Accumath

Version 1 catalog

Capabilities

Every capability planned for the January 2, 2027 launch, in plain language.

Estimated ship date: January 2, 2027 · updated 2026-10-29

Click a category to expand it.

Algebra
  • Polynomial arithmetic
  • Polynomial factorization
  • Transcendental equations with exact real root isolation
  • User-defined rewrite rules and rulesets
  • Table and map over expressions
  • Groebner bases
  • Polynomial system solving with Groebner bases
  • Noncommutative algebra (free algebras, NC polynomials, NC Groebner / rewriting)
  • Permutation & matrix groups (actions, orbits, stabilizers, Schreier–Sims style)
  • Finite groups: character tables (small orders), homomorphisms, quotients
  • Lie algebras / representation basics (structure constants, adjoint, weights — SoftFloat exact where possible)
  • Algebraic geometry limb (ideals, varieties, primary decomposition hooks, dimension/degree)
  • Puiseux series / plane-curve desingularization
  • Finite fields GF(pⁿ): arithmetic and polynomial factorization
  • Permutation groups through degree 12 (Mathieu M11, M12)
  • Matrix groups by Schreier–Sims through order 120, plus GL(n,q)
  • Character tables over ℚ(ζₘ), m = exponent of the group
  • Explicit irreducible representation matrices
  • Lie algebras through dimension 8 (sl(3))
  • Root systems and the Weyl character formula, rank ≤ 3
  • Noncommutative Gröbner past degree 8 and basis size 24
  • Primary decomposition: multiplicity and complex rational points
  • Plane-curve Puiseux series with a residual tail
  • Smith and Hermite normal forms
  • Quaternions
  • Jordan Canonical Form
  • Resultants
  • Discriminants
  • Minimal polynomials
  • Sparse polynomials
  • Partial fractions
  • LLL lattice reduction
  • Periodic trig solution families
Linear algebra
  • Exact matrix algebra (determinant, inverse, rank, characteristic polynomial, eigenvalues, LUP)
  • SoftFloat dense linear algebra (solve, determinant, inverse, QR, SVD, least squares)
  • Sparse matrices and iterative solvers (CSR, conjugate gradient, ILU(0), 2D Poisson)
  • Matrix exponential
  • Matrix logarithm
  • Matrix square root
  • Matrix power
  • Schur form
  • Reduced row echelon form (RREF)
  • Cramer’s rule
  • Rank and nullity with nullspace basis
  • Gram-Schmidt orthogonalization over the rationals
  • Sylvester’s criterion (positive/negative definite, indefinite)
  • LDL^T and rational Cholesky decomposition
  • Matrix 1-norm, infinity-norm, and 1-condition number
Calculus
  • Symbolic differentiation
  • Differentiation of user-defined functions
  • Implicit differentiation
  • Rational integration (Risch)
  • Algebraic integration (Risch)
  • Transcendental integration (Risch)
  • Heuristic integration
  • Exact products
  • Gruntz limits
  • Oscillatory limits
  • Taylor series
  • Total-degree Taylor polynomials in several variables
  • Risch and heuristic integrator completeness (specials and algebraic extensions)
  • Padé approximants
  • Definite integrals
  • Iterated double integrals
  • Cartesian triple integrals
  • Explicit change of variables (Jacobian)
  • Leibniz rule for parametric integrals
  • Laurent series
  • Asymptotic series
  • Stirling series
  • WKB
  • Series convergence tests (ratio, root, comparison, limit comparison, alternating series, integral, divergence)
  • Radius of convergence
  • Fractional calculus (Riemann–Liouville, Caputo, fractional ODEs)
  • Integral equations (Fredholm and Volterra, first and second kind)
  • Volterra convolution equations
  • Perturbation methods (regular, singular, matched asymptotics, multiple scales)
  • Arc length and plane curvature
  • Frenet frame (velocity, acceleration, unit tangent, principal normal, binormal, torsion)
  • Axis-aligned quadrics and conic classification
  • Inverse-function Jacobian and implicit systems
Vector calculus & complex analysis
  • Divergence
  • Curl
  • Laplacian
  • Contour integrals past rational poles and perfect-square quadratics
  • Non-conservative line integrals
  • Contour integrals
  • Surface and flux integrals
  • Green, Stokes, and divergence theorem automation
  • Curvilinear gradient
Integral transforms
  • Laplace transforms
  • Fourier transforms
  • Z transforms
  • Mellin transforms
  • Hankel transforms
  • Distributional transform forms
  • Z-Transform tables through n¹²
  • Convolution
  • Fourier series
Summation, recurrences & combinatorics
  • Gosper summation
  • Summation and product completeness (definite sums, multi-sums)
  • Zeilberger algorithm
  • Zeilberger recurrences of order 4 and higher (binomial sums)
  • q-Zeilberger
  • q-analogues (q-Pochhammer symbols, q-factorials, Gaussian binomials)
  • Sister Celine / WZ method
  • WZ method with three summation indices (including signed summands)
  • Generating functions
  • Lagrange inversion
  • Linear recurrence solving (Petkovšek)
  • Combinatorics (binomial, multinomial, derangements, Catalan, Stirling numbers, Bell numbers, partitions)
  • Finite automata (DFA and NFA)
  • Graph data structures + algorithms (BFS/DFS, shortest paths, MST, connectivity, components)
  • Graph spectra, Laplacians, random walks, centrality, matching, flows
  • Combinatorial graph generation and isomorphism
Ordinary differential equations
  • First-order ODEs
  • Constant-coefficient second-order ODEs
  • Second-order linear ODEs with harder forcing terms (phase-shifted trig, distinct-frequency products)
  • Kovacic algorithm
  • Frobenius method
  • Abel equations
  • Lie symmetries for ODEs
  • Hamiltonian mechanics and Noether currents
  • Integrating factors
  • Sturm-Liouville eigenvalue problems with constant coefficients (up to 16 eigenvalues)
  • Second-order linear ODEs with irregular singular points (Airy form)
  • Magnus expansion
  • Time-varying linear ODE systems of dimension 4 (Magnus expansion, block form)
  • Stochastic differential equations (Euler-Maruyama, Milstein) and Itô calculus rules
  • Delay differential equations (constant delays; pantograph delays with an exact series for the linear case, fourth-order in general)
  • Higher-order Euler-Lagrange equations
  • Automated trajectory solving from Euler-Lagrange equations
  • IVP state dimension past 32, matched to 2D method of lines
  • Sturm–Liouville with variable p, q, w and eigenvalues past 16
  • Time-varying linear ODE systems past dimension 4
  • Symplectic integrators for Hamiltonian systems
  • DAE index-0/1 solvers (symbolic index reduction and SoftFloat BDF/Radau)
  • DAE index reduction past index 1
  • Stiff and adaptive ODE suites (Rosenbrock, implicit Runge–Kutta, events, mass matrices)
  • Bifurcation analysis
  • Lane-Emden polytrope solver
  • Kerr spinning geodesic integration and the Forest–Ruth integrator
Partial differential equations
  • Charpit first-order PDEs (x,y-dependent, inhomogeneous, semi-linear)
  • Robin and mixed Dirichlet/Neumann boundary conditions on second-order PDEs
  • Multi-dimensional Green's functions (free space in 1D, 2D, 3D; unit disk)
  • PDE separation of variables
  • PDE classification (elliptic, parabolic, hyperbolic via B²−4AC)
  • d’Alembert solution for the wave equation
  • Rankine-Hugoniot shock speed and entropy condition
  • Natural boundary conditions and isoperimetric equations (variational)
  • Nonlinear PDEs (Burgers, KdV, eikonal)
  • Modal drumhead
  • Circular drumhead azimuthal modes m > 0
  • One-dimensional method of lines
  • 2D method of lines (parabolic and hyperbolic PDEs)
