1Introduction to Mathematical Physics
787.1 Fundamental Concepts
21.1 The Role of Mathematics in Physics
797.2 Euler-Lagrange Equations
31.2 Historical Overview
807.3 Variational Principles in Physics
41.3 Fundamental Concepts
817.4 Constrained Variations
51.4 Mathematical Preliminaries
827.5 Isoperimetric Problems
61.5 Units and Dimensions
837.6 Applications in Physics
71.6 Problem-Solving Strategies
84Tensor Analysis
8Vector Analysis
858.1 Tensor Algebra
92.1 Vectors and Vector Operations
868.2 Tensor Calculus
102.2 Vector Fields
878.3 Covariant Differentiation
112.3 Gradient, Divergence, and Curl
888.4 Curvature Tensors
122.4 Line Integrals
898.5 Applications in General Relativity
132.5 Surface Integrals
908.6 Applications in Continuum Mechanics
142.6 Integral Theorems (Green’s, Stokes’, and Gauss’s Theorems)
91Fourier Analysis
152.7 Applications in Physics
929.1 Fourier Series
16Matrices and Linear Algebra
939.2 Fourier Transforms
173.1 Matrices and Matrix Operations
949.3 Convolution and Correlation
183.2 Systems of Linear Equation
959.4 Discrete Fourier Transform (DFT)
193.3 Vector Spaces
969.5 Fast Fourier Transform (FFT)
203.4 Linear Transformations
979.6 Applications in Physics
213.5 Eigenvalues and Eigenvectors
98Probability and Statistics
223.6 Diagonalization
9910.1 Probability Theory
233.7 Applications in Physics
10010.2 Random Variables and Distributions
24Complex Analysis
10110.3 Sampling and Data Analysis
254.1 Complex Numbers and Functions
10210.4 Parameter Estimation
264.2 Analytic Functions
10310.5 Hypothesis Testing
274.3 Cauchy-Riemann Conditions
10410.6 Applications in Physics
284.4 Complex Integration
105Computational Methods
294.4.1 Complex Line Integrals
10611.1 Introduction to Scientific Computing
304.4.2 Cauchy-Goursat Theorem: 4.4.3 Cauchy’s Integral Formula
10711.2 Numerical Integration
314.5 Cauchy’s Integral Theorem and Formulas
10811.2.1 Riemann Sum Approximation
324.5.1 Cauchy’s Integral Theorem
10911.2.2 Trapezoid Rule
334.5.2 Cauchy’s Integral Formula: 4.5.3 Cauchy’s Derivative Formulas
11011.2.3 Simpson’s Rule
344.6 Laurent Series and Residue Theory
11111.2.4 Gaussian Quadrature
354.6.1 Laurent Series
11211.2.5 Adaptive Quadrature
364.6.2 Residue Theory: 4.6.3 Applications of Residue Theory
11311.3 Numerical Differentiation
374.7 Applications in Physics
11411.3.1 Finite Difference Approximations
384.7.1 Electromagnetism
11511.3.2 Higher-Order Finite Difference Formulas: 11.3.3 Finite Difference Formulas for Higher Derivatives
394.7.2 Quantum Mechanics
11611.4 Root-Finding Algorithms
404.7.3 Fluid Dynamics and Aerodynamics
11711.4.1 Bracketing Methods
414.7.4 Signal Processing and Communication Theory
11811.4.2 Newton-Raphson Method
42Ordinary Differential Equations
11911.4.3 Secant Method
435.1 First-Order Differential Equations
12011.4.4 Fixed-Point Iteration
445.1.1 Separable Differential Equations
12111.4.5 Muller’s Method
455.1.2 Linear Differential Equations: 5.1.3 Exact Differential Equations
12211.5 Interpolation and Approximation
465.2 Second-Order Linear Differential Equations
12311.5.1 Polynomial Interpolation
475.2.1 Homogeneous Equations
12411.5.2 Spline Interpolation
485.2.2 Non-Homogeneous Equations: 5.2.3 Cauchy-Euler Equations
12511.5.3 Least Squares Approximation
495.3 Series Solutions
12611.5.4 Fourier Series Approximation
505.3.1 Power Series Solutions: 5.3.2 Frobenius Method
12711.6 Monte Carlo Methods
515.4 Laplace Transforms
12811.6.1 Monte Carlo Integration
525.4.1 Definition and Properties
12911.6.2 Monte Carlo Simulation: 11.6.3 Markov Chain Monte Carlo (MCMC) Methods
535.4.2 Solving Differential Equations: 5.4.3 Transfer Functions and System Analysis
13011.7 Applications in Physics
545.5 Systems of Differential Equations
13111.7.1 Numerical Solution of Differential Equations: 11.7.2 Molecular Dynamics Simulations
555.5.1 Linear Systems: 5.5.2 Nonlinear Systems
132Group Theory
565.6 Numerical Methods for ODEs
13312.1 Introduction to Groups
575.6.1 Euler’s Method
13412.2 Representations and Characters
585.6.2 Runge-Kutta Methods: 5.6.3 Other Numerical Methods
13512.3 Lie Groups and Lie Algebras
595.7 Applications in Physics
13612.4 Symmetries in Physics
605.7.1 Classical Mechanics
13712.5 Applications in Quantum Mechanics
615.7.2 Electromagnetism
13812.6 Applications in Particle Physics
625.7.3 Quantum Mechanics
139Mathematical Modeling
635.7.4 Relativity
14013.1 Modeling Principles
64Partial Differential Equations
14113.2 Dimensional Analysis
656.1 Introduction to PDEs
14213.3 Scaling and Perturbation Theory
666.1.1 Classification of PDEs
14313.4 Dynamical Systems
676.1.2 Initial and Boundary Conditions: 6.1.3 Well-Posed Problems
14413.5 Chaos and Fractals
686.2 Separation of Variables
14513.6 Modeling in Specific Areas of Physics
696.2.1 Solving PDEs by Separation of Variables: 6.2.2 Applications in Physics
146Special Functions
706.3 Fourier Series and Transforms
14714.1 Gamma and Beta Functions
716.3.1 Fourier Series
14814.2 Legendre Functions
726.3.2 Fourier Transforms: 6.3.3 Applications in Physics
14914.3 Bessel Functions
736.4 Wave Equation
15014.4 Hermite and Laguerre Functions
746.5 Heat Equation
15114.5 Hypergeometric Functions
756.6 Laplace’s Equation
15214.6 Applications in Physics
766.7 Numerical Methods for PDEs
153Glossary
77Calculus of Variations
154Index
