_{i}^{2}=np,\quad V_{bi}=V_{T}\ln\dfrac{N_{A}N_{D}}{n_{i}^{2}},\quad V_{T}=\dfrac{kT}q\approx26\ \text{mV at 300 K}\)
\(I_{D}=I_{S}\left(e^{V_{D}/nV_{T}}-1\right)\ \text{(Shockley)},\quad J_{\text{drift}}=q\mu nE,\quad J_{\text{diff}}=qD\dfrac{dn}{dx},\quad D=\mu V_{T}\ \text{(Einstein)}\)
\(W=\sqrt{\dfrac{2\varepsilon(V_{bi}-V)}q\left(\dfrac1{N_{A}}+\dfrac1{N_{D}}\right)},\quad C_{ox}=\dfrac{\varepsilon_{ox}}{t_{ox}},\quad g_{m}=\dfrac{\partial I_{D}}{\partial V_{GS}}\)
Power systems and machines
\(\text{Per-unit: }X_{pu}=\dfrac{X_{\text{actual}}}{X_{\text{base}}},\quad Z_{\text{base}}=\dfrac{V_{\text{base}}^{2}}{S_{\text{base}}};\quad \text{3-phase }V_{L}=\sqrt3V_{\phi}\ (\text{Y})\)
\(n_{s}=\dfrac{120f}{p}\ \text{(rpm)},\quad s=\dfrac{n_{s}-n}{n_{s}},\quad P_{\text{mech}}=\tau\omega,\quad E_{a}=K\phi\omega\ (\text{DC machine}),\quad V_{t}=E_{a}+I_{a}R_{a}\)
\(\text{Transformer: }\dfrac{V_{1}}{V_{2}}=\dfrac{N_{1}}{N_{2}},\quad \text{efficiency}=\dfrac{P_{\text{out}}}{P_{\text{out}}+P_{\text{core}}+P_{\text{cu}}},\quad \text{voltage regulation}=\dfrac{V_{nl}-V_{fl}}{V_{fl}}\)
\(S=P+jQ,\quad P=|V||I|\cos\phi,\quad Q=|V||I|\sin\phi,\quad \text{power-flow: }P=\dfrac{|V_{1}||V_{2}|}{X}\sin\delta\)
Power electronics
\(\text{Buck: }V_{o}=DV_{in};\quad \text{boost: }V_{o}=\dfrac{V_{in}}{1-D};\quad \text{buck-boost: }V_{o}=-\dfrac{D}{1-D}V_{in}\)
\(\Delta i_{L}=\dfrac{(V_{in}-V_{o})DT}{L}\ (\text{buck}),\quad \Delta v_{o}=\dfrac{\Delta i_{L}}{8fC}\ (\text{buck ripple}),\quad \text{CCM/DCM boundary at }I_{L}=\tfrac12\Delta i_{L}\)
\(\text{Rectifier (full-wave, resistive): }V_{dc}=\dfrac{2V_{m}}\pi;\quad \text{half-wave: }V_{dc}=\dfrac{V_{m}}\pi;\quad \text{inverter: }V_{1,\text{peak}}=m_{a}\dfrac{V_{dc}}2\)
Embedded systems, sensors and robotics
\(\text{PWM duty }D=\dfrac{t_{on}}{T};\ \text{average voltage}=DV;\quad \text{ADC: }\text{code}=\dfrac{V_{in}}{V_{\text{ref}}}(2^{n}-1)\)
\(\text{Divider: }V_{o}=V_{in}\dfrac{R_{2}}{R_{1}+R_{2}};\quad \text{debounce, pull-up resistors, interrupts vs polling}\)
\(\text{Differential drive: }v=\dfrac{v_{R}+v_{L}}2,\ \omega=\dfrac{v_{R}-v_{L}}{L};\quad \text{2-link arm: }x=l_{1}\cos\theta_{1}+l_{2}\cos(\theta_{1}+\theta_{2})\)
Planning: A* uses f = g + h with an admissible heuristic. Kalman filter: predict then update; gain K = P H^T (H P H^T + R)^-1. Measurement uncertainty combines in quadrature; average N samples to cut noise by sqrt(N).
Aerodynamics
\(L=\tfrac12\rho V^{2}SC_{L},\quad D=\tfrac12\rho V^{2}SC_{D},\quad C_{D}=C_{D0}+\dfrac{C_{L}^{2}}{\pi eAR},\quad AR=\dfrac{b^{2}}S\)
\(\text{Thin airfoil: }C_{l}=2\pi(\alpha-\alpha_{L=0});\quad \text{finite wing: }C_{L}=\dfrac{a_{0}\,\alpha}{1+a_{0}/(\pi eAR)};\quad \text{Prandtl-Glauert: }C_{p}=\dfrac{C_{p0}}{\sqrt{1-Ma^{2}}}\)
\(\text{Normal shock (ideal gas): }Ma_{2}^{2}=\dfrac{1+\frac{k-1}2Ma_{1}^{2}}{kMa_{1}^{2}-\frac{k-1}2},\quad \dfrac{p_{2}}{p_{1}}=1+\dfrac{2k}{k+1}(Ma_{1}^{2}-1)\)
\(\text{Mach angle }\mu=\sin^{-1}\dfrac1{Ma},\quad q_{\infty}=\tfrac12\rho V^{2},\quad \text{stall at }C_{L,\max}\)
Propulsion and rocketry
\(F=\dot m_{e}v_{e}+(p_{e}-p_{a})A_{e},\quad I_{sp}=\dfrac F{\dot m g_{0}},\quad \Delta v=I_{sp}g_{0}\ln\dfrac{m_{0}}{m_{f}}\ \text{(Tsiolkovsky)}\)
\(\text{Jet: }\eta_{p}=\dfrac{2}{1+v_{e}/v_{0}},\quad \eta_{\text{th}}=\dfrac{\Delta KE}{\dot m_{f}Q_{R}},\quad c^{*}=\dfrac{p_{c}A_{t}}{\dot m},\quad C_{F}=\dfrac F{p_{c}A_{t}}\)
\(\text{Choked flow: }\dot m=\dfrac{p_{0}A^{*}}{\sqrt{T_{0}}}\sqrt{\dfrac k R}\left(\dfrac2{k+1}\right)^{\frac{k+1}{2(k-1)}}\)
Orbital mechanics and flight dynamics
\(v_{c}=\sqrt{\dfrac\mu r},\quad T=2\pi\sqrt{\dfrac{a^{3}}\mu},\quad \varepsilon=-\dfrac\mu{2a},\quad \mu_{\oplus}=3.986\times10^{5}\ \text{km}^{3}/\text{s}^{2}\)
\(\text{Hohmann: }\Delta v_{1}=\sqrt{\dfrac\mu{r_{1}}}\left(\sqrt{\dfrac{2r_{2}}{r_{1}+r_{2}}}-1\right),\quad \Delta v_{2}=\sqrt{\dfrac\mu{r_{2}}}\left(1-\sqrt{\dfrac{2r_{1}}{r_{1}+r_{2}}}\right)\)
\(r=\dfrac{p}{1+e\cos\nu},\quad p=a(1-e^{2}),\quad r_{p}=a(1-e),\quad r_{a}=a(1+e),\quad \Delta v_{\text{plane}}=2v\sin\dfrac{\Delta i}2\)
Aircraft modes: short period (fast pitch oscillation), phugoid (slow altitude/speed exchange), Dutch roll (yaw-roll coupled), spiral, roll subsidence. Static stability needs a negative pitch-moment slope Cm_alpha.
