Solution 11.3.2 Mathematics Methods for Physics Arfken

From ProofWiki

Jump to navigation Jump to search

George Arfken: Mathematical Methods for Physicists (2nd Edition)

Published $\text {1970}$, Academic Press

ISBN 0-12-059832-9.

Contents

preface to the second edition

preface to the first edition

acknowledgments

introduction

Chapter 1. VECTOR ANALYSIS

1.1 Definitions, elementary approach

1.2 Rotation of coordinates

1.3 Scalar or dot product

1.4 Vector or cross product

1.5 Triple scalar product, triple vector product

1.6 Gradient, $\nabla$

1.7 Divergence, $\nabla \cdot$

1.8 Curl, $\nabla \times$

1.9 Successive applications of $\nabla$

1.10 Vector integration

1.11 Gauss's theorem

1.12 Stokes's theorem

1.13 Potential theory

1.14 Gauss's law, Poisson's equation

1.15 Helmholtz's theorem

References

Chapter 2. COORDINATE SYSTEMS

2.1 Curvilinear coordinates

2.2 Differential vector operations

2.3 Special coordinate systems -- rectangular cartesian coordinates

2.4 Spherical polar coordinates $\tuple {r, \theta, \varphi}$

2.5 Separation of variables

2.6 Circular cylindrical coordinates $\tuple {\rho, \varphi, z}$

2.7 Elliptic cylindrical coordinates $\tuple {u, v, z}$

2.8 Parabolic cylindrical coordinates $\tuple {\xi, \eta, z}$

2.9 Bipolar coordinates $\tuple {\xi, \eta, z}$

2.10 Prolate spheroidal coordinates $\tuple {u, v, \varphi}$

2.11 Oblate spheroidal coordinates $\tuple {u, v, \varphi}$

2.12 Parabolic coordinates $\tuple {\xi, \eta, \varphi}$

2.13 Toroidal coordinates $\tuple {\xi, \eta, \varphi}$

2.14 Bispherical coordinates $\tuple {\xi, \eta, \varphi}$

2.15 Confocal ellipsoidal coordinates $\tuple {\xi_1, \xi_2, \xi_3}$

2.16 Confocal coordinates $\tuple {\xi_1, \xi_2, \xi_3}$

2.17 Confocal parabolic coordinates $\tuple {\xi_1, \xi_2, \xi_3}$

References

Chapter 3. TENSOR ANALYSIS

3.1 Introduction, definitions

3.2 Contraction, direct product

3.3 Quotient rule

3.4 Pseudotensors, dual tensors

3.5 Dyadics

3.6 Theory of elasticity

3.7 Lorentz covariance of Maxwell's equations

References

Chapter 4. DETERMINANTS, MATRICES, AND GROUP THEORY

4.1 Determinants

4.2 Matrices

4.3 Orthogonal matrices

4.4 Oblique coordinates

4.5 Hermitian matrices, unitary matrices

4.6 Diagonalization of matrices

4.7 Introduction to group theory

4.8 Discrete groups

4.9 Continuous groups

4.10 Generators

4.11 $\map {\mathrm {SU} } 2$, $\map {\mathrm {SU} } 3$ and nuclear particles

4.12 Homogeneous Lorentz Group

References

Chapter 5. INFINITE SERIES

5.1 Fundamental concepts

5.2 Convergence tests

5.3 Alternating series

5.4 Algebra of series

5.5 Series of functions

5.6 Taylor's expansion

5.7 Power series

5.8 Elliptic integrals

5.9 Bernoulli numbers

5.10 Infinite products

5.11 Asymptotic or semiconvergent series

References

Chapter 6. FUNCTIONS OF A COMPLEX VARIABLE I. ANALYTIC PROPERTIES, CONFORMAL MAPPING

6.1 Complex algebra

6.2 Cauchy-Riemann conditions

6.3 Cauchy's integral theorem

6.4 Cauchy's integral formula

6.5 Laurent expansion

6.6 Mapping

6.7 Conformal mapping

6.8 Schwarz-Christoffel transformation

References

Chapter 7. FUNCTIONS OF A COMPLEX VARIABLE II. CALCULUS OF RESIDUES

7.1 Singularities

7.2 Calculus of residues

7.3 Applications of the calculus of residues

7.4 The method of steepest descents

Chapter 8. SECOND-ORDER DIFFERENTIAL EQUATIONS

8.1 Partial differential equations of theoretical physics

8.2 Separation of variables -- ordinary differential equations