  • Charpit beyond the landed x,y-dependent contract
  • Nonlinear PDE families past Burgers, KdV, and eikonal
  • FEM: mesh generation (Delaunay), h/p-adaptive refinement, unstructured mesh Poisson/heat/wave/elasticity
  • PDE method library expansion (spectral, finite-volume 1D/2D, operator splitting)
  • Moving-boundary / Stefan / free-boundary PDE helpers
  • Sensitivity / adjoint ODE-PDE (forward + adjoint SoftFloat)
  • FEM depth (vector elements, mixed methods, nonlinear materials, modal analysis)
  • FEM multiphysics coupling (thermal–structural, basic CFD Poisson/Stokes)
  • Jackson-level electromagnetism and optics
Special functions
  • Polygamma
  • Elliptic K
  • Elliptic E
  • Elliptic Pi
  • Riemann zeta
  • Hurwitz zeta
  • Polylogarithm
  • Jacobi elliptic functions (sn, cn, dn, cd, ns, and amplitude)
  • Inverse Jacobi elliptic functions (arcsn, arccn, arcdn)
  • Spherical harmonics and associated Legendre functions
  • Hypergeometric 2F1
  • Error function (erf)
  • Exponential integral
  • Gamma function
  • Digamma
  • Bessel J
  • Bessel Y
  • Bessel I
  • Bessel K
  • Meijer G
  • Fresnel S and C, Si, Ci, logarithmic integral, erfc, incomplete gamma
  • Clausen Cl₂(θ), Cl₃(θ)
  • Barnes G-function G(z)
  • ₂F₁ analytic continuation outside unit disk (z≤−1, non-terminating z>1)
  • Complete elliptic Π (3rd kind) with analytic continuation n≥1
  • Analytic continuation of incomplete elliptic integrals |x|>1, K(m≥1), E(m>1)
  • Fox H-function H_{p,q}^{m,n}(z)
  • Special-function identity automation (Meijer-G / Fox-H / hypergeometric rewrite pipelines)
  • Orthogonal polynomials completeness (Jacobi/Laguerre/Hermite/Chebyshev families + recurrence/ops)
  • Bessel / Airy / Scorer / Kelvin / Struve / Anger–Weber numeric+symbolic SoftFloat limbs
  • Wright ω, generalized Lambert, polylog multi-valued branches (analytic continuation)
  • Mathieu ceₙ, seₙ, and the characteristic exponent for general q
  • Heun / Lamé / spheroidal wave special ODE functions
  • Clebsch-Gordan coefficients and 3j/6j symbols
  • Painlevé transcendents
  • Mathieu characteristic values
  • Irregular singular series past the Airy-first contract
  • Hypergeometric pFq
  • Complex-argument special functions (Bessel, Airy, Carlson elliptic, 2F1, erf, gamma)
Number theory
  • Number theory basics (modular arithmetic, primality, factorization)
  • Integer factorization and primality toolkit depth (ECM and quadratic sieve class methods, proven primality)
  • PSLQ integer relation detection
  • p-adic numbers
  • Number fields, rings of integers, ideal arithmetic (native)
  • Algebraic number theory extras (class group helpers, unit group, Dedekind domains)
  • Modular forms + elliptic curve arithmetic (L-series, Heegner points, class groups)
  • L-functions / Dirichlet characters / modular arithmetic toolkit expansion
  • Diophantine solvers (linear, Pell, selected elliptic Thue)
Cryptography
  • Cryptographic primitives (RSA, Diffie-Hellman, discrete log, Legendre symbol)
  • BSGS discrete logarithm for non-generator bases (least exponent, any modulus, base must be a unit)
  • secp256k1 point arithmetic and k·G, checked against published vectors
Logic & real algebra
  • Univariate inequality solving
  • Cylindrical algebraic decomposition (core cases)
  • Quantifier elimination for closed sentences
  • Quantifier elimination with one free variable
  • Boolean algebra and SAT (truth tables, CNF/DNF, DPLL)
  • Cylindrical algebraic decomposition and real quantifier elimination
  • Boolean satisfiability past 16 variables
Descriptive statistics
  • Symbolic moments past order 8
  • Numeric descriptive statistics
  • Welch t-test
  • ANOVA
  • Ordinary least squares (OLS)
  • Exact rational linear regression
  • Exact rational polynomial regression
  • Robust (HC1) regression standard errors
  • Newey-West errors
  • Logistic regression
  • Poisson regression
  • Probit regression
  • Information theory (entropy, KL divergence, mutual information, channel capacity)
Probability distributions
  • Normal distribution (PDF, moments, generating functions, entropy)
  • Exponential distribution
  • Binomial distribution
  • Uniform distribution
  • Poisson distribution
  • Gamma distribution
  • Beta distribution
  • Chi-squared distribution
  • Student t distribution
  • F distribution (CDF, mean and variance)
  • Symbolic distribution formulas (PDF, CDF, moment generating function, mean, variance, entropy)
Advanced statistics & machine learning
  • Beta, Gamma, Weibull, LogNormal, Multivariate Normal, Dirichlet, Wishart, and MLE
  • Goodness-of-fit: KS, Anderson–Darling, Shapiro–Wilk
  • Wilcoxon, Mann–Whitney U, Kruskal–Wallis
  • Survival analysis: Kaplan–Meier, log-rank, Cox
  • ARIMA/ARMA, ACF/PACF, STL, Kalman filter
  • System identification (ARX/OE/BJ, subspace)
  • Bayesian MCMC (Metropolis–Hastings, Gibbs, HMC/NUTS, ADVI, R-hat/ESS)
  • GLM, nonlinear regression, and mixed models
  • PCA, clustering, and Gaussian mixtures
  • Design of experiments and ANOVA depth
  • Neural network primitives and reverse-mode autodiff
  • Discrete Markov chains (transition matrices, stationary distributions, absorbing chains)
  • Probability and stochastic processes (Brownian motion, Itô calculus, martingales, Markov processes)
Optimization
  • Exact linear programming (simplex over the rationals)
  • Unconstrained and equality-constrained optimization (Newton, Lagrange multipliers)
  • Nonlinear inequality KKT optimization
  • Global optimization (simulated annealing, basin-hopping)
  • Convex optimization (QP/SOCP/SDP)
  • Integer programming
  • Multiobjective optimization (Pareto, differential evolution)
Signal processing & control
  • Analog filter design (Butterworth, Chebyshev I and II, elliptic)
  • FFT, power spectrum, DCT/DST, window functions
  • State-space control basics (transfer functions, controllability and observability, Ackermann, margins, PID, MPC)
  • Bilinear transform (s→z plane)
  • LP→HP, LP→BP, LP→BS frequency transformations
  • Classical control depth (root locus, Nichols, Bode/Nyquist, margins, lead/lag)
  • Kautsky–Nichols robust eigenstructure assignment
  • Full MIMO state-space design
  • Realistic aircraft dynamics G(s)
  • LQR/LQG/H₂/H∞ synthesis
  • Model reduction (balanced truncation, Hankel singular values)
  • STFT, spectrograms, CWT, and DWT
  • Signal processing depth (FIR/IIR, resampling, filter banks, Hilbert, cepstrum)
  • Communications primitives (constellations, sync, matched filters)
Numerical methods
  • Arbitrary-precision software floats with correct rounding, exact rationals, and big integers
  • Numeric root finding (single equations and systems)
  • Adaptive numeric quadrature (adaptive Simpson, Gauss, tanh-sinh)
  • SoftFloat virtual machine
  • Parallel SoftFloat batch evaluation
  • Interval arithmetic (certified enclosures)
  • 2D cubature (adaptive Simpson, Gauss-Legendre product, simplex rules)
  • 3D tensor product cubature rules
  • One-dimensional Monte Carlo
  • Interpolation (Lagrange, nearest, Chebyshev) and natural cubic splines
  • Forward-mode automatic differentiation (dual numbers)
  • Sparse grid cubature (Smolyak)
  • Multi-dimensional Monte Carlo with importance sampling
  • GPU acceleration for dense SoftFloat arrays, bit-identical to the CPU path
  • First-order uncertainty propagation
  • Numerical analysis depth (conditioning, stability, error analysis, Richardson extrapolation)