Machine design
\(\text{Goodman: }\dfrac{\sigma_{a}}{S_{e}}+\dfrac{\sigma_{m}}{S_{ut}}=\dfrac1n,\quad S_{e}=k_{a}k_{b}k_{c}k_{d}k_{e}S_{e}',\quad S_{e}'\approx0.5S_{ut}\)
\(\text{Gears: }m=\dfrac dN,\quad \text{ratio}=\dfrac{N_{2}}{N_{1}}=\dfrac{\omega_{1}}{\omega_{2}},\quad \text{spur }W_{t}=\dfrac{2T}{d},\quad \text{center distance}=\dfrac{d_{1}+d_{2}}2\)
\(\text{Bearing life: }L_{10}=\left(\dfrac CP\right)^{a}\times10^{6}\ \text{rev}\ (a=3\text{ ball},\ 10/3\text{ roller});\quad \text{bolt preload}\approx0.75\text{-}0.9\,S_{p}A_{t}\)
\(\text{Shaft: }d=\left(\dfrac{32n}\pi\sqrt{\left(\dfrac{K_{f}M_{a}}{S_{e}}\right)^{2}+\tfrac34\left(\dfrac{T_{m}}{S_{y}}\right)^{2}}\right)^{1/3}\ \text{(simplified)}\)
Fatigue, fracture and mechanical behavior
\(\dfrac{da}{dN}=C(\Delta K)^{m},\quad \Delta K=Y\Delta\sigma\sqrt{\pi a},\quad N_{f}=\int_{a_{i}}^{a_{c}}\dfrac{da}{C(\Delta K)^{m}}\ \text{(Paris law)}\)
\(\text{Miner: }\sum\dfrac{n_{i}}{N_{i}}=1;\quad \text{Basquin: }\sigma_{a}=\sigma_{f}'(2N)^{b};\quad a_{c}=\dfrac1\pi\left(\dfrac{K_{Ic}}{Y\sigma}\right)^{2}\)
\(\tau_{\text{CRSS}}=\sigma\cos\phi\cos\lambda\ \text{(Schmid)},\quad \Delta\sigma_{y}=\dfrac{Gb}{L}\ \text{(Orowan)},\quad \dot\varepsilon=A\sigma^{n}e^{-Q/RT}\ \text{(creep)}\)
\(\sigma_{T}=\sigma(1+\varepsilon),\quad \varepsilon_{T}=\ln(1+\varepsilon),\quad \sigma_{T}=K\varepsilon_{T}^{n},\quad K_{t}=\dfrac{\sigma_{\max}}{\sigma_{\text{nom}}}\)
Finite element method and vibrations
\([K]\{u\}=\{F\},\quad K^{e}=\int B^{T}DB\,dV,\quad \text{bar element: }K=\dfrac{AE}{L}\begin{pmatrix}1&-1\\-1&1\end{pmatrix}\)
\(\text{Beam element: }K=\dfrac{EI}{L^{3}}\begin{pmatrix}12&6L&-12&6L\\6L&4L^{2}&-6L&2L^{2}\\-12&-6L&12&-6L\\6L&2L^{2}&-6L&4L^{2}\end{pmatrix}\)
\([M]\{\ddot u\}+[C]\{\dot u\}+[K]\{u\}=\{F\},\quad \det(K-\omega^{2}M)=0\ \text{(natural frequencies)},\quad \text{forced: }X=\dfrac{F_{0}/k}{\sqrt{(1-r^{2})^{2}+(2\zeta r)^{2}}},\ r=\dfrac\omega{\omega_{n}}\)
Composites and aerospace structures
\(E_{c}=V_{f}E_{f}+V_{m}E_{m}\ \text{(Voigt, parallel)},\quad \dfrac1{E_{c}}=\dfrac{V_{f}}{E_{f}}+\dfrac{V_{m}}{E_{m}}\ \text{(Reuss, series)}\)
\(\text{Thin-wall torsion (closed): }q=\dfrac T{2A_{m}},\quad \tau=\dfrac q t,\quad \text{shear flow in beams: }q=\dfrac{VQ}I\)
\(\text{Plate buckling: }\sigma_{cr}=k\dfrac{\pi^{2}E}{12(1-\nu^{2})}\left(\dfrac tb\right)^{2};\quad \text{load factor }n=\dfrac Lw;\quad \text{margin of safety}=\dfrac{\text{allowable}}{\text{applied}}-1\)
Structural design: steel, concrete, timber, seismic
LRFD: factored load = 1.2D + 1.6L (basic combination); design strength phi*Rn must be at least the required strength. Steel: phi = 0.9 for yielding in tension/flexure, 0.75 for fracture, 0.9 for compression, 0.9 for shear (typical).
\(\text{Steel: }\phi P_{n}=\phi F_{y}A_{g};\quad \phi M_{n}=\phi F_{y}Z_{x};\quad \text{column: }F_{cr}=0.658^{F_{y}/F_{e}}F_{y}\ (F_{e}\ge0.44F_{y}),\ F_{e}=\dfrac{\pi^{2}E}{(KL/r)^{2}}\)
\(\text{RC beam: }a=\dfrac{A_{s}f_{y}}{0.85f_{c}'b},\quad M_{n}=A_{s}f_{y}\left(d-\dfrac a2\right),\quad \phi=0.9\ (\text{tension-controlled}),\quad \rho_{\min}=\dfrac{3\sqrt{f_{c}'}}{f_{y}}\ (\text{psi units})\)
\(\text{Seismic: }V=C_{s}W,\quad C_{s}=\dfrac{S_{DS}}{R/I_{e}},\quad T_{a}=C_{t}h_{n}^{x};\quad \text{response spectrum: }S_{a}\text{ vs }T\)
Timber: allowable stress design with adjustment factors (load duration, moisture, size, repetitive member). Prestressing counters tension with compression; losses include elastic shortening, creep, shrinkage, relaxation, friction.
Geotechnical engineering
\(e=\dfrac{V_{v}}{V_{s}},\quad n=\dfrac{e}{1+e},\quad S\,e=wG_{s},\quad \gamma=\dfrac{(G_{s}+Se)\gamma_{w}}{1+e}\)
\(\sigma'=\sigma-u,\quad \tau=c'+\sigma'\tan\phi',\quad K_{a}=\tan^{2}\left(45^{\circ}-\dfrac\phi2\right),\quad K_{p}=\tan^{2}\left(45^{\circ}+\dfrac\phi2\right)\)
\(\text{Terzaghi bearing: }q_{u}=cN_{c}+qN_{q}+\tfrac12\gamma BN_{\gamma};\quad \text{consolidation: }S=\dfrac{C_{c}H}{1+e_{0}}\log\dfrac{\sigma_{f}'}{\sigma_{0}'},\quad t=\dfrac{T_{v}H_{dr}^{2}}{c_{v}}\)
\(\text{Darcy: }v=ki,\quad Q=kiA,\quad i=\dfrac{\Delta h}L;\quad \text{Atterberg: PI}=\text{LL}-\text{PL}\)
Renewable and environmental energy
\(P_{\text{wind}}=\tfrac12\rho Av^{3}C_{p},\quad C_{p,\max}=\dfrac{16}{27}\approx0.593\ \text{(Betz)},\quad \text{TSR}=\dfrac{\omega R}{v}\)
\(\text{PV: }P=VI,\quad \text{fill factor FF}=\dfrac{P_{\max}}{V_{oc}I_{sc}},\quad \eta=\dfrac{P_{\max}}{GA},\quad I=I_{L}-I_{0}\left(e^{V/nV_{T}}-1\right),\quad G\approx1000\ \text{W/m}^{2}\ \text{(STC)}\)
\(\text{Capacity factor}=\dfrac{E_{\text{actual}}}{P_{\text{rated}}\times8760},\quad \text{LCOE}=\dfrac{\sum(I_{t}+M_{t})/(1+r)^{t}}{\sum E_{t}/(1+r)^{t}}\)
\(\text{Gaussian plume: }C=\dfrac Q{2\pi u\sigma_{y}\sigma_{z}}e^{-y^{2}/2\sigma_{y}^{2}}\left[e^{-(z-H)^{2}/2\sigma_{z}^{2}}+e^{-(z+H)^{2}/2\sigma_{z}^{2}}\right]\)
