Mathematical Tools for Physics

7 Operators and Matrices.......... 168
The Idea of an Operator
Definition of an Operator
Examples of Operators
Matrix Multiplication
Inverses
Areas, Volumes, Determinants
Matrices as Operators
Eigenvalues and Eigenvectors
Change of Basis
Summation Convention
Can you Diagonalize a Matrix?
Eigenvalues and Google

8 Multivariable Calculus........... 208
Partial Derivatives
Differentials
Chain Rule
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............. 248
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

Summation Convention

More Complicated Potentials

10 Partial Differential Equations....... 283

The Heat Equation

Separation of Variables

Oscillating Temperatures

Spatial Temperature Distributions

Specified Heat Flow

Electrostatics

11 Numerical Analysis............ 315

Interpolation

Solving equations

Differentiation

Integration

Differential Equations

Fitting of Data

Euclidean Fit

Differentiating noisy data

Partial Differential Equations

12 Tensors.................. 354

Examples

Components

Relations between Tensors

Non-Orthogonal Bases

Manifolds and Fields

Coordinate Systems

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