Calculus & Differential Equations
Calculus
- Foundations:
- Limits, continuity and sequences: Real Functions, Sequence Convergence, Squeeze Theorem, Supremum & Infimum, Lipschitz Continuity
- Elementary functions (polynomial, exponential, trigonometric, hyperbolic): Calculus of Simple Functions
- Differentiation (single variable):
- Derivative rules: product rule, Quotient Rule, Chain Rule
- Implicit Differentiation
- Mean value theorem, L’Hôpital’s rule
- Approximation: Taylor Series, Asymptotic Expansion
- Integration (single variable):
- Riemann Integral, fundamental theorem of calculus
- Techniques: substitution, integration by parts, partial fractions
- Improper integrals: Gaussian Integral, Gamma Function
- Differentiating under the integral: Leibniz Integral Rule
- Beyond Riemann: Lebesgue Integral
- Sequences and series:
- Geometric Series
- Power Series, Radius of Convergence, Abel's Theorem
- Convergence tests, Convergence of Functions
- Multivariable differentiation:
- Partial derivatives and multi-indices (see [[pde-fundamentals]])
- Gradient and directional derivatives
- Derivative matrices: Jacobian Matrix, Hessian Matrix, Jacobian and Hessian Matrices
- Multivariable chain rule and implicit function theorem
- Taylor’s theorem in several variables
- Multivariable integration:
- Multiple integrals and Fubini’s theorem
- Change of Variables, Change-of-Variables Formula (Jacobians, polar, cylindrical, spherical)
- Vector calculus:
- Vector Calculus
- Differential operators: Divergence, curl, Laplace Operator
- Line and surface integrals
- Integral theorems: Green, Stokes, divergence theorem
- Divergence
- Optimization:
- Optimization, critical points and the second derivative test
- Constrained optimization: Lagrange Multipliers
- Convex Function
- Calculus of variations:
- Integral transforms:
- Convolution
- Laplace Transform, Fourier series and transform
Ordinary Differential Equations (ODEs)
- Fundamentals: Ordinary Differential Equations
- Order, linearity, initial value problems, notation
- Existence and uniqueness (Picard-Lindelöf), see Lipschitz Continuity
- First-order ODEs:
- Separable equations
- Linear equations and the [[integrating-factor-method]]
- Exact equations (equations in full differentials)
- Homogeneous equations
- Bernoulli and Riccati equations
- Qualitative analysis: direction fields, equilibria, stability
- Higher-order linear ODEs:
- Homogeneous equations with constant coefficients (characteristic equation)
- Inhomogeneous equations: undetermined coefficients, variation of parameters
- Reduction of order, Wronskian
- Euler-Cauchy equations
- Systems of ODEs:
- Linear systems with constant coefficients: Eigenvectors and Eigenvalues, matrix exponential
- Phase plane analysis and stability
- Further solution methods:
- Power series: Power Series
- Laplace transform: Laplace Transform
- Green’s functions
- Numerical methods: Euler, Runge-Kutta, see Finite-Difference Methods
- Boundary value problems:
- Two-point boundary value problems
- Sturm-Liouville problems and eigenfunction expansions
- Related:
- Difference equations: Linear Difference Equations
- Stochastic versions: Stochastic Differential Equations
Partial Differential Equations (PDEs)
- Fundamentals:
- Partial Differential Equations (partial derivatives, order, linearity, homogeneity, classification)
- Well-posedness (existence, uniqueness, stability)
- Boundaries and Initial Values (Dirichlet, Neumann, Robin)
- Tools:
- Gradient, Divergence, Laplace Operator
- Change of Variables
- Gaussian Integral
- Divergence theorem and integration by parts
- First-order PDEs:
- PDE Characteristics
- Method of Characteristics
- Transport Equation
- Quasilinear Partial Differential Equations (Burgers’ equation, shocks)
- Nonlinear first-order PDEs: Hamilton-Jacobi Equation, Hamilton-Jacobi-Bellman Equation
- Hyperbolic PDEs:
- Wave Equation
- D’Alembert’s Formula
- Domain of dependence, energy methods
- Parabolic PDEs:
- Heat Equation (diffusion, Fick, Fourier, Newton)
- Heat kernel and maximum principle
- Applications: Fokker-Planck Equation, Kolmogorov Equations, Black-Scholes PDE, Feynman-Kac Formula
- Elliptic PDEs:
- Laplace Equation
- Poisson’s equation, harmonic functions, mean value property
- Maximum principle
- Solution methods:
- Separation of variables
- Fourier series and Fourier transform
- Laplace Transform
- Fundamental solutions and Green’s functions, Convolution
- Similarity solutions
- Duhamel’s principle (inhomogeneous equations)
- Advanced topics:
- Weak solutions and Sobolev spaces
- Viscosity solutions
- Numerical methods: Finite-Difference Methods