Bioengineering: transport and physiology
\(J=-D\dfrac{dC}{dx},\quad \dfrac{\partial C}{\partial t}=D\dfrac{\partial^{2}C}{\partial x^{2}}+R,\quad D=\dfrac{k_{B}T}{6\pi\eta r}\ \text{(Stokes-Einstein)},\quad \langle x^{2}\rangle=2Dt\)
\(E_{\text{ion}}=\dfrac{RT}{zF}\ln\dfrac{C_{o}}{C_{i}},\quad V_{m}=\dfrac{RT}{F}\ln\dfrac{P_{K}K_{o}+P_{Na}Na_{o}+P_{Cl}Cl_{i}}{P_{K}K_{i}+P_{Na}Na_{i}+P_{Cl}Cl_{o}}\ \text{(GHK)}\)
\(CO=HR\times SV,\quad MAP\approx DBP+\tfrac13(SBP-DBP),\quad Q=\dfrac{\Delta P\pi r^{4}}{8\mu L},\quad R=\dfrac{8\mu L}{\pi r^{4}}\)
\(\text{Starling: }J_{v}=K_{f}\left[(P_{c}-P_{i})-\sigma(\pi_{c}-\pi_{i})\right];\quad \tau_{m}=R_{m}C_{m};\quad J=P\Delta C\ \text{(membrane permeability)}\)
\(\text{Stress-strain for tissue: }\sigma=E\varepsilon\ (\text{small}),\quad \text{Kelvin-Voigt: }\sigma=E\varepsilon+\eta\dot\varepsilon,\quad \text{Maxwell: }\dot\varepsilon=\dfrac{\dot\sigma}{E}+\dfrac\sigma\eta\)
Bioreactors and bioprocesses
\(\mu=\dfrac{\mu_{\max}S}{K_{s}+S}\ \text{(Monod)},\quad \dfrac{dX}{dt}=\mu X,\quad X=X_{0}e^{\mu t},\quad t_{d}=\dfrac{\ln2}{\mu}\)
\(Y_{X/S}=\dfrac{\Delta X}{-\Delta S},\quad \text{chemostat steady state: }\mu=D=\dfrac FV,\quad S=\dfrac{K_{s}D}{\mu_{\max}-D},\quad X=Y(S_{0}-S)\)
\(k_{L}a:\ \dfrac{dC}{dt}=k_{L}a(C^{*}-C)-q_{O_{2}}X,\quad \text{scale-up by constant }P/V,\ k_{L}a\text{ or tip speed}\)
Nanoengineering
\(\dfrac SV=\dfrac3r\ \text{(sphere)},\quad D=\dfrac{K\lambda}{\beta\cos\theta}\ \text{(Scherrer, }K\approx0.9),\quad \Delta P=\dfrac{2\gamma}r\ \text{(Young-Laplace)}\)
\(E_{n}=\dfrac{n^{2}h^{2}}{8mL^{2}},\quad \text{quantum dot gap: }E\approx E_{g}+\dfrac{\hbar^{2}\pi^{2}}{2\mu R^{2}},\quad \lambda_{D}=\sqrt{\dfrac{\varepsilon k_{B}T}{2N_{A}e^{2}I}}\ \text{(Debye length)}\)
DLVO: colloid stability is van der Waals attraction plus electrostatic repulsion. Gold nanoparticle color depends on size (plasmon resonance). Top-down (lithography, etching) vs bottom-up (self-assembly, CVD, sol-gel).
\(\text{Stoichiometric scaling: surface atoms fraction rises as particle size falls; melting point depression }T_{m}(r)=T_{m,\infty}\left(1-\dfrac{\alpha}{r}\right)\)
Chemical engineering: reactors, separations, process control
\(\text{Batch: }t=N_{A0}\int_{0}^{X}\dfrac{dX}{(-r_{A})V};\quad \text{PFR: }\tau=C_{A0}\int\dfrac{dX}{-r_{A}};\quad \text{CSTR: }\tau=\dfrac{C_{A0}X}{-r_{A}};\quad \text{series CSTRs approach PFR}\)
\(\text{Residence time }\tau=\dfrac VQ;\quad \text{selectivity }S=\dfrac{r_{\text{desired}}}{r_{\text{undesired}}};\quad \text{Thiele modulus }\phi=L\sqrt{\dfrac kD}\)
\(\text{Raoult/relative volatility: }\alpha_{AB}=\dfrac{y_{A}/x_{A}}{y_{B}/x_{B}},\quad y=\dfrac{\alpha x}{1+(\alpha-1)x};\quad \text{Fenske: }N_{\min}=\dfrac{\ln\left[\frac{x_{D}}{1-x_{D}}\frac{1-x_{B}}{x_{B}}\right]}{\ln\alpha}\)
\(\text{McCabe-Thiele: operating lines from material balances; }q\text{-line slope }\dfrac{q}{q-1};\quad \text{absorber (Kremser): }\dfrac{y_{N+1}-y_{1}}{y_{N+1}-mx_{0}}=\dfrac{A^{N+1}-A}{A^{N+1}-1}\)
\(\text{FOPDT: }G(s)=\dfrac{Ke^{-\theta s}}{\tau s+1};\quad \text{Ziegler-Nichols PID from }K_{u},P_{u}:\ K_{c}=0.6K_{u},\ \tau_{I}=P_{u}/2,\ \tau_{D}=P_{u}/8\)
Graduate engineering
Convex and numerical optimization
\(\min f_{0}(x)\ \text{s.t. }f_{i}(x)\le0,\ Ax=b;\quad L(x,\lambda,\nu)=f_{0}+\sum\lambda_{i}f_{i}+\nu^{T}(Ax-b),\quad g(\lambda,\nu)=\inf_{x}L\)
\(\text{KKT: }\nabla f_{0}+\sum\lambda_{i}\nabla f_{i}+A^{T}\nu=0,\ \ f_{i}\le0,\ \ \lambda_{i}\ge0,\ \ \lambda_{i}f_{i}=0;\quad \text{strong duality holds under Slater's condition}\)
\(\text{Gradient descent (convex, }L\text{-smooth): }f(x_{k})-f^{*}\le\dfrac{L\|x_{0}-x^{*}\|^{2}}{2k};\quad \text{strongly convex: linear rate};\quad \text{Newton: }x^{+}=x-[\nabla^{2}f]^{-1}\nabla f\)
\(\text{Proximal gradient: }x^{+}=\operatorname{prox}_{tg}(x-t\nabla f);\quad \text{ADMM alternates updates of }x,\ z\text{ and the dual variable};\quad \text{LP/QP/SOCP/SDP are convex}\)
Estimation and stochastic systems
\(\hat x_{LS}=(A^{T}A)^{-1}A^{T}b,\quad I(\theta)=E\left[\left(\dfrac{\partial\ln p}{\partial\theta}\right)^{2}\right],\quad \operatorname{Var}(\hat\theta)\ge\dfrac1{I(\theta)}\ \text{(Cramer-Rao)}\)
\(\text{Kalman predict: }x^{-}=Ax,\ P^{-}=APA^{T}+Q;\quad \text{update: }K=P^{-}H^{T}(HP^{-}H^{T}+R)^{-1},\ x=x^{-}+K(z-Hx^{-}),\ P=(I-KH)P^{-}\)
\(\text{Likelihood ratio test: }\Lambda(x)=\dfrac{p(x|H_{1})}{p(x|H_{0})}\gtrless\eta;\quad \text{Neyman-Pearson maximizes }P_{D}\text{ for fixed }P_{FA};\quad \text{MAP: }\arg\max p(\theta|x)\)
Hidden Markov models: forward algorithm computes likelihood, Viterbi finds the best state path, Baum-Welch (EM) learns parameters. Wiener filter minimizes mean square error for stationary signals.