Physics, chemistry & biology
  • Chemistry (stoichiometry, 298 K thermochemistry, periodic table for Z = 1 to 118)
  • Quantum operator algebra and one- and two-qubit circuits
  • Hydrogen radial wavefunctions
  • Population dynamics (logistic growth, Lotka-Volterra, SIR models with R0)
  • Kepler two-body solver with anomaly equations
  • Functional path integrals and measures
  • Dirac bra-ket operator algebra
  • Pharmacokinetic models
  • Thermodynamics (Carnot, ideal and van der Waals gases, Maxwell relations, entropy)
  • Electromagnetism (field tensor, Lorentz force, Poynting vector, Maxwell equations)
  • Mechanics law pack (kinematics, dynamics, energy, momentum, rotation, gravitation)
  • Thermodynamics law pack (laws, cycles, heat transfer, entropy, free energy)
  • Electromagnetism and circuits law pack (E&M fields, DC/AC circuits, RC/RL/RLC transients)
  • Waves and optics law pack (wave equation, interference, diffraction, geometric optics, polarization)
  • Chemistry I law pack (stoichiometry, gases, quantum, bonding, thermochemistry)
  • Chemistry II law pack (kinetics, equilibrium, acids/bases, electrochemistry, thermodynamics)
  • Quantum mechanics core (Schrödinger equation, operators, angular momentum, perturbation theory, scattering)
  • Engineering statics (forces, moments, equilibrium, trusses, friction, centroids)
Geometry
  • Curved geometry engine (metrics, Christoffel symbols, curvature tensors, geodesics)
  • Differential geometry extras (differential forms, Hodge star, exterior calculus automation)
  • Tensor algebra / index gymnastics (Einstein summation, symmetries)
  • Linearized gravitational waves and metric perturbations
  • Computational geometry (convex hulls, Delaunay/Voronoi, Boolean polygons)
  • Conic sections (standard forms, foci, directrices, eccentricity, rotations)
Finance
  • Finance (Black-Scholes, implied volatility, binomial options, bonds, NPV, portfolio moments, VaR)
Proofs & certificates
  • Nullstellensatz Groebner combination matrix recovery and radical membership certificates
  • Lyapunov certificate (discrete-time, exact up to 3×3)
  • Formal certificates (WZ, Nullstellensatz, Risch, Sturm, Pratt, Farkas, Lyapunov, Cayley-Hamilton, and more)
  • Lean 4 proof export
Language, input & units
  • Assumption and domain system (real, complex, and integer predicates that drive simplify and solve)
  • Expression simplifier
  • Basic assumptions engine
  • Piecewise, Abs, and Sign calculus (branchwise differentiation, domain-aware rewriting)
  • LaTeX output for results
  • Piecewise, Abs, and Sign: integrate, limit, and solve
  • Math parser
  • LaTeX math input
  • Step-by-step solutions
  • Wolfram Language syntax input (derivatives and integrals)
  • Unit and quantity system with dimensional analysis and offline CODATA conversion tables
  • SI, cgs, and natural units (full unit layer)
  • Coordinate conversion (Cartesian, polar, cylindrical, spherical)
  • Derivation trees from law to working equation, with per-step dimensional checking
  • Law registry (searchable index of all 669 laws and methods)
  • Physical constants exposed in the notebook as named, unit-carrying values
  • Temperature units (affine Celsius and Fahrenheit to kelvin)
  • Data import and export (CSV, JSON, Excel, native HDF5 read and write)
  • Code generation (Rust emitter producing SoftFloat code)
  • Parallel SoftFloat array evaluation (CPU; parallel results match sequential bit for bit)
  • Discrete mathematics and combinatorics (sets, logic, graph theory, recurrence, generating functions)
Notebook & desktop
  • Offline natural-language math parser
  • Local offline entity store
  • Curated physics, chemistry, and math tables, with a query API
  • Image and audio processing
  • MATLAB .mat import
  • Notebook core (documents, cell state, dependency graph, job queue, engine interface; headless open-source build)
  • Local-only notebook IPC
  • Notebook desktop shell
  • 2D plotting
  • Polar and parametric curves (plotting, differentiation, arc length, area)
  • Clipboard and headless PDF/LaTeX
  • 3D plotting
  • Vector fields, streamlines, and phase portraits
  • Plot themes, annotations, dual axes, and PNG/SVG/PDF export
  • Scatter, bar, and histogram plots
  • Region and inequality shading
  • Complex-plane plots
  • Manipulate / Dynamic sliders
  • CDF portable interactive documents
STEM curriculum — 669 laws & methods, law by law
Freshman — 131 laws
Calc I
  • Limit definition (epsilon-delta)
  • Limit laws and squeeze theorem
  • Continuity; IVT and EVT
  • Derivative definition
  • Linearity and power rule
  • Product rule
  • Quotient rule
  • Chain rule
  • Derivatives of elementary functions
  • Implicit differentiation
  • L'Hopital's rule
  • Mean value theorem / Rolle
  • Optimization / first and second derivative tests
Calc II
  • Definite integral as Riemann sum
  • Fundamental theorem of calculus (parts 1 and 2)
  • Antiderivative / standard integral table
  • u-substitution
  • Integration by parts
  • Partial fractions
  • Trigonometric substitution / trig integrals
  • Improper integrals
  • Arc length
  • Areas and volumes (disk/washer/shell), surface of revolution
  • Parametric derivative
  • Polar derivative and polar area
  • Geometric and p-series
  • Divergence test (nth-term)
  • Integral test
  • Comparison and limit comparison tests
  • Ratio and root tests
  • Alternating series test; absolute/conditional convergence
  • Power series radius/interval of convergence
  • Taylor and Maclaurin series
  • Taylor remainder / error bound
  • Standard Maclaurin series (e^x, sin, cos, ln(1+x), 1/(1-x), (1+x)^k)
  • Trigonometric and exponential/log identities
  • Separable and linear first-order ODEs
Physics I
  • 1D kinematics (constant acceleration)
  • 2D / projectile motion
  • Rotational kinematics
  • Newton's second law
  • Work-energy theorem
  • Conservation of energy and momentum; collisions
  • Torque and angular momentum
  • Moment of inertia and parallel-axis theorem
  • Newton's law of gravitation
  • Fluids: continuity, hydrostatics and Bernoulli
  • Simple harmonic motion
  • Ideal gas law
  • First law of thermodynamics
  • Second law, entropy and Carnot efficiency
  • Wave equation and traveling waves
  • Superposition, standing waves, beats, Doppler
Physics II
  • Coulomb's law
  • Electric field of charge distributions
  • Gauss's law
  • Electric potential
  • Capacitance and stored energy
  • Ohm's law and power
  • Kirchhoff's laws
  • RC transient
  • RL transient
  • RLC series transient
  • Magnetic force (Lorentz)
  • Biot-Savart law
  • Ampere's law
  • Faraday's law and Lenz's law
  • Maxwell's four equations
  • Snell's law
  • Thin lens and mirror equation
  • Interference and diffraction
  • Electromagnetic waves
Chem I
  • Mole concept and molar mass
  • Balancing and stoichiometry (conservation of atoms)
  • Periodic law (element properties by Z)
  • Molarity and dilution
  • Gas laws (Boyle, Charles, Gay-Lussac, Avogadro)
  • Dalton's law of partial pressures
  • Graham's law of effusion
  • van der Waals equation
  • Calorimetry
  • Hess's law (standard enthalpy of reaction)
  • Planck-Einstein relation and photoelectric effect
  • de Broglie wavelength
  • Bohr model and Rydberg formula
  • Uncertainty principle and [x,p]=i hbar
  • Formal charge, octet rule, VSEPR
Chem II
  • Rate law and reaction order
  • Integrated rate laws (zero, first, second order)
  • Half-life
  • Arrhenius equation