8.3 Singular points

8.4 Series solutions -- Frobenius' method

8.5 A second solution

8.6 Nonhomogeneous equation -- Green's function

8.7 Numerical solutions

References

Chapter 9. STURM-LIOUVILLE THEORY -- ORTHOGONAL FUNCTIONS

9.1 Self-adjoint differential equations

9.2 Hermitian (self-adjoint) operators

9.3 Schmidt Orthogonalization

9.4 Completeness of eigenfunctions

References

Chapter 10. THE GAMMA FUNCTION (FACTORIAL FUNCTION)

10.1 Definitions, simple properties

10.2 Digamma and polygamma functions

10.3 Stirling's series

10.4 The beta function

10.5 The incomplete gamma functions and related functions

References

Chapter 11. BESSEL FUNCTIONS

11.1 Bessel functions of the first kind $\map {J_\nu} x$

11.2 Orthogonality

11.3 Neumann functions, Bessel functions of the second kind, $\map {N_\nu} x$

11.4 Hankel functions

11.5 Modified Bessel functions, $\map {I_\nu} x$ and $\map {K_\nu} x$

11.6 Asymptotic expansions

11.7 Spherical Bessel functions

References

Chapter 12. LEGENDRE FUNCTIONS

12.1 Generating function

12.2 Recurrence relations and special properties

12.3 Orthogonality

12.4 Alternate definitions of Legendre polynomials

12.5 Associated Legendre function

12.6 Spherical harmonics

12.7 Angular momentum and ladder operators

12.8 The addition theorem for spherical harmonics

12.9 Integrals of the product of three spherical harmonics

12.10 Legendre functions of the second kind, $\map {Q_n} x$

12.11 Application to spheroidal coordinate systems

12.12 Vector spherical harmonics

References

Chapter 13. SPECIAL FUNCTIONS

13.1 Hermite functions

13.2 Laguerre functions

13.3 Chebyshev (Tschebyscheff) polynomials

13.4 Hypergeometric functions

13.5 Confluent hypergeometric functions

References

Chapter 14. FOURIER SERIES

14.1 General properties

14.2 Advantages, uses of Fourier series

14.3 Applications of Fourier series

14.4 Properties of Fourier series

14.5 Gibbs phenomenon

References

Chapter 15. INTEGRAL TRANSFORMS

15.1 Integral transforms

15.2 Development of the Fourier integral

15.3 Fourier transforms -- inversion theorem

15.4 Fourier transforms of derivatives

15.5 Convolution theorem

15.6 Momentum representation

15.7 Elementary Laplace transforms

15.8 Laplace transform of derivatives

15.9 Other properties

15.10 Convolution or Faltung theorem

15.11 Inverse Laplace transformation

References

Chapter 16. INTEGRAL EQUATIONS

16.1 Introduction

16.2 Integral transforms, generating functions

16.3 Neumann series, separable (degenerate) kernels

16.4 Hilbert-Schmidt theory

16.5 Green's function -- one dimension

16.6 Green's functions -- two and three dimensions

References

Chapter 17. CALCULUS OF VARIATIONS

17.1 One dependent and one independent variable

17.2 Applications of the Euler equation

17.3 Generalizations, several dependent variables

17.4 Several independent variables

17.5 More than one dependent, more than one independent variable

17.6 Lagrangian multipliers

17.7 Variation subject to constraints

17.8 Rayleigh-Ritz variational technique

References

GENERAL REFERENCES

index

Next

Further Editions

• 1966: George Arfken: Mathematical Methods for Physicists

Source work progress

• 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $1$ Vector Analysis $1.1$ Definitions, Elementary Approach: Exercise $1.1.1$

Solution 11.3.2 Mathematics Methods for Physics Arfken

Source: https://proofwiki.org/wiki/Book:George_Arfken/Mathematical_Methods_for_Physicists/Second_Edition

Share this post