1# Mathematical Methods Example Files
2
3This directory contains 12 standalone Python scripts demonstrating mathematical methods commonly used in physics and engineering (based on Mary Boas' "Mathematical Methods in the Physical Sciences").
4
5## Files Overview
6
7### 01_infinite_series.py
8- Series convergence tests (ratio, root, comparison)
9- Partial sums and convergence visualization
10- Taylor and Maclaurin series expansions
11- Euler summation technique
12- Power series radius of convergence
13
14### 02_complex_numbers.py
15- Complex arithmetic operations
16- Polar and exponential forms
17- Euler's formula
18- Roots of unity
19- Conformal mappings (zĀ², 1/z, exp(z))
20- Simple Mandelbrot set visualization
21
22### 03_linear_algebra.py
23- Matrix operations (addition, multiplication, transpose)
24- Determinants and matrix inverse
25- Eigenvalues and eigenvectors
26- Matrix diagonalization
27- Singular Value Decomposition (SVD)
28- Solving linear systems
29- Matrix exponential
30
31### 05_vector_analysis.py
32- Gradient of scalar fields
33- Divergence of vector fields
34- Curl of vector fields
35- Line integrals
36- Surface integrals
37- Green's, Stokes', and Divergence theorem verification
38
39### 06_fourier.py
40- Fourier series coefficients
41- Fast Fourier Transform (FFT)
42- Spectral analysis
43- Filtering in frequency domain
44- Parseval's theorem
45- Windowing techniques (Hanning, Hamming, Blackman)
46
47### 07_ode.py
48- Euler's method
49- Runge-Kutta 4th order (RK4)
50- scipy.integrate.solve_ivp
51- Harmonic oscillator
52- Damped oscillator
53- Lorenz system (chaotic dynamics)
54- Phase portraits
55
56### 08_special_functions.py
57- Bessel functions J_n(x) and Y_n(x)
58- Legendre polynomials P_n(x)
59- Hermite polynomials H_n(x)
60- Laguerre polynomials L_n(x)
61- Spherical harmonics Y_l^m(Īø,Ļ)
62- Gamma function Ī(x)
63- Orthogonality properties
64
65### 10_pde.py
66- Heat equation (parabolic): finite difference method
67- Wave equation (hyperbolic): explicit scheme
68- Laplace equation (elliptic): Jacobi relaxation
69- Poisson equation with source term
70- Time evolution visualization
71
72### 11_complex_analysis.py
73- Cauchy integral formula (numerical)
74- Residue theorem and computation
75- Laurent series expansion
76- Poles and essential singularities
77- Analytic continuation concepts
78- Conformal mapping visualization
79
80### 12_laplace_transform.py
81- Laplace transform pairs
82- Inverse Laplace transform (Bromwich integral)
83- Solving ODEs using Laplace transform
84- Transfer functions
85- Step response
86- Frequency response (Bode plots)
87- Convolution theorem
88
89### 14_calculus_of_variations.py
90- Euler-Lagrange equation
91- Brachistochrone problem (fastest descent)
92- Catenary curve (hanging chain)
93- Geodesics on surfaces
94- Lagrangian mechanics (pendulum, spring-mass)
95- Minimal surface of revolution (catenoid)
96
97### 15_tensors.py
98- Tensor operations with numpy
99- Index notation and Einstein summation (np.einsum)
100- Metric tensor (Euclidean, polar, spherical)
101- Christoffel symbols
102- Coordinate transformations (Cartesian ā polar)
103- Raising and lowering indices
104- Levi-Civita tensor and cross product
105
106## Requirements
107
108All scripts are standalone and can be run independently. They require:
109
110- **Required**: `numpy`
111- **Optional**: `scipy` (for advanced numerical methods)
112- **Optional**: `matplotlib` (for visualizations)
113
114If scipy or matplotlib are not available, the scripts will run with reduced functionality and skip visualizations.
115
116## Usage
117
118Run any script directly:
119
120```bash
121python 01_infinite_series.py
122python 02_complex_numbers.py
123# ... etc
124```
125
126Each script:
127- Prints detailed numerical results to console
128- Generates visualizations (saved as PNG files) if matplotlib is available
129- Includes docstrings explaining the mathematical concepts
130
131## File Naming Convention
132
133Files are numbered according to the chapter structure in Boas' textbook:
134- Chapter 1: Infinite Series (01)
135- Chapter 2: Complex Numbers (02)
136- Chapter 3: Linear Algebra (03)
137- Chapter 5: Vector Analysis (05)
138- Chapter 6: Fourier Analysis (06)
139- Chapter 7: ODEs (07)
140- Chapter 8: Special Functions (08)
141- Chapter 10: PDEs (10)
142- Chapter 11: Complex Analysis (11)
143- Chapter 12: Laplace Transform (12)
144- Chapter 14: Calculus of Variations (14)
145- Chapter 15: Tensors (15)
146
147Note: Not all chapters have example files (e.g., Chapter 4 was incorporated into Chapter 3).
148
149## License
150
151These examples are part of the 03_Study project and are released under the MIT License.