  • Equilibrium constants Kc, Kp and reaction quotient
  • Water autoionization, pH and pOH
  • Weak acid/base equilibria (Ka, Kb)
  • Henderson-Hasselbalch
  • Solubility product
  • Gibbs free energy
  • dG = -RT ln K
  • Entropy statistical definition / third law
  • Nernst equation and cell potential
  • Faraday's laws of electrolysis
Track (LinAlg/Discrete/Statics)
  • Matrix multiplication and transpose
  • Determinant and cofactor expansion
  • Matrix inverse
  • Cramer's rule
  • Gaussian elimination, RREF, rank, nullspace
  • Eigenvalues and characteristic equation
  • Eigenvectors and diagonalization
  • Gram-Schmidt orthogonalization
  • Dot product, cross product, projection, norm
  • Linear independence, basis and dimension
  • Boolean algebra laws and truth tables
  • Permutations and combinations; binomial theorem
  • Inclusion-exclusion and pigeonhole
  • Recurrence relations
  • Mathematical induction / basic proofs
  • Modular arithmetic, gcd and Euclid
  • 3D static force equilibrium
  • 3D moment equilibrium
  • Trusses and friction (method of joints/sections; F<=mu N)
  • Centroid and center of mass
Cross-cutting (units/coords/derivation)
  • SI base units and dimensional homogeneity
  • Unit conversion (prefixes, derived units, temperature)
  • cgs / Gaussian unit system
  • Natural units
  • Physical constants (CODATA)
  • Cartesian <-> polar/cylindrical/spherical point conversion
  • Curvilinear scale factors and grad/div/curl/Laplacian (cyl, sph)
  • Jacobian and volume element
  • Derivation tree (law -> working equation) with verification
  • Measurement uncertainty propagation
Sophomore
Mathematics
Multivariable calculus
  • Partial derivatives, mixed partials (Clairaut)
  • Multivariable chain rule, total differential, Jacobian matrix
  • Gradient and directional derivative
  • Tangent plane and linear approximation
  • Hessian and second-derivative test for critical points
  • Lagrange multipliers (constrained extrema)
  • Double integrals over general regions (Fubini, iterated)
  • Triple integrals; cylindrical and spherical coordinates
  • Change of variables and Jacobian in multiple integrals
  • Vector-valued functions: velocity, acceleration, arc length, curvature, TNB frame
  • Mass, center of mass and moments of inertia by integration
  • Scalar and vector line integrals
  • Conservative fields and potential functions
  • Surface integrals and flux
  • Green's theorem
  • Stokes' theorem
  • Divergence (Gauss) theorem
  • Curvilinear gradient, divergence, curl, Laplacian
  • Vector identities (curl grad = 0, div curl = 0, Helmholtz decomposition)
  • Differentiation under the integral sign (Leibniz rule)
  • Taylor series in several variables
  • Quadric surfaces and conic sections (classification, foci)
  • Implicit and inverse function theorems
Ordinary differential equations
  • First-order ODEs: separable, linear, exact, Bernoulli, Riccati
  • Existence and uniqueness (Picard-Lindelof), Lipschitz conditions
  • Autonomous equations, equilibria and 1-D stability
  • Second-order linear constant-coefficient (homogeneous, characteristic equation)
  • Undetermined coefficients and variation of parameters
  • Cauchy-Euler equations
  • Higher-order linear ODEs with constant coefficients (order 3-6)
  • Linear systems: eigenvalue method, fundamental matrix, matrix exponential
  • Laplace transform method for IVPs (step, impulse, convolution)
  • Series solutions at ordinary and regular singular points (Frobenius)
  • Bessel, Legendre, Hermite, Laguerre equations and solutions
  • Phase plane, linearization and classification of equilibria
  • Lyapunov stability and Lyapunov functions
  • Limit cycles, Poincare-Bendixson, bifurcations
Linear algebra II
  • Vector spaces, subspaces, basis, dimension, rank-nullity
  • Linear transformations, matrix of a map, change of basis, similarity
  • Inner products, orthogonal projection, Gram-Schmidt
  • Least squares and normal equations
  • QR decomposition
  • LU factorization with pivoting
  • Eigenvalues, eigenvectors and diagonalization
Probability
  • Axioms, conditional probability, Bayes' theorem, independence
  • Combinatorial probability and counting
  • Random variables: pmf, pdf, cdf, expectation, variance
  • Binomial, Poisson, uniform, exponential, gamma, beta distributions
  • Normal distribution (pdf, cdf, quantile, moments)
  • Geometric, negative binomial, hypergeometric, Weibull, lognormal, Cauchy
Physics
Modern physics
  • Photons, blackbody radiation (Planck law, Wien, Stefan-Boltzmann), photoelectric and Compton effects
  • Matter waves, de Broglie, wave-particle duality, uncertainty
  • Particle in a box, harmonic oscillator, tunneling through barriers
  • Hydrogen atom: quantum numbers, orbitals, spectrum, spin, exclusion principle
  • Nuclear decay law, binding energy, fission and fusion energetics
  • Special relativity basics: Lorentz transformation, time dilation, velocity addition, E=mc^2
Chemistry
Organic chemistry
  • Bonding, hybridization, resonance, formal charge, molecular geometry
  • Nomenclature and functional-group recognition
  • Stereochemistry: chirality, R/S (CIP), enantiomers, diastereomers, meso
  • Conformations: Newman projections, cyclohexane chair, A-values
  • Acid-base chemistry: pKa, resonance and inductive effects
  • Nucleophilic substitution SN1/SN2 and elimination E1/E2
  • Electrophilic addition to alkenes and alkynes (Markovnikov, anti-Markovnikov)
  • Aromaticity (Huckel 4n+2) and electrophilic aromatic substitution
  • Carbonyl chemistry: addition, acyl substitution, enolates, aldol, Claisen
  • Spectroscopy: IR, 1H and 13C NMR (shift, integration, n+1 splitting), mass spectrometry
Engineering
Circuits
  • Node and mesh analysis, KCL/KVL, superposition, Thevenin and Norton equivalents
  • First-order (RC, RL) and second-order (RLC) transients
  • AC steady state: phasors, impedance, complex power, resonance, power factor
  • Op-amp circuits, diodes and transistors (nonlinear device equations)
  • Digital logic: Boolean algebra, Karnaugh maps, combinational and sequential design
Fluid mechanics
  • Hydrostatics, buoyancy, pressure distribution
  • Control-volume relations: Reynolds transport, continuity, momentum, energy, Bernoulli
Engineering thermodynamics
  • Properties of pure substances, steam tables, ideal gas, compressibility
  • First law for closed systems and control volumes
  • Second law, entropy balance, isentropic efficiency
Statics and dynamics
  • Particle kinematics and kinetics: Newton, work-energy, impulse-momentum
  • Equilibrium of rigid bodies, trusses, friction, centroids, moments of inertia
  • Internal forces, shear and moment diagrams, virtual work
Mechanics of materials
  • Stress and strain, Hooke's law, axial loading, thermal stress
  • Torsion of shafts, beam bending, flexure formula, beam deflection
Computer science
Algorithms
  • Asymptotic notation, recurrences, Master theorem
  • Sorting, order statistics, correctness and lower bounds
  • Data structures: heaps, search trees, hashing
Cross-cutting
Foundations
  • Unit systems and dimensional homogeneity (SI)
  • Coordinate systems and point/vector conversion
  • Derivation trees and step-by-step solutions
  • Physical constants and reference data
  • Measurement uncertainty and error propagation
Junior
Mathematics
Ordinary differential equations
  • Sturm-Liouville problems and eigenfunction expansions
  • Boundary-value problems and shooting