Advanced control and systems theory
\(\dot x=Ax+Bu,\ y=Cx+Du;\quad \text{controllable iff }\operatorname{rank}[B\ AB\cdots A^{n-1}B]=n;\quad \text{observable iff }\operatorname{rank}\begin{bmatrix}C\\CA\\\vdots\\CA^{n-1}\end{bmatrix}=n\)
\(\text{Pole placement: }u=-Kx,\ \text{eig}(A-BK)\text{ chosen};\quad \text{LQR: }J=\int(x^{T}Qx+u^{T}Ru)\,dt,\ K=R^{-1}B^{T}P,\ A^{T}P+PA-PBR^{-1}B^{T}P+Q=0\)
\(\text{Lyapunov: }V>0,\ \dot V<0\Rightarrow\text{stable};\quad \text{linear: }A^{T}P+PA=-Q,\ P>0\iff A\text{ Hurwitz}\)
\(S=\dfrac1{1+L},\quad T=\dfrac L{1+L},\quad S+T=1;\quad \text{Nyquist: }Z=N+P;\quad \|G\|_{\infty}=\sup_{\omega}\bar\sigma(G(j\omega))\)
Pontryagin's principle: optimal u minimizes the Hamiltonian H = L + lambda^T f with costate dynamics. Dynamic programming: Hamilton-Jacobi-Bellman equation. Small-gain theorem: loop is stable if product of gains is less than 1. Passivity: V-dot is at most u^T y.
Continuum, elasticity and fracture (graduate)
\(\varepsilon_{ij}=\tfrac12(u_{i,j}+u_{j,i}),\quad \sigma_{ij,j}+b_{i}=0,\quad \sigma_{ij}=\lambda\varepsilon_{kk}\delta_{ij}+2\mu\varepsilon_{ij},\quad \lambda=\dfrac{E\nu}{(1+\nu)(1-2\nu)},\ \mu=G\)
\(I_{1}=\sigma_{kk},\quad J_{2}=\tfrac12s_{ij}s_{ij},\quad \sigma_{vm}=\sqrt{3J_{2}},\quad \text{Tresca: }\tau_{\max}=\tfrac12(\sigma_{1}-\sigma_{3})=\tfrac{\sigma_{y}}2,\quad d\varepsilon^{p}_{ij}=d\lambda\dfrac{\partial f}{\partial\sigma_{ij}}\)
\(K_{I}=Y\sigma\sqrt{\pi a},\quad G=\dfrac{K_{I}^{2}}{E'},\quad \sigma_{f}=\sqrt{\dfrac{2E\gamma}{\pi a}}\ \text{(Griffith)},\quad \sigma_{ij}=\dfrac{K}{\sqrt{2\pi r}}f_{ij}(\theta),\quad E'=E\ (\text{plane stress}),\ \tfrac E{1-\nu^{2}}\ (\text{plane strain})\)
\(W_{\text{neo-Hookean}}=\tfrac\mu2(I_{1}-3),\quad F=\dfrac{\partial x}{\partial X},\quad \lambda_{1}\lambda_{2}\lambda_{3}=1\ \text{(incompressible)},\quad c_{L}=\sqrt{\dfrac{\lambda+2\mu}\rho},\ c_{S}=\sqrt{\dfrac\mu\rho}\)
Compressible flow, combustion and plasma (graduate)
\(\dfrac{dA}{A}=(Ma^{2}-1)\dfrac{dV}V,\quad \dfrac{A}{A^{*}}=\dfrac1{Ma}\left[\dfrac{2}{k+1}\left(1+\dfrac{k-1}2Ma^{2}\right)\right]^{\frac{k+1}{2(k-1)}},\quad \nu(Ma)=\sqrt{\tfrac{k+1}{k-1}}\tan^{-1}\sqrt{\tfrac{k-1}{k+1}(Ma^{2}-1)}-\tan^{-1}\sqrt{Ma^{2}-1}\)
\(\text{Damkohler }Da=\dfrac{\tau_{\text{flow}}}{\tau_{\text{chem}}},\quad \text{Arrhenius }k=Ae^{-E_{a}/RT},\quad \text{adiabatic flame: }\sum n_{i}h_{i}(T_{ad})=\sum n_{i}h_{i}(T_{0})\)
\(\omega_{c}=\dfrac{|q|B}{m},\quad r_{L}=\dfrac{mv_{\perp}}{|q|B},\quad \mathbf v_{E\times B}=\dfrac{\mathbf E\times\mathbf B}{B^{2}},\quad \omega_{p}=\sqrt{\dfrac{ne^{2}}{\varepsilon_{0}m}},\quad \lambda_{D}=\sqrt{\dfrac{\varepsilon_{0}k_{B}T}{ne^{2}}}\)
\(\text{Ideal MHD: }\dfrac{\partial\mathbf B}{\partial t}=\nabla\times(\mathbf v\times\mathbf B),\quad \beta=\dfrac{nk_{B}T}{B^{2}/2\mu_{0}},\quad \text{Saha: }\dfrac{n_{i}n_{e}}{n_{n}}\propto T^{3/2}e^{-E_{i}/k_{B}T}\)
Heat, mass and multiphase transport (graduate)
\(\rho c\dfrac{\partial T}{\partial t}=\nabla\cdot(k\nabla T)+\dot q''',\quad \dfrac{T-T_{s}}{T_{i}-T_{s}}=\operatorname{erf}\dfrac{x}{2\sqrt{\alpha t}}\ \text{(semi-infinite solid)}\)
\(\dfrac{\delta_{th}}{\delta}\sim Pr^{-1/3},\quad Sh=0.332Re^{1/2}Sc^{1/3}\ \text{(laminar flat plate)},\quad D_{\text{eff}}=D\left(1+\dfrac{Pe^{2}}{48}\right)\ \text{(Taylor dispersion, }Pe=Ua/D)\)
\(q_{12}=\dfrac{\sigma(T_{1}^{4}-T_{2}^{4})}{\dfrac1{\varepsilon_{1}}+\dfrac1{\varepsilon_{2}}-1}\ \text{(parallel plates)},\quad B_{\lambda}=\dfrac{2hc^{2}}{\lambda^{5}(e^{hc/\lambda k_{B}T}-1)},\quad Q_{12}=A_{1}F_{12}\sigma(T_{1}^{4}-T_{2}^{4})\)
\(\text{Boiling: }q''_{\max}=0.131h_{fg}\rho_{v}^{1/2}\left[\sigma g(\rho_{l}-\rho_{v})\right]^{1/4}\ \text{(critical heat flux)},\quad \text{Stokes flow }Re\ll1:\ \mu\nabla^{2}\mathbf u=\nabla p\)
Photonics, optics and EM (graduate)
\(\nabla^{2}\mathbf E=\mu\varepsilon\dfrac{\partial^{2}\mathbf E}{\partial t^{2}},\quad r_{s}=\dfrac{n_{1}\cos\theta_{i}-n_{2}\cos\theta_{t}}{n_{1}\cos\theta_{i}+n_{2}\cos\theta_{t}},\quad R=|r|^{2}\)
\(\text{Gaussian beam: }w(z)=w_{0}\sqrt{1+(z/z_{R})^{2}},\quad z_{R}=\dfrac{\pi w_{0}^{2}}\lambda,\quad \theta=\dfrac\lambda{\pi w_{0}}\)
\(\text{Fiber: }V=\dfrac{2\pi a}\lambda NA,\ \text{single-mode if }V<2.405,\quad NA=\sqrt{n_{1}^{2}-n_{2}^{2}},\quad \text{responsivity }\mathcal R=\dfrac{\eta q\lambda}{hc}\)
\(\text{SHG phase matching: }\Delta k=k_{2\omega}-2k_{\omega}=0;\quad \text{far field}=\text{Fourier transform of aperture field};\quad \text{laser threshold: }g_{th}=\alpha+\dfrac1{2L}\ln\dfrac1{R_{1}R_{2}}\)
Analog, RF and VLSI circuits
\(g_{m}=\dfrac{2I_{D}}{V_{ov}}=\sqrt{2\mu C_{ox}\dfrac WLI_{D}},\quad r_{o}=\dfrac1{\lambda I_{D}},\quad A_{v,\text{CS}}=-g_{m}(r_{o}\parallel R_{D}),\quad A_{\text{intrinsic}}=g_{m}r_{o}\)
