Applied Mathematician Course

Lesson 1 • Logic and Proof Techniques

Covers propositional logic, quantifiers, and standard proof strategies. Establishes the formal reasoning backbone used throughout the course.

Lesson 2 • Sets, Relations, and Functions

Introduces set theory, binary relations, and function types. Provides the language for describing mathematical structures precisely.

Lesson 3 • Number Systems and Algebraic Structures

Surveys integers, rationals, reals, and complex numbers alongside groups and rings. Connects abstract algebra to computational practice.

Lesson 1 • Eigenvalues and Eigenvectors

Derives eigenvalue theory and diagonalization. Connects to differential equations, stability analysis, and principal component analysis.

Lesson 2 • Vectors and Vector Spaces

Defines vector spaces, subspaces, and linear independence. Establishes the geometric and algebraic intuition needed for matrix theory.

Lesson 3 • Matrix Operations and Systems

Covers matrix arithmetic, Gaussian elimination, and solution sets of linear systems. Directly enables numerical and symbolic problem-solving.

Lesson 1 • Integral Calculus and Measure Basics

Develops Riemann and Lebesgue integration concepts and the fundamental theorem. Supports probability theory and differential equations.

Lesson 2 • Limits, Continuity, and Convergence

Formalizes epsilon-delta definitions and sequence convergence. Provides the analytical foundation for differentiation and integration.

Lesson 3 • Multivariable Calculus

Extends differentiation and integration to functions of several variables. Directly supports optimization, physics modeling, and vector calculus.

Lesson 1 • Statistical Inference and Estimation

Presents maximum likelihood, Bayesian estimation, and hypothesis testing. Connects probability theory to practical data analysis workflows.

Lesson 2 • Expectation, Variance, and Distributions

Derives moments, common distributions, and moment-generating functions. Enables characterization and comparison of probabilistic models.

Lesson 3 • Probability Axioms and Random Variables

Establishes Kolmogorov axioms, sample spaces, and random variable types. Forms the theoretical basis for all stochastic modeling.

Lesson 1 • Higher-Order Linear ODEs

Solves constant-coefficient and variable-coefficient linear ODEs using operator methods. Supports mechanical and electrical system modeling.

Lesson 2 • First-Order ODEs and Solution Methods

Covers separable, linear, and exact equations with integrating factors. Establishes solution techniques applied throughout applied mathematics.

Lesson 3 • Systems of ODEs and Phase Plane Analysis

Analyzes linear and nonlinear ODE systems using matrix methods and phase portraits. Enables stability and bifurcation analysis of dynamic models.

Lesson 1 • Root Finding and Nonlinear Equations

Covers bisection, Newton-Raphson, and secant methods with convergence analysis. Provides tools for solving nonlinear algebraic equations numerically.

Lesson 2 • Numerical ODE Solvers

Implements Euler, Runge-Kutta, and multistep methods for initial value problems. Prepares students for simulation of dynamic systems.

Lesson 3 • Numerical Integration and Differentiation

Develops quadrature rules and finite difference schemes with error bounds. Supports numerical ODE and PDE solvers introduced in later chapters.

Lesson 4 • Numerical Linear Algebra

Implements iterative and direct solvers for large linear systems. Connects to matrix decompositions from linear algebra in computational settings.

Lesson 1 • Unconstrained Optimization

Covers gradient descent, Newton's method, and convergence theory for smooth objectives. Builds the algorithmic foundation for machine learning and engineering design.

Lesson 2 • Convex Optimization

Analyzes convex sets, functions, and algorithms including interior-point methods. Provides the theoretical framework for modern data science optimization.

Lesson 3 • Linear and Integer Programming

Presents simplex method, duality, and branch-and-bound for integer programs. Supports operations research and resource allocation applications.

Lesson 1 • Numerical Methods for PDEs

Implements finite difference and finite element methods for elliptic and parabolic PDEs. Prepares students for large-scale computational simulation.

Lesson 2 • Fourier Series and Transform Methods

Develops Fourier series, transforms, and their application to PDEs. Enables spectral solution techniques for heat, wave, and Laplace equations.

Lesson 3 • Classification and Well-Posedness of PDEs

Classifies second-order PDEs as elliptic, parabolic, or hyperbolic and examines well-posedness. Sets the framework for choosing appropriate solution methods.

Lesson 4 • Separation of Variables and Eigenfunction Expansions

Applies separation of variables to canonical PDEs on bounded domains. Connects Sturm-Liouville theory to eigenfunction solution methods.