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Mathematical Tools for Physics
"I wrote this text for a one semester course at the sophomore-junior level. Our experience with
students taking our junior physics courses is that even if they've had the mathematical prerequisites,
they usually need more experience using the mathematics to handle it eciently and to possess usable
intuition about the processes involved. If you've seen in nite series in a calculus course, you may have
no idea that they're good for anything. If you've taken a di erential equations course, which of the
scores of techniques that you've seen are really used a lot? The world is (at least) three dimensional so you clearly need to understand multiple integrals, but will everything be rectangular?"
Real-time TrafficCalculus | Calculus Of Variations | Complex Algebra | Differential Equations | Fourier Analysis | Fourier Series | Infinite Series | Matrices | Numerical Analysis | Partial Differential Equations | Tensors | Vector Calculus | Vector Spaces |
| Added | 16 Apr 2009 |
| Updated | 16 Apr 2009 |
| Authors | James Nearing |
Introduction
1 Basic Stuff
Trigonometry
Parametric Differentiation
Gaussian Integrals
erf and Gamma
Differentiating
Integrals
Polar Coordinates
Sketching Graphs
2 Infinite Series
The Basics
Deriving Taylor Series
Convergence
Series of Series
Power series, two variables
Stirling's Approximation
Useful Tricks
Diffraction
Checking Results
3 Complex Algebra
Complex Numbers
Some Functions
Applications of Euler's Formula
Geometry
Series of cosines
Logarithms
Mapping
4 Differential Equations
Linear Constant-Coefficient
Forced Oscillations
Series Solutions
Some General Methods
Trigonometry via ODE's
Green's Functions
Separation of Variables
Circuits
Simultaneous Equations
Simultaneous ODE's
Legendre's Equation
5 Fourier Series
Examples
Computing Fourier Series
Choice of Basis
Musical Notes
Periodically Forced ODE's
Return to Parseval
Gibbs Phenomenon
6 Vector Spaces
The Underlying Idea
Axioms
Examples of Vector Spaces
Linear Independence
Norms
Scalar Product
Bases and Scalar Products
Gram-Schmidt Orthogonalization
Cauchy-Schwartz inequality
Infinite Dimensions
7 Operators and Matrices
The Idea of an Operator
Definition of an Operator
Examples of Operators
Matrix Multiplication
Inverses
Rotations, 3-d
Areas, Volumes, Determinants
Matrices as Operators
Eigenvalues and Eigenvectors
Change of Basis
Summation Convention
Can you Diagonalize a Matrix?
Eigenvalues and Google
Special Operators
8 Multivariable Calculus
Partial Derivatives
Chain Rule
Differentials
Geometric Interpretation
Gradient
Electrostatics
Plane Polar Coordinates
Cylindrical, Spherical Coordinates
Vectors: Cylindrical, Spherical Bases
Gradient in other Coordinates
Maxima, Minima, Saddles
Lagrange Multipliers
Solid Angle
Rainbow
3D Visualization
9 Vector Calculus 1
Fluid Flow
Vector Derivatives
Computing the divergence
Integral Representation of Curl
The Gradient
Shorter Cut for div and curl
Identities for Vector Operators
Applications to Gravity
Gravitational Potential
Index Notation
More Complicated Potentials
10 Partial Differential Equations
The Heat Equation
Separation of Variables
Oscillating Temperatures
Spatial Temperature Distributions
Specified Heat Flow
Electrostatics
Cylindrical Coordinates
11 Numerical Analysis
Interpolation
Solving equations
Differentiation
Integration
Differential Equations
Fitting of Data
Euclidean Fit
Differentiating noisy data
Partial Differential Equations
12 Tensors
Examples
Components
Relations between Tensors
Birefringence
Non-Orthogonal Bases
Manifolds and Fields
Coordinate Bases
Basis Change
13 Vector Calculus 2
Integrals
Line Integrals
Gauss's Theorem
Stokes' Theorem
Reynolds' Transport Theorem
Fields as Vector Spaces
14 Complex Variables
Differentiation
Integration
Power (Laurent) Series
Core Properties
Branch Points
Cauchy's Residue Theorem
Branch Points
Other Integrals
Other Results
15 Fourier Analysis
Fourier Transform
Convolution Theorem
Time-Series Analysis
Derivatives
Green's Functions
Sine and Cosine Transforms
Wiener-Khinchine Theorem
16 Calculus of Variations
Examples
Functional Derivatives
Brachistochrone
Fermat's Principle
Electric Fields
Discrete Version
Classical Mechanics
Endpoint Variation
Kinks
Second Order
17 Densities and Distributions
Density
Functionals
Generalization
Delta-function Notation
Alternate Approach
Differential Equations
Using Fourier Transforms
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