Mathematical Methods in the Physical Sciences - Overview¶

Introduction¶

Systematic mathematical tools are essential for solving core problems in physics and engineering. This course is based on Mary L. Boas's Mathematical Methods in the Physical Sciences and systematically covers the most frequently used mathematical methodologies in physical sciences.

Starting with infinite series and complex numbers, and proceeding through linear algebra, partial differentiation, vector analysis, Fourier analysis, differential equations, special functions, complex analysis, integral transforms, calculus of variations, and tensor analysis — we comprehensively cover the mathematical tools that form the theoretical foundation of modern physics and engineering.

Each lesson provides rigorous mathematical theory along with Python (NumPy, SciPy, SymPy, Matplotlib) implementations, enabling direct computation and visualization of abstract formulas.

File List¶

No.	Filename	Topic	Main Content
00	00_Overview.md	Overview	Course introduction and study guide
01	01_Infinite_Series.md	Infinite series and convergence	Convergence tests, power series, Taylor series, asymptotic series
02	02_Complex_Numbers.md	Complex numbers	Complex algebra, polar/exponential representation, De Moivre's theorem, Euler's formula
03	03_Linear_Algebra.md	Linear algebra	Matrices, determinants, systems of equations, eigenvalues/eigenvectors, diagonalization, quadratic forms
04	04_Partial_Differentiation.md	Partial differentiation	Partial derivatives, chain rule, Lagrange multipliers, exact differentials, Taylor series
05	05_Vector_Analysis.md	Vector analysis	Gradient, divergence, curl, line/surface integrals, Stokes, Gauss, Green theorems
06	06_Curvilinear_Coordinates.md	Curvilinear coordinates and multiple integrals	Cylindrical/spherical coordinates, Jacobian, coordinate transformation, volume/area elements
07	07_Fourier_Series.md	Fourier series	Fourier coefficients, convergence conditions, Gibbs phenomenon, Parseval's theorem
08	08_Fourier_Transforms.md	Fourier transforms	Continuous Fourier transform, DFT, FFT, convolution theorem, applications
09	09_ODE_First_Second_Order.md	Ordinary differential equations (1st/2nd order)	Separable/exact/linear ODEs, integrating factor, characteristic equation
10	10_Higher_Order_ODE_Systems.md	Higher-order ODEs and systems	Variation of parameters, systems of ODEs, phase plane, stability
11	11_Series_Solutions_Special_Functions.md	Series solutions and special functions	Frobenius method, Bessel, Legendre, Hermite, Laguerre, spherical harmonics
12	12_Sturm_Liouville_Theory.md	Sturm-Liouville theory	Eigenvalue problems, orthogonal functions, completeness, Rayleigh quotient, comparison theorem
13	13_Partial_Differential_Equations.md	Partial differential equations	PDE classification, separation of variables, Helmholtz equation, uniqueness theorems
14	14_Complex_Analysis.md	Complex analysis	Analytic functions, residue theorem, 4 types of real integrals, analytic continuation
15	15_Laplace_Transform.md	Laplace transform	Definition and properties, inverse transform, solving ODEs/circuit problems, transfer functions
16	16_Greens_Functions.md	Green's functions	Delta function, Green's function construction, boundary value problems, physics applications
17	17_Calculus_of_Variations.md	Calculus of variations	Euler-Lagrange equation, constraints, Lagrangian mechanics
18	18_Tensor_Analysis.md	Tensor analysis	Index notation, metric tensor, covariant derivative, physics applications

Required Libraries¶

pip install numpy scipy matplotlib sympy

NumPy: Numerical computation, array operations, linear algebra
SciPy: Special functions, integration, ODE/PDE solvers, FFT
Matplotlib: Function graphs, vector fields, contour visualization
SymPy: Symbolic calculus, series expansion, Laplace transform

Recommended Study Path¶

Phase 1: Foundational Tools (01-06) — 3-4 weeks¶

01 Infinite series → 02 Complex numbers → 03 Linear algebra → 04 Partial differentiation
                                                                       │
                                          05 Vector analysis → 06 Curvilinear coordinates

Methods for determining convergence and divergence of series
Algebraic and geometric properties of complex numbers
Matrices, eigenvalues, quadratic forms (foundation for ODE/S-L/tensors)
Partial derivatives, Lagrange multipliers, thermodynamic relations
Differentiation and integration of vector fields (grad, div, curl)
Operations in various coordinate systems

Goal: Acquire mathematical tools that form the foundation for all subsequent topics

Phase 2: Fourier Analysis (07-08) — 1-2 weeks¶

07 Fourier series → 08 Fourier transform

Frequency decomposition of periodic functions
Continuous/discrete Fourier transforms and FFT
Core tools for signal processing and PDE solving