  • Green's functions for ODEs
  • Lie symmetry methods, Abel and Kovacic solvers
  • Floquet theory and Mathieu equation
  • Perturbation methods (regular, singular, matched asymptotics, multiple scales)
  • Integral equations (Fredholm, Volterra)
Partial differential equations
  • Classification of second-order PDEs (elliptic, parabolic, hyperbolic)
  • First-order PDEs: method of characteristics
  • d'Alembert solution of the wave equation
  • Separation of variables (rectangular, polar, cylindrical, spherical)
  • Fourier series (coefficients, even/odd, half-range)
  • Fourier convergence (Dirichlet conditions, Gibbs), Parseval
  • Heat equation: Dirichlet, Neumann, Robin boundary conditions
  • Wave equation on string and membrane (normal modes)
  • Laplace and Poisson equations: harmonic functions, mean value, maximum principle
  • Integral-transform solutions of PDEs (Fourier, Laplace, Hankel)
  • Green's functions and fundamental solutions for PDEs
Calculus of variations
  • Euler-Lagrange equations (first and higher order)
  • Constrained variational problems, isoperimetric, natural boundary conditions
Linear algebra II
  • Spectral theorem for symmetric and Hermitian matrices
  • Quadratic forms, definiteness and Sylvester's criterion
  • Singular value decomposition, pseudoinverse, Eckart-Young
  • Jordan canonical form and generalized eigenvectors
  • Minimal polynomial and Cayley-Hamilton
  • Matrix functions (exp, log, sqrt, power) and Schur form
  • Cholesky and LDL^T factorizations
  • Matrix norms and condition numbers
  • Bilinear forms, dual space, adjoint operator
  • Tensor products, Kronecker product, multilinear algebra
  • Smith and Hermite normal forms (modules over a PID)
  • Perron-Frobenius and stochastic matrices
Real analysis
  • Sequences, limits, Cauchy criterion, monotone convergence
  • Series convergence tests, absolute and conditional convergence
  • Completeness, sup and inf, Bolzano-Weierstrass, Heine-Borel
  • Continuity, uniform continuity, compactness theorems
  • Differentiability, mean value theorem, Taylor with remainder
  • Riemann and Riemann-Stieltjes integrability
  • Pointwise vs uniform convergence, Weierstrass M-test
  • Power series, radius of convergence, analytic functions
  • Gamma and Beta functions, Stirling's formula
Complex analysis
  • Complex arithmetic, polar form, roots of unity, elementary functions
  • Analytic functions, Cauchy-Riemann equations, harmonic conjugates
  • Branches, multivalued functions and Riemann surfaces
  • Cauchy's theorem and integral formula
  • Residue theorem and real integrals by residues
  • Laurent series and classification of singularities
  • Argument principle, Rouche's theorem, Liouville, maximum modulus
Abstract algebra
  • Groups, subgroups, cosets, Lagrange
  • Symmetric, alternating, dihedral groups; cycle notation and sign
  • Homomorphisms, kernels, quotient groups, isomorphism theorems
  • Group actions, orbit-stabilizer, Burnside counting
  • Rings, ideals, quotient rings, ring homomorphisms
  • Euclidean domains, PIDs, UFDs; polynomial gcd and factorization
Number theory
  • Divisibility, Euclid, Bezout, CRT
  • Primes, fundamental theorem of arithmetic, factorization
  • Fermat, Euler, totient, Mobius, divisor functions
  • Quadratic residues, Legendre/Jacobi symbols, quadratic reciprocity
  • Primitive roots, multiplicative order, discrete logarithm
Probability
  • Chi-squared, Student t, F distributions
  • Moment generating, characteristic functions, cumulants
  • Joint, marginal, conditional distributions; covariance and correlation
  • Transformations of random variables and sums (convolution)
  • Multivariate normal and conditional normal
  • Order statistics
  • Law of large numbers and central limit theorem
  • Inequalities: Markov, Chebyshev, Chernoff, Hoeffding
Numerical analysis
  • Floating-point arithmetic, rounding error, conditioning
  • Root finding: bisection, Newton, secant, systems
  • Interpolation: Lagrange, Chebyshev nodes, splines, error bounds
  • Numerical differentiation, Richardson extrapolation, automatic differentiation
  • Quadrature: trapezoid, Simpson, Gauss, adaptive, Romberg
  • Direct linear solvers, LU with pivoting, QR, error analysis
  • Iterative solvers: Jacobi, Gauss-Seidel, SOR, conjugate gradient, GMRES
Optimization
  • Linear programming, simplex and duality
  • Unconstrained optimization: gradient, Newton, quasi-Newton
Topology
  • Topological spaces, bases, continuity, homeomorphism
  • Subspace, product and quotient topologies
  • Connectedness and compactness
Statistics/Data
Statistics
  • Descriptive statistics, quantiles, skewness, kurtosis
  • Sampling distributions of the mean and variance
  • Point estimation: method of moments, maximum likelihood
  • t-tests (one-sample, paired, two-sample/Welch)
  • One-way and two-way ANOVA
  • Chi-squared goodness-of-fit and independence tests
  • Linear regression: OLS, standard errors, inference
Physics
Classical mechanics
  • Lagrangian mechanics: Euler-Lagrange equations, generalized coordinates
  • Constraints, Lagrange multipliers for holonomic constraints, generalized forces
  • Conservation laws and Noether's theorem
  • Central force problem and Kepler orbits
  • Scattering cross sections and Rutherford scattering
  • Small oscillations and normal modes
  • Rigid-body kinematics, inertia tensor, principal axes
  • Rigid-body dynamics: Euler equations, torque-free motion, gyroscope
  • Non-inertial frames: Coriolis, centrifugal forces
  • Hamiltonian mechanics: Legendre transform, Hamilton's equations, Poisson brackets
Electromagnetism
  • Maxwell's equations (vacuum and macroscopic), boundary conditions at interfaces
  • Electrostatics: Coulomb, Gauss, potential, Poisson and Laplace equations
  • Boundary-value problems: separation of variables with Legendre, Bessel expansions
  • Method of images
  • Multipole expansion, dipole and quadrupole moments
  • Dielectrics, polarization, bound charge, D and E fields
  • Magnetostatics: Biot-Savart, Ampere, vector potential, magnetic multipoles
  • Faraday induction, inductance, quasi-static fields
  • Magnetic media: M, H, boundary conditions, hysteresis
Optics
  • Geometric optics, ray-transfer (ABCD) matrices, Fermat's principle
  • Interference: Young, thin films, Michelson, Fabry-Perot
  • Fraunhofer and Fresnel diffraction, gratings, resolution
Quantum mechanics
  • Wave function, Schrodinger equation, probability current, normalization
  • Infinite and finite square wells, delta potential, bound and scattering states
  • Harmonic oscillator: ladder operators, Hermite functions
  • Hilbert-space formalism, Hermitian operators, generalized uncertainty principle
  • Angular momentum operators and commutation relations
  • Spherical harmonics and orbital angular momentum eigenfunctions
  • Spin-1/2, Pauli matrices, spin in a magnetic field
  • Addition of angular momenta, Clebsch-Gordan, 3j and 6j symbols
  • Hydrogen atom: radial equation, spectrum, degeneracy
Statistical mechanics
  • Thermodynamic potentials, Legendre transforms and Maxwell relations
  • Second and third laws, entropy, heat engines and Carnot cycle
  • Microcanonical, canonical, grand-canonical ensembles; partition function
  • Ideal gas statistical mechanics, Maxwell-Boltzmann distribution, equipartition
  • Planck distribution, blackbody radiation, photon gas
Chemistry
Organic chemistry
  • Pericyclic reactions: Diels-Alder, Woodward-Hoffmann rules, frontier orbitals