\(\overline{i_{n}^{2}}=4k_{B}T\gamma g_{m}\Delta f,\quad \overline{v^{2}}=\dfrac{k_{B}T}C\ (kT/C),\quad C_{M}=C(1+|A_{v}|)\ \text{(Miller)},\quad PM=180^{\circ}+\angle L(j\omega_{u})\)
\(NF=10\log F,\ F=1+\dfrac{T_{e}}{T_{0}},\quad F_{\text{cascade}}=F_{1}+\dfrac{F_{2}-1}{G_{1}}+\cdots,\quad IIP_{3}\approx P_{1dB}+9.6\ \text{dB},\quad PAE=\dfrac{P_{out}-P_{in}}{P_{DC}}\)
\(\text{PLL: }\text{phase noise }\mathcal L(\Delta f)=10\log\dfrac{P_{\text{sideband}}(1\ \text{Hz})}{P_{\text{carrier}}};\quad \text{CMOS dynamic power }P=\alpha CV_{DD}^{2}f;\quad \text{delay}\propto\dfrac{CV_{DD}}{I}\)
Wireless, networks and queueing (graduate)
\(C_{\text{MIMO}}=\log_{2}\det\left(I+\dfrac{SNR}{N_{t}}HH^{H}\right),\quad PL=PL_{0}+10n\log_{10}\dfrac d{d_{0}},\quad P_{r}=P_{t}+G_{t}+G_{r}-PL\ \text{(dB)}\)
\(\dfrac{E_{b}}{N_{0}}=SNR\cdot\dfrac{B}{R_{b}},\quad P_{e,\text{BPSK/QPSK}}=Q\!\left(\sqrt{\tfrac{2E_{b}}{N_{0}}}\right),\quad \text{OFDM: }\Delta f=\dfrac BN,\ \text{cyclic prefix}>\text{delay spread}\)
\(\text{M/M/1: }\rho=\dfrac\lambda\mu,\ L=\dfrac\rho{1-\rho},\ W=\dfrac1{\mu-\lambda};\quad \text{Little: }L=\lambda W;\quad \text{ALOHA: }S=Ge^{-2G}\ (\max\tfrac1{2e}),\ \text{slotted: }S=Ge^{-G}\ (\max\tfrac1e)\)
Spatial point processes (Poisson) model random node locations; stochastic geometry gives coverage probability. Routing: shortest path, distance vector (Bellman-Ford), link state (Dijkstra).
Machine learning theory and deep learning (graduate)
\(P(|\bar X-\mu|\ge\varepsilon)\le2e^{-2n\varepsilon^{2}}\ \text{(Hoeffding)},\quad \text{generalization gap}\lesssim\sqrt{\dfrac{d\ln(m/d)+\ln(1/\delta)}m}\ \text{(VC)}\)
\(\text{SVM: }\min\tfrac12\|w\|^{2}\ \text{s.t. }y_{i}(w\cdot x_{i}+b)\ge1;\quad \text{ridge: }\hat w=(X^{T}X+\lambda I)^{-1}X^{T}y;\quad \text{softmax-CE gradient}=p-y\)
\(\text{ELBO}=E_{q}[\ln p(x|z)]-D_{KL}(q(z|x)\|p(z)),\quad \text{attention}=\operatorname{softmax}\!\left(\dfrac{QK^{T}}{\sqrt{d_{k}}}\right)V,\quad \text{Adam: }m,v\ \text{moving averages of }g,\ g^{2}\)
EM algorithm: E-step computes posterior responsibilities, M-step maximizes expected complete-data likelihood. PCA: top singular vectors of centered data. Bias-variance tradeoff and double descent describe model complexity.
Complexity, algorithms and cryptography (graduate)
\(P\subseteq NP\subseteq PSPACE\subseteq EXP;\quad \text{Savitch: }NSPACE(s)\subseteq DSPACE(s^{2});\quad BPP\subseteq\Sigma_{2}\cap\Pi_{2}\ \text{(PH)}\)
\(\text{Markov: }P(X\ge a)\le\dfrac{E[X]}a;\quad \text{Chebyshev: }P(|X-\mu|\ge k\sigma)\le\dfrac1{k^{2}};\quad \text{Chernoff: }P(X\ge(1+\delta)\mu)\le e^{-\delta^{2}\mu/3}\ (0<\delta\le1)\)
Approximation ratio: algorithm value is within factor alpha of optimal. Dynamic programming needs optimal substructure and overlapping subproblems. Potential method gives amortized cost = actual + change in potential.
Crypto security: IND-CPA (no information from chosen plaintexts), reductions show breaking the scheme solves a hard problem (DDH, LWE, factoring). Zero-knowledge: completeness, soundness, zero-knowledge. Lattices: LLL finds short vectors approximately.
Structural dynamics, earthquake and stability (graduate)
\((K-\omega^{2}M)\phi=0,\quad \phi_{i}^{T}M\phi_{j}=0\ (i\ne j),\quad C=aM+bK,\quad \zeta_{n}=\tfrac12\left(\dfrac a{\omega_{n}}+b\omega_{n}\right),\quad DAF=\dfrac1{\sqrt{(1-r^{2})^{2}+(2\zeta r)^{2}}}\)
\(\text{Newmark: }\gamma=\tfrac12,\beta=\tfrac14\ (\text{average acceleration, unconditionally stable});\quad \text{SRSS: }R=\sqrt{\sum R_{n}^{2}};\quad \text{Duhamel: }u(t)=\dfrac1{m\omega_{d}}\int p(\tau)e^{-\zeta\omega(t-\tau)}\sin\omega_{d}(t-\tau)d\tau\)
\(P_{cr}=\dfrac{\pi^{2}EI}{(KL)^{2}}\ (K=0.5\ \text{fixed-fixed},\ 0.7\ \text{fixed-pinned},\ 1.0\ \text{pinned},\ 2.0\ \text{fixed-free});\quad D=\dfrac{Eh^{3}}{12(1-\nu^{2})}\ \text{(plate)}\)
\(T_{\text{eff}}=2\pi\sqrt{\dfrac W{k_{\text{eff}}g}}\ \text{(isolation)},\quad \beta=\dfrac{\mu_{g}}{\sigma_{g}},\ P_{f}=\Phi(-\beta)\ \text{(reliability)},\quad FS=\dfrac{\tau_{f}}{\tau}\ \text{(slopes)},\quad \tau_{f}=c'+\sigma'\tan\phi'\)
Biomedical engineering (graduate)
\(\omega_{0}=\gamma B_{0}\ (\gamma/2\pi=42.58\ \text{MHz/T for }^{1}\text{H}),\quad M_{z}=M_{0}(1-e^{-t/T_{1}}),\quad M_{xy}=M_{0}e^{-t/T_{2}}\)
\(I=I_{0}e^{-\mu x},\quad HU=1000\dfrac{\mu-\mu_{w}}{\mu_{w}},\quad Z=\rho c,\quad R=\left(\dfrac{Z_{2}-Z_{1}}{Z_{2}+Z_{1}}\right)^{2}\ \text{(ultrasound)}\)
\(C_{m}\dfrac{dV}{dt}=-\sum g_{i}(V-E_{i})+I\ \text{(Hodgkin-Huxley)},\quad \lambda=\sqrt{\dfrac{r_{m}}{r_{i}}}\ \text{(length constant)},\quad \dfrac{d\mathbf x}{dt}=S\mathbf v,\ \text{flux balance: }S\mathbf v=0\)
Regulatory (US): Class I (low risk), II (510(k) clearance), III (PMA approval). Tissue engineering: oxygen diffusion limits constructs to about 200 micrometers without vascularization. One-compartment pharmacokinetics: C = C0 exp(-kt).
