Mathematical Tools for Physics

by Nearing, James

• ISBN13: 9780486477725
• ISBN10: 048647772X
• Edition: Reprint
• Format: Paperback
• Publisher: Dover Publications

Summary

Having the right answer doesn't guarantee understanding. This book helps physics students learn to take an informed and intuitive approach to solving problems. It assists undergraduates in developing their skills and provides them with grounding in important mathematical methods.

Table of Contents

Introduction	p. v
Bibliography	p. vii
Basic Stuff	p. 1
Trigonometry
Parametric Differentiation
Gaussian Integrals
erf and Gamma
Differentiating
Integrals
Polar Coordinates
Sketching Graphs
Infinite Series	p. 25
The Basics
Deriving Taylor Series
Convergence
Series of Series
Power series, two variables
Stirling's Approximation
Useful Tricks
Diffraction
Checking Results
Complex Algebra	p. 57
Complex Numbers
Some Functions
Applications of Euler's Formula
Geometry
Series of cosines
Logarithms
Mapping
Differential Equations	p. 73
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
Asymptotic Behavior
Fourier Series	p. 109
Examples
Computing Fourier Series
Choice of Basis
Musical Notes
Periodically Forced ODE's
Return to Parseval
Gibbs Phenomenon
Vector Spaces	p. 135
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
Operators and Matrices	p. 157
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
Multivariable Calculus	p. 196
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
Vector Calculus 1	p. 235
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
Partial Differential Equations	p. 268
The Heat Equation
Separation of Variables
Oscillating Temperatures
Spatial Temperature Distributions
Specified Heat Flow
Electrostatics
Cylindrical Coordinates
Numerical Analysis	p. 296
Interpolation
Solving equations
Differentiation
Integration
Differential Equations
Fitting of Data
Euclidean Fit
Differentiating noisy data
Partial Differential Equations
Tensors	p. 327
Examples
Components
Relations between Tensors
Birefringence
Non-Orthogonal Bases
Manifolds and Fields
Coordinate Bases
Basis Change
Vector Calculus 2	p. 360
Integrals
Line Integrals
Gauss's Theorem
Stokes' Theorem
Reynolds Transport Theorem
Fields as Vector Spaces
Complex Variables	p. 385
Differentiation
Integration
Power (Laurent) Series
Core Properties
Branch Points
Cauchy's Residue Theorem
Branch Points
Other Integrals
Other Results
Fourier Analysis	p. 412
Fourier Transform
Convolution Theorem
Time-Series Analysis
Derivatives
Green's Functions
Sine and Cosine Transforms
Wiener-Khinchine Theorem
Calculus of Variations	p. 427
Examples
Functional Derivatives
Brachistochrone
Fermat's Principle
Electric Fields
Discrete Version
Classical Mechanics
Endpoint Variation
Kinks
Second Order
Densities and Distributions	p. 455
Density
Functionals
Generalization
Delta-function Notation
Alternate Approach
Differential Equations
Using Fourier Transforms
More Dimensions
Index	p. 477
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