  • Radical reactions, carbenes, organometallics (Grignard, Gilman)
  • Hammond postulate, kinetic vs thermodynamic control, Curtin-Hammett
  • Huckel molecular-orbital theory (pi systems)
Physical chemistry
  • Real gases, virial equation, van der Waals, compression factor
  • Kinetic theory of gases: Maxwell distribution, mean free path, collision frequency
  • First law: internal energy, enthalpy, heat capacity, Kirchhoff's law
  • Second and third laws: entropy changes, Helmholtz and Gibbs energies
  • Chemical potential, Gibbs-Duhem, phase rule, Clapeyron and Clausius-Clapeyron
  • Mixtures: Raoult, Henry, colligative properties, activities
  • Chemical equilibrium: K, van't Hoff equation, temperature and pressure dependence
  • Electrochemistry: Nernst, cell potentials, Debye-Huckel, conductivity
  • Rate laws, integrated rate laws, half-life, Arrhenius
  • Reaction mechanisms: steady-state, pre-equilibrium, Lindemann, chain reactions
Quantum chemistry
  • Particle in a box, rigid rotor, harmonic oscillator for molecules
  • Hydrogenic atoms, many-electron atoms, term symbols, spin-orbit coupling
  • Molecular orbital theory: LCAO, diatomic MO diagrams, bond order
  • Molecular symmetry: point groups, character tables, reducible representations
Analytical chemistry
  • Acid-base titration curves, buffers, polyprotic equilibria, activity coefficients
  • Complexation and EDTA titrations, solubility, redox titrations
  • Beer-Lambert law and spectrophotometry
  • Calibration curves, regression, detection limits, error propagation
  • Statistical treatment of measurements: t-tests, Q-test, Grubbs, confidence intervals
Biochemistry
  • Bioenergetics: Delta G prime, coupled reactions, ATP hydrolysis
  • Amino acids and proteins: titration, pI, Henderson-Hasselbalch
  • Enzyme kinetics: Michaelis-Menten, Lineweaver-Burk, inhibition, kcat/KM
Engineering
Circuits
  • Two-port networks, transformers, three-phase power
  • Transmission lines: telegrapher's equations, reflection coefficient, Smith chart
Signals and systems
  • LTI systems, convolution, impulse response
  • Fourier series and continuous-time Fourier transform
  • Laplace transform, region of convergence, transfer functions
  • Z-transform and discrete-time systems
  • DFT, FFT, windowing, spectral estimation
  • Sampling theorem, aliasing, reconstruction
  • Analog filter design: Butterworth, Chebyshev, elliptic
Control systems
  • Modeling: transfer functions, block diagrams, state-space conversion
  • Time response: first and second-order specifications, steady-state error
  • Stability: Routh-Hurwitz, pole locations
  • Root-locus analysis and design
  • Frequency response: Bode, Nyquist, gain and phase margins, lead-lag
  • PID control and tuning (Ziegler-Nichols)
Fluid mechanics
  • Differential relations: continuity, Navier-Stokes, stream function, vorticity
  • Dimensional analysis, Buckingham pi, similarity (Re, Fr, Ma)
  • Viscous flow in ducts: laminar, Moody chart, Darcy-Weisbach, minor losses
  • Flow past immersed bodies: boundary layers, drag, lift
Heat transfer
  • Conduction rate equation, thermal resistance networks, composite walls
  • Heat diffusion equation and boundary conditions
  • 1-D steady conduction with generation, radial systems, extended surfaces (fins)
  • Transient conduction: lumped capacitance, Heisler charts, semi-infinite solid
  • Convection: boundary layers, Nusselt, Reynolds, Prandtl correlations (external, internal, free)
  • Heat exchangers: LMTD and effectiveness-NTU
  • Thermal radiation: Stefan-Boltzmann, view factors, radiosity
Engineering thermodynamics
  • Exergy (availability) analysis
  • Power cycles: Rankine, Brayton, Otto, Diesel; refrigeration and heat pumps
  • Gas mixtures, psychrometrics, air conditioning
Statics and dynamics
  • Planar rigid-body kinematics and kinetics, instantaneous center
  • Three-dimensional rigid-body kinetics, gyroscopic motion
  • Mechanical vibrations: SDOF free/forced, damping, resonance, transmissibility
  • MDOF vibrations and modal analysis
Mechanics of materials
  • Stress transformation, principal stresses, Mohr's circle
  • Failure criteria (Tresca, von Mises), column buckling (Euler), energy methods
Computer science
Algorithms
  • Graph traversal: BFS, DFS, connected components
  • Shortest paths (Dijkstra, Bellman-Ford) and minimum spanning trees
  • Maximum flow, matchings, bipartite graphs
  • Dynamic programming and greedy algorithms
  • Eulerian and Hamiltonian paths, graph coloring, chromatic polynomial
Theory of computation
  • Regular languages: DFA, NFA, regular expressions, minimization
  • Context-free languages, pushdown automata, pumping lemma
Discrete mathematics
  • Propositional and predicate logic, quantifier elimination, SAT
  • Induction and proof techniques, formal proof export
  • Generating functions, enumeration, partitions, Stirling and Catalan numbers
Cross-cutting
Foundations
  • Dimensional analysis, Buckingham pi theorem, nondimensionalization and scaling
  • Cartesian and general tensors: index notation, Levi-Civita identities, contraction
  • Special-function library (Gamma, Bessel, Legendre, hypergeometric, elliptic, orthogonal polynomials)
  • Integral transforms (Laplace, Fourier, Hankel, Z; Mellin is state 5)
  • Operators on function spaces: adjoints, spectral decomposition, bra-ket notation
Senior
Mathematics
Ordinary differential equations
  • Delay and stochastic differential equations
Partial differential equations
  • Nonlinear PDEs: Burgers, KdV, eikonal
  • Conservation laws, shocks, Rankine-Hugoniot, entropy conditions
Linear algebra II
  • Unitary, normal matrices and Hermitian operators on complex spaces
Real analysis
  • Differential forms and Stokes on manifolds
  • Stone-Weierstrass, Arzela-Ascoli, equicontinuity
Complex analysis
  • Conformal maps, Mobius transformations, Schwarz-Christoffel
  • Analytic continuation and monodromy
  • Entire functions, Hadamard and Weierstrass factorization, Mittag-Leffler
Abstract algebra
  • Sylow theorems, p-groups, solvable and nilpotent groups
  • Direct and semidirect products; structure of finite abelian groups
  • Free groups, presentations, Todd-Coxeter
  • Groebner bases, elimination, polynomial system solving
  • Modules over a PID and the structure theorem
  • Field extensions, minimal polynomials, degree and tower law
  • Finite fields GF(p^n)
  • Galois theory (Galois groups, fundamental theorem, solvability by radicals)
  • Splitting fields, cyclotomic fields, ruler-and-compass constructibility
  • Quaternions and Clifford algebras
Number theory
  • Diophantine equations: linear, Pell, sums of squares
  • Continued fractions and rational approximation
  • Dirichlet characters, L-functions, arithmetic functions
  • Primality tests (Miller-Rabin, AKS) and factoring (ECM, quadratic sieve)
Probability
  • Modes of convergence (in probability, a.s., L^p, in distribution)
  • Conditional expectation, tower property, martingales
  • Markov chains: stationary distribution, absorbing, classification
  • Poisson processes, renewal theory, queues
  • Shannon entropy, KL divergence, mutual information, channel capacity
  • Monte Carlo and MCMC (Metropolis-Hastings, Gibbs, HMC)
Numerical analysis
  • Eigenvalue algorithms: power, QR algorithm, Lanczos, Arnoldi
  • Numerical ODEs: Euler, Runge-Kutta, multistep, adaptive step, stability regions
  • Stiff problems, implicit Runge-Kutta, DAEs
  • Symplectic and geometric integrators