Nanoscale and materials science (graduate)
\(G=H-TS,\quad \sum x_{i}d\mu_{i}=0\ (T,P\ \text{const, Gibbs-Duhem}),\quad \dfrac{n_{v}}N=e^{-Q_{v}/k_{B}T},\quad \Delta G^{*}=\dfrac{16\pi\gamma^{3}}{3\Delta G_{v}^{2}},\ r^{*}=\dfrac{2\gamma}{\Delta G_{v}}\)
\(C(x,t)=\dfrac M{\sqrt{\pi Dt}}e^{-x^{2}/4Dt}\ \text{(thin film)},\quad X=1-e^{-kt^{n}}\ \text{(Avrami)},\quad E_{\text{disl}}\approx\alpha Gb^{2}\ \text{per length}\)
\(W_{\text{vdW}}=-\dfrac{A_{H}}{12\pi D^{2}}\ \text{(plates, per area)},\quad i=i_{0}\left[e^{\alpha_{a}F\eta/RT}-e^{-\alpha_{c}F\eta/RT}\right]\ \text{(Butler-Volmer)},\quad i=nFAc\sqrt{\dfrac D{\pi t}}\ \text{(Cottrell)}\)
\(E=\hbar v_{F}|k|\ (v_{F}\approx10^{6}\ \text{m/s, graphene}),\quad R_{g}^{2}=\dfrac{Nb^{2}}6\ \text{(ideal chain)},\quad \text{SQ limit}\approx33\%\ \text{at }E_{g}\approx1.3\ \text{eV}\)
Reaction, catalysis and thermodynamics (graduate)
\(\theta=\dfrac{KP}{1+KP}\ \text{(Langmuir)},\quad \phi=L\sqrt{\dfrac kD},\quad \eta=\dfrac{\tanh\phi}\phi\ \text{(slab, first order)},\quad E(t)_{\text{CSTR}}=\dfrac{e^{-t/\tau}}\tau\)
\(\mu_{i}=\mu_{i}^{\circ}+RT\ln(\gamma_{i}x_{i}),\quad \ln\gamma_{1}=Ax_{2}^{2}\ \text{(Margules)},\quad f_{i}=\phi_{i}y_{i}P,\quad G^{E}=RT\sum x_{i}\ln\gamma_{i}\)
Mechanisms: rate-determining step, pseudo-steady-state on intermediates, Langmuir-Hinshelwood (both adsorbed) vs Eley-Rideal (one adsorbed). Nonideal reactors: dead volume and bypassing show in the residence time distribution.
Beyond STEM: other majors and GEs
Economics: micro and macro
\(E_{d}=\dfrac{\%\Delta Q_{d}}{\%\Delta P},\quad \text{elastic if }|E_{d}|>1;\quad TR=PQ,\quad MR=\dfrac{\Delta TR}{\Delta Q},\quad ATC=\dfrac{TC}{Q}\)
Profit max: produce where MR = MC. Perfect competition: P = MR = MC; long run P = minimum ATC, zero economic profit. Monopoly: MR < P, produces less and charges more, deadweight loss. Consumer surplus is area under demand above price; producer surplus is above supply below price.
\(HHI=\sum s_{i}^{2},\quad \text{Lerner index}=\dfrac{P-MC}{P},\quad \text{GDP}=C+I+G+NX,\quad \text{real GDP}=\dfrac{\text{nominal GDP}}{\text{deflator}/100}\)
\(\text{Inflation}=\dfrac{CPI_{t}-CPI_{t-1}}{CPI_{t-1}},\quad r\approx i-\pi\ \text{(Fisher)},\quad MV=PY\ \text{(quantity theory)},\quad \text{multiplier}=\dfrac1{1-MPC}\)
Game theory: a dominant strategy is best regardless of the opponent; Nash equilibrium means no player gains by changing alone. Prisoner's dilemma: both defect even though both cooperating is better. Comparative advantage: specialize where opportunity cost is lowest.
\(\text{OLS: }\hat\beta=\dfrac{\operatorname{Cov}(X,Y)}{\operatorname{Var}(X)},\quad R^{2}=1-\dfrac{SSR}{SST},\quad t=\dfrac{\hat\beta}{SE(\hat\beta)},\quad \text{omitted variable bias}\ne0\ \text{if it correlates with }X\text{ and }Y\)
Accounting and corporate finance
\(A=L+E,\quad NI=\text{revenue}-\text{expenses},\quad \text{current ratio}=\dfrac{\text{current assets}}{\text{current liabilities}},\quad ROE=\dfrac{NI}{E},\quad \text{gross margin}=\dfrac{\text{revenue}-COGS}{\text{revenue}}\)
\(\text{CAPM: }E[R]=R_{f}+\beta(R_{m}-R_{f}),\quad P_{0}=\dfrac{D_{1}}{r-g}\ \text{(Gordon)},\quad WACC=\dfrac EV r_{E}+\dfrac DV r_{D}(1-T),\quad NPV=\sum\dfrac{C_{t}}{(1+r)^{t}}\)
Debits increase assets and expenses; credits increase liabilities, equity, and revenue. Statements: income statement (period), balance sheet (point in time), cash flow (operating, investing, financing). Accrual accounting records revenue when earned.
Psychology and cognitive science methods
Experiment: manipulate the independent variable, measure the dependent variable, randomly assign participants, control confounds. Correlation does not imply causation. Reliability (consistency) vs validity (measures what it claims).
Choosing a test: two group means t-test; same people twice paired t-test; 3+ group means ANOVA; two categorical variables chi-square; two continuous variables correlation or regression.
\(d=\dfrac{M_{1}-M_{2}}{s_{\text{pooled}}}\ (0.2\text{ small},\ 0.5\text{ medium},\ 0.8\text{ large});\quad r:\ 0.1,\ 0.3,\ 0.5;\quad p<\alpha\Rightarrow\text{reject }H_{0}\)
Type I error: false positive (rate alpha). Type II: false negative (rate beta). Power = 1 - beta; rises with sample size, effect size, and alpha.
\(d'=z(\text{hit rate})-z(\text{false alarm rate}),\quad \dfrac{\Delta I}{I}=k\ \text{(Weber)},\quad RT=a+b\log_{2}(n+1)\ \text{(Hick)},\quad MT=a+b\log_{2}\dfrac{2D}{W}\ \text{(Fitts)}\)
Conditioning: classical pairs a neutral stimulus with an unconditioned one. Operant: reinforcement increases behavior (positive adds, negative removes a stimulus); punishment decreases it. Piaget: sensorimotor, preoperational, concrete operational, formal operational. Big Five: openness, conscientiousness, extraversion, agreeableness, neuroticism. Working memory holds about 4 chunks.
Biology: cell, genetics and physiology
Organelles: nucleus (DNA), ribosome (translation), rough ER (protein synthesis), smooth ER (lipids), Golgi (modify and sort), mitochondria (ATP), lysosome (digestion), chloroplast (photosynthesis), peroxisome (H2O2 breakdown). Prokaryotes lack a nucleus.
Mitosis: prophase, metaphase, anaphase, telophase (PMAT), gives 2 identical diploid cells. Meiosis: 2 divisions, 4 haploid cells; crossing over in prophase I and independent assortment create variation.
Mendel: monohybrid cross Aa x Aa gives 3:1 phenotype, 1:2:1 genotype. Dihybrid AaBb x AaBb gives 9:3:3:1. Testcross gives 1:1 if heterozygous. Chi-square uses df = categories - 1.
\(\text{Recombination frequency}=\dfrac{\text{recombinants}}{\text{total}}\times100\ (1\%=1\ \text{cM}),\quad \text{PCR copies}=2^{n},\quad N=N_{0}2^{n}\ \text{(bacterial growth)}\)
\(\Psi=\Psi_{S}+\Psi_{P},\quad \Psi_{S}=-iCRT\ \text{(water moves to lower }\Psi),\quad \text{surface area/volume falls as cells grow}\)
Photosynthesis: light reactions (thylakoid, make ATP and NADPH, release O2), Calvin cycle (stroma, fix CO2). Respiration: glycolysis (cytosol), pyruvate oxidation and Krebs (matrix), electron transport (inner membrane).