Goal: Acquire frequency domain analysis capabilities

Phase 3: Differential Equations (09-13) — 3-4 weeks¶

09 ODE (1st/2nd order) → 10 Higher-order ODE/systems
                          │
11 Series solutions/special functions → 12 S-L theory → 13 PDE

Analytical methods for ordinary differential equations
Special functions and orthogonal function systems (Bessel, Legendre, spherical harmonics)
Separation of variables for partial differential equations, Helmholtz equation

Goal: Ability to analytically solve core equations of physics

Phase 4: Advanced Topics (14-18) — 2-3 weeks¶

14 Complex analysis → 15 Laplace transform
                      │
16 Green's functions → 17 Calculus of variations → 18 Tensor analysis

Complex integration and residue theorem, 4 types of real integrals
Initial value problems using Laplace transform
Green's functions and boundary value problems
Euler-Lagrange equation and Lagrangian mechanics
Tensors and fundamentals of general relativity

Goal: Acquire sophisticated mathematical tools for handling advanced physics and engineering problems

Prerequisites¶

Required¶

Calculus: Differentiation, integration, partial derivatives, chain rule
Linear algebra: Vectors, matrices, eigenvalues, determinants
Python basics: Functions, loops, lists

Recommended¶

NumPy basics: Array creation and operations
College physics: Mechanics, electromagnetism fundamentals (helpful for understanding application examples)

Math_for_AI: ML/DL perspective mathematics (complementary)
Numerical_Simulation: Numerical methods (complement to analytical methods)
Statistics: Probability and statistics

Learning Objectives¶

Upon completing this course, you will be able to:

Series convergence tests: Apply various tests to determine series convergence/divergence
Complex number applications: Derive trigonometric identities using complex exponentials, find polynomial roots
Vector field analysis: Calculate divergence and curl of physical fields, apply integral theorems
Coordinate transformations: Choose and transform coordinate systems matching problem symmetry
Fourier analysis: Analyze frequency components of signals, filtering, PDE solving
ODE analytical solutions: Find general and particular solutions for various types of ordinary differential equations
Special function understanding: Properties and physical applications of Bessel, Legendre, etc.
PDE solving: Solve heat equation, wave equation, Laplace equation using separation of variables
Complex integration: Calculate real integrals using residue theorem
Variational problems: Solve optimization problems using Euler-Lagrange equation
Tensor operations: Apply index notation and tensor transformation rules
Physics problem solving: Mathematically formulate and solve real physics/engineering problems by synthesizing the above tools

Relationship with Existing Courses¶

Mathematical_Methods          Math_for_AI              Numerical_Simulation
──────────────────────────────────────────────────────────────────────────────
Analytical & general math     ML/DL specialized math   Numerical computation
Physics/engineering focus     Optimization, probability Numerical ODE/PDE solvers
Based on Boas textbook        Deep learning math       Simulation applications

Mathematical_Methods: How should we solve it (analytical methods)
Numerical_Simulation: How do we compute it (numerical methods)
Math_for_AI: How to apply to AI (ML/DL perspective)

References¶

Textbooks¶

Boas, M. L. (2005). Mathematical Methods in the Physical Sciences, 3rd ed. Wiley.
Main reference for this course
Arfken, G. B., Weber, H. J., & Harris, F. E. (2012). Mathematical Methods for Physicists, 7th ed. Academic Press.
Graduate-level reference
Kreyszig, E. (2011). Advanced Engineering Mathematics, 10th ed. Wiley.
Comprehensive engineering mathematics reference
Riley, K. F., Hobson, M. P., & Bence, S. J. (2006). Mathematical Methods for Physics and Engineering, 3rd ed. Cambridge University Press.
Another standard textbook for physics/engineering mathematics

Online Resources¶

MIT OCW 18.04: Complex Variables with Applications
MIT OCW 18.03: Differential Equations
3Blue1Brown: Fourier Transform visualization
Paul's Online Math Notes: ODE/PDE reference

Tools¶

Wolfram Alpha: Formula verification
Desmos: Function visualization
SymPy Live: Online symbolic computation

Version Information¶

Initial version: 2026-02-08
Author: Claude (Anthropic)
Based on textbook: Boas, Mathematical Methods in the Physical Sciences, 3rd ed.
Python version: 3.8+
Main library versions:
NumPy >= 1.20
SciPy >= 1.7
Matplotlib >= 3.4
SymPy >= 1.9

License¶

This material can be freely used for educational purposes. Please cite the source for commercial use.

Next step: Begin with 01. Infinite Series and Convergence.