  • Finite differences for PDEs: consistency, stability, CFL, von Neumann analysis
  • Approximation: Chebyshev, Pade, minimax, least squares
  • FFT and spectral differentiation
Optimization
  • KKT conditions, nonlinear programming
  • Convex optimization: QP, SOCP, SDP, duality
  • Interior-point methods
Topology
  • Separation axioms, Urysohn, metrization, Tychonoff
  • Fundamental group and covering spaces
  • Seifert-van Kampen and classification of surfaces
Differential geometry
  • Curves: Frenet-Serret, curvature and torsion
  • Surfaces: first and second fundamental forms, Gauss and mean curvature
  • Theorema Egregium, geodesics on surfaces, Gauss-Bonnet
Statistics/Data
Statistics
  • Sufficiency, completeness, Rao-Blackwell, Lehmann-Scheffe
  • Fisher information and Cramer-Rao bound
  • Confidence intervals and pivotal quantities
  • Neyman-Pearson lemma, likelihood ratio tests, power
  • Nonparametric tests (Wilcoxon, Mann-Whitney, KS, Anderson-Darling)
  • Regression diagnostics, leverage, residual analysis, multicollinearity
  • Robust and HAC standard errors
  • Generalized linear models (logistic, Poisson, probit)
  • Asymptotic theory: delta method, asymptotic normality of MLE
Physics
Classical mechanics
  • Canonical transformations, generating functions, symplectic condition
  • Hamilton-Jacobi theory, action-angle variables, Liouville's theorem
Electromagnetism
  • Poynting theorem, energy and momentum conservation, Maxwell stress tensor
  • Potentials, gauge transformations (Lorenz, Coulomb), retarded potentials
  • Plane waves, polarization, reflection and refraction (Fresnel equations)
  • Waves in conductors, skin depth, dispersion, Kramers-Kronig
  • Waveguides, resonant cavities, optical fibers
  • Radiation: Larmor formula, dipole and multipole radiation, antennas
  • Scattering and diffraction: Rayleigh, Thomson, Mie, Kirchhoff
  • Covariant electrodynamics: field tensor, Lorentz force, transformation of E and B
Optics
  • Polarization: Jones and Mueller calculus, birefringence
  • Fourier optics, coherence, Gaussian beams and laser cavities
Quantum mechanics
  • Time-independent perturbation theory (non-degenerate and degenerate), fine structure
  • Variational principle and Rayleigh-Ritz
  • WKB approximation, Bohr-Sommerfeld quantization, tunneling
  • Time-dependent perturbation theory, Fermi golden rule, adiabatic theorem
  • Scattering theory: Born approximation, partial waves, cross sections
  • Identical particles, bosons and fermions, helium, Slater determinants
  • Symmetries, parity, time reversal, rotation matrices (Wigner D)
  • Qubits, quantum gates, entanglement, Bell states
  • Minimal coupling, Aharonov-Bohm effect, Landau levels
Statistical mechanics
  • Fermi-Dirac and Bose-Einstein statistics, ideal quantum gases
  • Einstein and Debye solids, heat capacity
  • Phase equilibria, Clausius-Clapeyron, van der Waals critical point
  • Interacting systems: virial expansion, mean-field theory, Landau theory
  • Ising model: 1-D transfer matrix, mean-field, Onsager 2-D
  • Kinetic theory, Boltzmann equation, H-theorem, transport coefficients
Relativity
  • Lorentz transformations, 4-vectors, relativistic kinematics and collisions
  • Relativistic Doppler, aberration, Thomas precession
  • Relativistic Lagrangian and Hamiltonian particle dynamics
Solid-state physics
  • Crystal structure, Bravais lattices, reciprocal lattice, Bragg diffraction
  • Phonons: lattice vibrations, dispersion, density of states
  • Free electron Fermi gas, Fermi energy, electronic heat capacity
  • Bloch theorem, nearly free electron model and energy bands
  • Tight-binding model and band structure
  • Semiconductors: carrier concentration, doping, p-n junction
Chemistry
Organic chemistry
  • Heterocycles, carbohydrates, amino acids and peptides, polymers
  • Retrosynthesis and asymmetric synthesis
Physical chemistry
  • Collision theory, transition-state (Eyring) theory, reaction dynamics
  • Statistical thermodynamics of molecules: partition functions and thermodynamic functions
  • Surfaces and catalysis: Langmuir, BET isotherms, adsorption kinetics
  • Transport: diffusion (Fick), viscosity, Stokes-Einstein
  • Molecular interactions, intermolecular potentials, solids and crystals
Quantum chemistry
  • Molecular spectroscopy: rotational, vibrational, electronic; selection rules, Franck-Condon
  • Magnetic resonance: NMR chemical shift, J-coupling, spin Hamiltonians, EPR
Analytical chemistry
  • Chromatography: retention, plate theory, resolution, van Deemter equation
  • Electroanalytical methods: potentiometry, voltammetry, Cottrell, Randles-Sevcik
  • Atomic and molecular spectrometry, mass spectrometry, FT methods
Biochemistry
  • Cooperativity and binding: Hill equation, Scatchard, allostery
  • Membrane potentials, transport, Nernst and Goldman equations
  • Metabolic networks: mass-action ODEs, flux balance
  • Pharmacokinetic models (one- and two-compartment, AUC, bioavailability)
  • Population and epidemic models (logistic, Lotka-Volterra, SIR)
Engineering
Signals and systems
  • Digital filter design: bilinear transform, FIR/IIR, frequency transformations
  • Time-frequency analysis: STFT, wavelets, spectrograms
  • Modulation and communication systems: AM, FM, matched filter, constellations
Control systems
  • State-space: controllability, observability, pole placement (Ackermann)
  • Lyapunov stability for linear systems
Fluid mechanics
  • Potential flow: stream and potential functions, superposition, conformal methods
  • Compressible flow: isentropic relations, shocks, nozzles, Fanno and Rayleigh flow
  • Open-channel flow and turbomachinery
Heat transfer
  • Boiling and condensation
  • Mass transfer: Fick's law, diffusion, convection mass transfer
Engineering thermodynamics
  • Combustion, chemical and phase equilibrium
  • Thermodynamic property relations (Maxwell, Clapeyron, Joule-Thomson)
Mechanics of materials
  • 3-D elasticity: stress and strain tensors, generalized Hooke's law, plane stress/strain
Materials science
  • Crystal structure, packing, defects, dislocations and strengthening
  • Diffusion in solids (Fick's laws), Arrhenius temperature dependence
  • Phase diagrams, lever rule, phase transformations, TTT curves
  • Fracture mechanics, fatigue (S-N, Paris law), creep
  • Composites (rule of mixtures), polymers, ceramics, electrical and thermal properties
Computer science
Algorithms
  • Graph spectra, Laplacians, random walks, isomorphism
  • NP-completeness, reductions, approximation algorithms
  • Computational geometry: convex hull, Voronoi/Delaunay
Theory of computation
  • Turing machines, decidability, reducibility
Discrete mathematics
  • Polya enumeration and Burnside counting
  • Ramsey theory, extremal combinatorics, design theory
Cryptography
  • Number-theoretic hardness: RSA, Diffie-Hellman, discrete logarithm
  • Elliptic-curve cryptography: group law, scalar multiplication, secp256k1
  • Perfect secrecy, one-time pad, private-key encryption, block ciphers (AES), modes
  • Hash functions, MACs, authenticated encryption, digital signatures
Coding theory
  • Linear block codes: generator and parity-check matrices, Hamming codes, syndrome decoding
  • Cyclic codes, BCH and Reed-Solomon codes
  • Channel capacity and source coding (Shannon theorems, Huffman)
Numerical linear algebra
  • Conditioning and stability of algorithms; backward error
  • Householder QR, SVD algorithms, least-squares conditioning