Taxonomy: Domain, Kingdom, Phylum, Class, Order, Family, Genus, Species. Natural selection needs variation, heritability, and differential reproduction. Gram-positive stains purple (thick peptidoglycan).
Earth and ocean science
\(f=2\Omega\sin\varphi\ (\Omega=7.292\times10^{-5}\ \text{s}^{-1}),\quad fV=-\dfrac1\rho\dfrac{\partial p}{\partial n}\ \text{(geostrophic)},\quad p=p_{0}+\rho gz\)
\(T_{e}=\left[\dfrac{S(1-\alpha)}{4\sigma}\right]^{1/4}\approx255\ \text{K},\quad \text{seawater: }S\approx35\ \text{psu},\ \rho\approx1025\ \text{kg/m}^{3};\quad +1\ \text{atm per 10 m depth}\)
\(c_{\text{deep}}=\sqrt{\dfrac{g\lambda}{2\pi}}=\dfrac{gT}{2\pi},\quad c_{\text{shallow}}=\sqrt{gh},\quad E\propto10^{1.5M}\ \text{(earthquake energy)},\quad \text{each magnitude}=10\times\text{amplitude}\)
Plate tectonics: divergent (spreading, new crust), convergent (subduction, mountains, trenches), transform (slip). Plates move a few cm per year. Rock cycle: igneous, sedimentary, metamorphic. Thermohaline circulation is driven by temperature and salinity density differences.
Public health and epidemiology
\(\text{Incidence}=\dfrac{\text{new cases}}{\text{person-time at risk}},\quad \text{prevalence}=\dfrac{\text{existing cases}}{\text{population}},\quad RR=\dfrac{a/(a+b)}{c/(c+d)},\quad OR=\dfrac{ad}{bc}\)
\(\text{Sensitivity}=\dfrac{TP}{TP+FN},\quad \text{specificity}=\dfrac{TN}{TN+FP},\quad PPV=\dfrac{TP}{TP+FP},\quad NPV=\dfrac{TN}{TN+FN}\)
\(R_{0}=\text{average secondary cases},\ \text{epidemic grows if }R_{0}>1;\quad \text{herd immunity threshold}=1-\dfrac1{R_{0}},\quad NNT=\dfrac1{ARR}\)
Study designs: RCT (strongest for causation), cohort (follow exposed vs unexposed, gives RR), case-control (start from outcome, gives OR, good for rare disease), cross-sectional (snapshot, prevalence). Confounder is associated with exposure and outcome but is not on the causal path.
Music theory
Major scale steps: whole, whole, half, whole, whole, whole, half. Natural minor: whole, half, whole, whole, half, whole, whole. Circle of fifths, sharps added in order F C G D A E B; flats B E A D G C F.
Intervals in semitones: m2 1, M2 2, m3 3, M3 4, P4 5, tritone 6, P5 7, m6 8, M6 9, m7 10, M7 11, octave 12. Triads: major 0-4-7, minor 0-3-7, diminished 0-3-6, augmented 0-4-8. Dominant seventh 0-4-7-10.
\(f_{n}=440\cdot2^{(n-49)/12}\ \text{Hz (piano key }n),\quad \text{semitone ratio}=2^{1/12},\quad \text{just intervals: octave }2{:}1,\ \text{fifth }3{:}2,\ \text{fourth }4{:}3\)
Roman numerals: uppercase major, lowercase minor, o diminished. Common progressions I-V-vi-IV and ii-V-I. Cadences: authentic V-I, plagal IV-I, half ends on V, deceptive V-vi.
Languages and linguistics
Spanish present tense endings: -ar verbs o, as, a, amos, ais, an; -er verbs o, es, e, emos, eis, en; -ir verbs o, es, e, imos, is, en. French -er verbs: e, es, e, ons, ez, ent.
German articles (nominative, accusative, dative, genitive): masculine der, den, dem, des; feminine die, die, der, der; neuter das, das, dem, des; plural die, die, den, der. Mandarin has 4 tones plus neutral; pinyin is the romanization.
Linguistics: phoneme (meaningful sound unit), morpheme (smallest meaning unit), syntax rule S -> NP VP, NP -> Det N. IPA consonants are described by place (bilabial, alveolar, velar), manner (stop, fricative, nasal), and voicing.
Language learning: spaced repetition, comprehensible input, speaking early, and tracking high-frequency vocabulary (the top 1000 words cover most everyday speech).
Writing, reading and study skills
Argument essay: hook, context, thesis, then paragraphs built as claim, evidence, analysis, link (PEEL/PEAL), counterargument, conclusion that says why it matters. Revise for structure first, sentences second, typos last.
Citations: MLA (Author Page), APA (Author, Year), Chicago (footnotes or author-date). Always cite quotes, paraphrases, data, and ideas. Paraphrase means restructure the sentence, not swap words.
Study methods that work: active recall (self-test), spaced repetition (review at about 1, 3, 7, 14, 30 days), interleaving problem types, Feynman technique (explain it simply), SQ3R for reading, Cornell notes, Pomodoro (25 min work, 5 min break).
Exams: read the whole test first, do easy points first, show your work, budget time per point, and keep 5 minutes to check units and signs. Common fallacies: straw man, ad hominem, false dilemma, slippery slope, circular reasoning, post hoc.
Visual arts, film and design
\(\text{Golden ratio }\varphi=\dfrac{1+\sqrt5}2\approx1.618,\quad \text{contrast ratio}=\dfrac{L_{1}+0.05}{L_{2}+0.05}\ (\text{WCAG AA text}\ge4.5{:}1),\quad \text{rule of thirds: lines at }\tfrac13,\tfrac23\)
Color: additive RGB (screens), subtractive CMY(K) (print). Complementary colors sit opposite on the wheel; analogous colors are adjacent. Hue, saturation, value describe a color.
\(\text{Exposure triangle: aperture f-stops }1.4,\ 2,\ 2.8,\ 4,\ 5.6,\ 8,\ 11,\ 16\ (\text{each step halves light});\ \text{shutter}\approx\dfrac1{\text{focal length}};\ \text{ISO doubles = 1 stop};\ \text{180° shutter: }\dfrac1{2\cdot\text{fps}}\)
Design process: empathize, define, ideate, prototype, test. Principles: contrast, repetition, alignment, proximity (CRAP), hierarchy, and whitespace.
