Physical Math Course

Lesson 1 • Scalars, Vectors, and Coordinate Systems

Introduces scalar and vector quantities and standard coordinate frames. Provides the geometric foundation for all force and motion analysis.

Lesson 2 • Numbers, Units, and Dimensional Analysis

Covers SI units, significant figures, and dimensional consistency checks. Establishes the quantitative precision required throughout the course.

Lesson 3 • Functions and Physical Graphs

Reviews functional relationships and graphical interpretation relevant to physics. Students learn to extract physical meaning from slope, intercept, and curvature.

Lesson 1 • Series Expansions and Approximations

Covers Taylor and Maclaurin series as approximation tools in physics. Students apply small-angle and weak-field approximations to simplify equations.

Lesson 2 • Integration and Accumulated Quantities

Covers definite and indefinite integrals as tools for computing displacement, work, and flux. Builds intuition for integration as summation over continuous distributions.

Lesson 3 • Differentiation and Physical Rates

Applies derivative rules to position, velocity, and acceleration. Grounds abstract calculus in concrete kinematic and dynamic interpretations.

Lesson 4 • Multiple Integrals and Physical Applications

Introduces double and triple integrals for computing mass, charge, and volume. Connects coordinate choice to computational efficiency in symmetric geometries.

Lesson 1 • Second-Order Linear ODEs

Develops characteristic equation methods for constant-coefficient second-order ODEs. Covers overdamped, critically damped, and underdamped solution families.

Lesson 2 • First-Order ODEs and Physical Decay

Solves separable and linear first-order ODEs modeling radioactive decay and RC circuits. Establishes the link between differential equations and exponential behavior.

Lesson 3 • Driven Oscillators and Resonance

Adds forcing terms to the harmonic oscillator equation and finds particular solutions. Students analyze amplitude-frequency response and resonance conditions.

Lesson 1 • Matrices and Linear Transformations

Defines matrices as linear maps and covers multiplication, transpose, and inverse. Connects matrix operations to coordinate rotations and physical transformations.

Lesson 2 • Tensors and Index Notation

Introduces rank-0, rank-1, and rank-2 tensors with Einstein summation convention. Students apply tensor notation to stress, strain, and inertia tensors.

Lesson 3 • Eigenvalues and Eigenvectors

Derives the characteristic polynomial and solves for eigenvalues and eigenvectors. Students interpret eigenvalues as principal values in physical applications.

Lesson 4 • Inner Product Spaces and Orthogonality

Defines inner products, norms, and orthogonal bases in physical vector spaces. Prepares students for Hilbert space methods used in quantum mechanics.

Lesson 1 • Curvilinear Coordinate Systems

Derives gradient, divergence, and curl in cylindrical and spherical coordinates. Enables efficient solution of field problems with matching geometric symmetry.

Lesson 2 • Gauss's and Stokes's Theorems

States and proves the divergence theorem and Stokes's theorem with physical examples. Students use these theorems to convert between integral and differential field equations.

Lesson 3 • Line Integrals and Work

Evaluates line integrals of vector fields along parameterized paths. Students compute work done by force fields and identify path-independent cases.

Lesson 4 • Surface and Volume Integrals

Extends integration to surfaces and volumes for flux and charge calculations. Builds the integral machinery needed for Gauss's and Stokes's theorems.

Lesson 1 • Fourier Series and Eigenfunction Expansions

Expands initial and boundary data in Fourier series to complete PDE solutions. Students compute Fourier coefficients and assess convergence for discontinuous data.

Lesson 2 • Laplace Equation and Potential Theory

Solves Laplace's equation in rectangular, cylindrical, and spherical geometries. Students apply results to electrostatic and gravitational potential problems.

Lesson 3 • Fourier Transform Methods for PDEs

Extends Fourier series to the Fourier transform for problems on unbounded domains. Students solve the heat and wave equations on infinite lines using transform pairs.

Lesson 4 • Classification and Physical Origins of PDEs

Classifies PDEs as elliptic, parabolic, or hyperbolic and links each type to physical phenomena. Establishes the role of boundary and initial conditions in determining unique solutions.

Lesson 1 • Complex Numbers and Functions

Reviews complex arithmetic and introduces analytic functions via Cauchy-Riemann equations. Connects analyticity to the existence of harmonic conjugate pairs in 2D fields.

Lesson 2 • Laurent Series and Singularities

Expands functions in Laurent series and classifies isolated singularities. Students identify poles, essential singularities, and removable singularities in physical functions.

Lesson 3 • Conformal Mapping in Physics

Uses conformal maps to transform complex geometries into solvable standard domains. Students apply Joukowski and Schwarz-Christoffel maps to fluid and electrostatic problems.

Lesson 4 • Residue Theorem and Real Integrals

Applies the residue theorem to evaluate improper real integrals and trigonometric integrals. Students close contours strategically and apply Jordan's lemma.

Lesson 1 • Monte Carlo Methods in Physics

Uses random sampling to simulate physical processes and estimate integrals. Students implement basic Monte Carlo algorithms and assess statistical convergence.

Lesson 2 • Probability Distributions in Measurement

Introduces Gaussian, Poisson, and binomial distributions as models for physical measurements. Students compute means, variances, and probabilities for realistic detector scenarios.

Lesson 3 • Error Propagation and Uncertainty

Derives error propagation formulas for functions of measured quantities. Students quantify combined uncertainties in multi-variable experimental results.

Lesson 4 • Least-Squares Fitting and Regression

Applies linear and nonlinear least-squares fitting to extract physical parameters. Students evaluate fit quality and interpret parameter uncertainties from covariance matrices.