Cross-cutting
Foundations
  • Group theory and representations as a language (symmetry, selection rules)
  • Asymptotic and perturbative methods (WKB, boundary layers, saddle point)
  • Variational principles and constrained optimization
Graduate / PhD
Mathematics
Partial differential equations
  • Sobolev spaces, weak solutions, Lax-Milgram
  • Galerkin and finite-element methods
  • Hamilton-Jacobi equations and viscosity solutions
  • Distributions, Fourier transforms of distributions, fundamental solutions
Calculus of variations
  • Second variation, direct methods, Noether's theorem
Real analysis
  • Lebesgue measure and integration (MCT, DCT, Fatou)
  • L^p, Hilbert and Banach spaces, spectral theory of operators
  • Metric spaces, contraction mapping, Baire category
Complex analysis
  • Gamma, zeta, theta and elliptic functions
  • Prime number theorem and zeta zeros
  • Riemann mapping theorem, Schwarz reflection, several complex variables
Abstract algebra
  • Representation theory of finite groups: characters and orthogonality
  • Lie algebras, root systems and the Weyl character formula
  • Hilbert Nullstellensatz, radical membership, primary decomposition
  • Algebraic geometry: varieties, resultants, Bezout, Puiseux expansions
  • Noncommutative algebra and NC Groebner bases
  • Homological algebra: Tor, Ext, group cohomology
  • Category theory: functors, adjoints, limits
  • Dedekind domains, number fields, ideal class groups, units
Number theory
  • Elliptic curves: group law, torsion, rank, modular forms
  • p-adic numbers and Hensel's lemma
  • Lattice reduction (LLL) and integer relations (PSLQ)
Probability
  • Brownian motion, Ito calculus, SDEs, Black-Scholes
  • Measure-theoretic probability: sigma-algebras, Kolmogorov, Radon-Nikodym
Numerical analysis
  • Finite volume, spectral and finite-element PDE methods
  • Numerical SDEs and Monte Carlo error analysis
  • Verified numerics: interval Newton, certified enclosures
Optimization
  • Integer and mixed-integer programming, branch and bound
  • Global and multiobjective optimization
  • Dynamic programming, network flows, optimal control (LQR, Pontryagin)
Topology
  • Simplicial and singular homology, Euler characteristic, Betti numbers
  • De Rham cohomology and closed vs exact forms
Differential geometry
  • Smooth manifolds, charts, tangent vectors, differential forms
  • Riemannian metrics, Levi-Civita connection, Christoffel symbols
  • Curvature: Riemann, Ricci, scalar curvature, Einstein tensor
  • Hodge star, codifferential, Laplace-de Rham
  • Lie groups, Lie algebras, exponential map, vector fields and flows
  • Symplectic manifolds and Hamiltonian flows
  • Fiber bundles, connections, characteristic classes
Statistics/Data
Statistics
  • Bayesian inference: conjugate priors, posteriors, MCMC
  • Bootstrap, permutation tests, cross-validation
  • Regularization (ridge, lasso), model selection (AIC, BIC)
  • Multivariate analysis: PCA, clustering, discriminant, mixtures (EM)
  • Time series: ARIMA, ACF/PACF, spectral analysis, Kalman filter
  • Survival analysis and experimental design
  • Statistical learning: SVM, trees, boosting, neural networks
  • Nonparametric regression, kernel smoothing and splines
Machine learning
  • Neural networks, backpropagation, reverse-mode automatic differentiation
Physics
Classical mechanics
  • Classical chaos, Lyapunov exponents, KAM, canonical perturbation theory
  • Lagrangian and Hamiltonian field theory, continuous systems
  • Symplectic integrators for Hamiltonian systems
  • Lane-Emden polytropes and stellar structure
Electromagnetism
  • Radiation by moving charges: Lienard-Wiechert, synchrotron, bremsstrahlung, Cherenkov
  • Radiation damping and classical models of charged particles
  • Gaussian and SI unit systems in electrodynamics
Quantum mechanics
  • Density matrices, open quantum systems, decoherence
  • Second quantization and many-body theory
  • Relativistic quantum mechanics: Klein-Gordon and Dirac equations
  • Path integrals and functional measures
Statistical mechanics
  • Brownian motion, Langevin and Fokker-Planck equations, fluctuation-dissipation
  • Renormalization group and critical exponents
  • Monte Carlo methods for statistical systems (Metropolis)
Relativity
  • Geodesic equation, parallel transport, equivalence principle
  • Riemann, Ricci and Einstein tensors; Bianchi identities
  • Einstein field equations and stress-energy tensors (perfect fluid, Maxwell)
  • Schwarzschild solution: horizon, photon sphere, ISCO, tidal forces, precession
  • Kerr, Reissner-Nordstrom, Kerr-Newman black holes; Weyl scalars and Petrov type
  • Geodesic orbit integration in Schwarzschild and Kerr
  • Linearized gravity and gravitational waves
  • FLRW cosmology, Friedmann equations, de Sitter space
  • QFT in curved spacetime, Hawking radiation
Solid-state physics
  • Fermi surfaces, Hall effect, semiclassical transport
  • Superconductivity: London, BCS, Meissner effect
  • Magnetism: dia/paramagnetism, Curie-Weiss, ferromagnetism, magnetic resonance
  • Dielectrics, plasmons, optical processes, excitons
  • Defects, dislocations, nanostructures
Quantum field theory
  • Canonical quantization of the scalar field, propagators
  • Dirac equation, gamma matrices, spinors
  • Feynman diagrams and rules, S-matrix, cross sections and decay rates
  • QED processes: Compton, pair production, vertex corrections
  • Renormalization and the renormalization group
  • Non-abelian gauge theories, Standard Model group theory
  • Spontaneous symmetry breaking, Goldstone and Higgs mechanisms
  • Functional integrals in QFT
  • Particle kinematics: Mandelstam variables, phase space, nuclear binding
Chemistry
Organic chemistry
  • Physical organic: Hammett, linear free energy, isotope effects, mechanism elucidation
Quantum chemistry
  • Hartree-Fock, SCF, basis sets, post-HF methods
  • Density functional theory
  • Perturbation and variation methods for molecules
Biochemistry
  • Protein folding thermodynamics, nucleic acid melting, biomolecular simulation
Engineering
Signals and systems
  • Random signals: power spectral density, Wiener filtering, Kalman filtering
Control systems
  • Optimal control: LQR, LQG, H2, H-infinity, Kalman-Bucy
  • Model reduction, MIMO design, robust and nonlinear control
Fluid mechanics
  • Turbulence modeling, CFD, Stokes flow, lubrication theory
Heat transfer
  • Numerical conduction: finite-difference and finite-element
Mechanics of materials
  • Finite-element structural analysis and plasticity
Computer science
Algorithms
  • Randomized and online algorithms, probabilistic analysis
Theory of computation
  • Complexity classes: P, NP, PSPACE, hierarchies
Cryptography
  • Lattice-based and quantum-secure cryptography
  • Zero-knowledge, secure multiparty computation, advanced public-key topics
Coding theory
  • Convolutional codes, Viterbi decoding, turbo and LDPC codes
  • Code bounds (Hamming, Singleton, Gilbert-Varshamov, Plotkin) and weight enumerators
Numerical linear algebra
  • Krylov methods: Arnoldi, Lanczos, GMRES, preconditioning
Cross-cutting
Foundations
  • Green's functions and distribution theory as a universal solver
  • Curated thermophysical, spectroscopic, pKa and materials tables
  • Law-library registry: law objects with assumptions, units, domain, references
  • Textbook-problem gold corpus and acceptance tests per course
  • Proof assistants: certificates and Lean export for theorems