Conversions and constants
Length
\(1\ \text{in}=2.54\ \text{cm}\)
\(1\ \text{ft}=30.48\ \text{cm}\)
\(1\ \text{yd}=0.9144\ \text{m}\)
\(1\ \text{mi}=1.609\ \text{km}\)
\(1\ \text{m}=3.281\ \text{ft}\)
Mass
\(1\ \text{lb}=453.6\ \text{g}\)
\(1\ \text{oz}=28.35\ \text{g}\)
\(1\ \text{kg}=2.205\ \text{lb}\)
Volume
\(1\ \text{gal}=3.785\ \text{L}\)
\(1\ \text{qt}=0.9464\ \text{L}\)
\(1\ \text{fl oz}=29.57\ \text{mL}\)
\(1\ \text{L}=1000\ \text{cm}^{3}\)
Pressure
\(1\ \text{atm}=101.3\ \text{kPa}=760\ \text{mmHg}=14.70\ \text{psi}\)
\(1\ \text{bar}=100\ \text{kPa}\)
Temperature
\(°\text{F}=\tfrac95\,°\text{C}+32\)
\(\text{K}=°\text{C}+273.15\)
Speed
\(1\ \text{mph}=0.4470\ \text{m/s}\)
\(1\ \text{km/h}=0.2778\ \text{m/s}\)
Energy and power
\(1\ \text{cal}=4.184\ \text{J}\)
\(1\ \text{kWh}=3.6\ \text{MJ}\)
\(1\ \text{hp}=745.7\ \text{W}\)
\(1\ \text{eV}=1.602\times10^{-19}\ \text{J}\)
Time
\(1\ \text{day}=86{,}400\ \text{s}\)
\(1\ \text{yr}=3.156\times10^{7}\ \text{s}\)
SI prefixes
| Prefix | Symbol | Factor |
|---|---|---|
| tera | T | \(10^{12}\) |
| giga | G | \(10^{9}\) |
| mega | M | \(10^{6}\) |
| kilo | k | \(10^{3}\) |
| centi | c | \(10^{-2}\) |
| milli | m | \(10^{-3}\) |
| micro | μ | \(10^{-6}\) |
| nano | n | \(10^{-9}\) |
| pico | p | \(10^{-12}\) |
| femto | f | \(10^{-15}\) |
Physical constants
| Symbol | Name | Value |
|---|---|---|
| \(c\) | speed of light | \(2.998\times10^{8}\ \text{m/s}\) |
| \(g\) | gravity (Earth) | \(9.807\ \text{m/s}^{2}\) |
| \(G\) | gravitational | \(6.674\times10^{-11}\ \text{N·m}^{2}/\text{kg}^{2}\) |
| \(h\) | Planck | \(6.626\times10^{-34}\ \text{J·s}\) |
| \(k_{B}\) | Boltzmann | \(1.381\times10^{-23}\ \text{J/K}\) |
| \(N_{A}\) | Avogadro | \(6.022\times10^{23}\ \text{mol}^{-1}\) |
| \(R\) | gas constant | \(8.314\ \text{J/(mol·K)}=0.08206\ \text{L·atm/(mol·K)}\) |
| \(e\) | elementary charge | \(1.602\times10^{-19}\ \text{C}\) |
| \(m_{e}\) | electron mass | \(9.109\times10^{-31}\ \text{kg}\) |
| \(m_{p}\) | proton mass | \(1.673\times10^{-27}\ \text{kg}\) |
| \(\varepsilon_{0}\) | permittivity | \(8.854\times10^{-12}\ \text{F/m}\) |
| \(F\) | Faraday | \(96{,}485\ \text{C/mol}\) |
| \(V_{m}\) | gas molar volume | \(22.41\ \text{L/mol}\ (0\,°\text{C},\ 1\ \text{atm})\) |
Reference tables
Greek alphabet
| Letter | Name | Letter | Name | Letter | Name |
|---|---|---|---|---|---|
| Α α | alpha | Η η | eta | Ν ν | nu |
| Β β | beta | Θ θ | theta | Ξ ξ | xi |
| Γ γ | gamma | Ι ι | iota | Ο ο | omicron |
| Δ δ | delta | Κ κ | kappa | Π π | pi |
| Ε ε | epsilon | Λ λ | lambda | Ρ ρ | rho |
| Ζ ζ | zeta | Μ μ | mu | Σ σ | sigma |
| Τ τ | tau | Υ υ | upsilon | Φ φ | phi |
| Χ χ | chi | Ψ ψ | psi | Ω ω | omega |
Resistor color code
| Color | Digit |
|---|---|
| black | 0 |
| brown | 1 |
| red | 2 |
| orange | 3 |
| yellow | 4 |
| green | 5 |
| blue | 6 |
| violet | 7 |
| gray | 8 |
| white | 9 |
| gold | 5% |
| silver | 10% |
Number systems
| dec | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| bin | 0000 | 0001 | 0010 | 0011 | 0100 | 0101 | 0110 | 0111 | 1000 | 1001 | 1010 | 1011 | 1100 | 1101 | 1110 | 1111 |
| hex | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F |
Computer science
\(1<\log n<n<n\log n<n^{2}<2^{n}<n!\)
\(\neg(A\wedge B)=\neg A\vee\neg B,\quad \neg(A\vee B)=\neg A\wedge\neg B\)
ASCII: '0' = 48, 'A' = 65, 'a' = 97
\(2^{4}=16,\ 2^{8}=256,\ 2^{10}=1{,}024,\ 2^{16}=65{,}536,\ 2^{32}\approx4.29\times10^{9}\)
Periodic trends
Atomic radius: larger going down a group, smaller going left to right.
Ionization energy and electronegativity: larger going up and to the right (F is highest).
Fill order: 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d 7p (s 2, p 6, d 10, f 14)
Engineering and science tables
SI derived units
| Quantity | Unit | In base units |
|---|---|---|
| force | newton (N) | \(\text{kg·m/s}^{2}\) |
| energy, work | joule (J) | \(\text{N·m}\) |
| power | watt (W) | \(\text{J/s}\) |
| pressure | pascal (Pa) | \(\text{N/m}^{2}\) |
| frequency | hertz (Hz) | \(\text{s}^{-1}\) |
| charge | coulomb (C) | \(\text{A·s}\) |
| potential | volt (V) | \(\text{J/C}\) |
| resistance | ohm (Ω) | \(\text{V/A}\) |
| capacitance | farad (F) | \(\text{C/V}\) |
| inductance | henry (H) | \(\text{Wb/A}\) |
| magnetic flux | weber (Wb) | \(\text{V·s}\) |
| magnetic field | tesla (T) | \(\text{Wb/m}^{2}\) |
Dimensionless numbers
| Name | Definition | Meaning |
|---|---|---|
| Reynolds | \(\rho VL/\mu\) | inertial / viscous |
| Prandtl | \(c_{p}\mu/k\) | momentum / thermal diffusivity |
| Nusselt | \(hL/k\) | convective / conductive heat transfer |
| Biot | \(hL_{c}/k_{s}\) | internal / external resistance |
| Fourier | \(\alpha t/L^{2}\) | dimensionless time |
| Froude | \(V/\sqrt{gL}\) | inertial / gravity |
| Mach | \(V/c\) | flow / sound speed |
| Grashof | \(g\beta\Delta TL^{3}/\nu^{2}\) | buoyancy / viscous |
| Weber | \(\rho V^{2}L/\sigma\) | inertial / surface tension |
| Schmidt | \(\mu/\rho D\) | momentum / mass diffusivity |
| Strouhal | \(fL/V\) | oscillation / flow |
Typical material properties (approximate)
| Material | E (GPa) | Yield (MPa) | Density (kg/m³) | k (W/m·K) |
|---|---|---|---|---|
| Steel (A36) | 200 | 250 | 7850 | ~50 |
| Aluminum 6061-T6 | 69 | 275 | 2700 | ~167 |
| Copper | 117 | ~70 annealed | 8960 | ~400 |
| Titanium Ti-6Al-4V | 114 | ~880 | 4430 | ~7 |
| Concrete (compression) | ~30 | f'c 20-40 | 2400 | ~1.4 |
| Glass | ~70 | brittle | 2500 | ~1 |
| Water | n/a | n/a | 1000 | 0.6 |
| Air (20 °C) | n/a | n/a | 1.20 | 0.026 |
More constants
| Quantity | Value |
|---|---|
| \(\hbar\) | \(1.055\times10^{-34}\ \text{J·s}\) |
| \(\mu_{0}\) | \(4\pi\times10^{-7}\ \text{N/A}^{2}\) |
| \(k_{e}\) | \(8.988\times10^{9}\ \text{N·m}^{2}/\text{C}^{2}\) |
| \(\sigma\) (Stefan-Boltzmann) | \(5.670\times10^{-8}\ \text{W/m}^{2}\text{K}^{4}\) |
| Atomic mass unit | \(1.661\times10^{-27}\ \text{kg}\) |
| Standard gravity | \(9.80665\ \text{m/s}^{2}\) |
| Earth mass / radius | \(5.972\times10^{24}\ \text{kg}\ /\ 6.371\times10^{6}\ \text{m}\) |
| Sun mass | \(1.989\times10^{30}\ \text{kg}\) |
| Astronomical unit | \(1.496\times10^{11}\ \text{m}\) |
| Light-year | \(9.461\times10^{15}\ \text{m}\) |
| Speed of sound (air, 20 °C) | \(343\ \text{m/s}\) |
| Specific heat of water | \(4.184\ \text{J/g·K}\) |
| Bohr radius | \(5.292\times10^{-11}\ \text{m}\) |