Mathematical Methods in the Physical Sciences, 3rd Edition

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ISBN: 9780471198260 | 0471198269
Cover: Hardcover
Copyright: 7/22/2005

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Now in its third edition, Mathematical Concepts in the Physical Sciences provides a comprehensive introduction to the areas of mathematical physics. It combines all the essential math concepts into one compact, clearly written reference.

Infinite Series, Power Series

(45)

The Geometric Series

(3)

Definitions and Notation

(2)

Applications of Series

(1)

Convergent and Divergent Series

(3)

Testing Series for Convergence; the Preliminary Test

(1)

Convergence Tests for Series of Positive Terms: Absolute Convergence

(7)

The Comparison Test

(1)

The Integral Test

(2)

The Ratio Test

(2)

A Special Comparison Test

(2)

Alternating Series

(1)

Conditionally Convergent Series

(1)

Useful Facts About Series

(1)

Power Series; Interval of Convergence

(3)

Theorems About Power Series

(1)

Expanding Functions in Power Series

(2)

Techniques for Obtaining Power Series Expansions

(8)

Multiplying a Series by a Polynomial or by Another Series

(1)

Division of Two Series or of a Series by a Polynomial

(1)

Binomial Series

(1)

Substitution of a Polynomial or a Series for the Variable in Another Series

(1)

Combination of Methods

(1)

Taylor Series Using the Basic Maclaurin Series

(1)

Using a Computer

(2)

Accuracy of Series Approximations

(3)

Some Uses of Series

(8)

Miscellaneous Problems

(2)

Complex Numbers

(36)

Introduction

(1)

Real and Imaginary Parts of a Complex Number

(1)

The Complex Plane

(2)

Terminology and Notation

(2)

Complex Algebra

(5)

Simplifying to x+iy form

(1)

Complex Conjugate of a Complex Expression

(1)

Finding the Absolute Value of z

(1)

Complex Equations

(1)

Graphs

(1)

Physical Applications

(1)

Complex Infinite Series

(2)

Complex Power Series; Disk of Convergence

(2)

Elementary Functions of Complex Numbers

(1)

Euler's Formula

(3)

Powers and Roots of Complex Numbers

(3)

The Exponential and Trigonometric Functions

(3)

Hyperbolic Functions

(2)

Logarithms

(1)

Complex Roots and Powers

(1)

Inverse Trigonometric and Hyperbolic Functions

(2)

Some Applications

(4)

Miscellaneous Problems

(2)

Linear Algebra

(106)

Introduction

(1)

Matrices; Row Reduction

(6)

Determinants; Cramer's Rule

(7)

Vectors

(10)

Lines and Planes

106

(8)

Matrix Operations

114

(10)

Linear Combinations, Linear Functions, Linear Operators

124

(8)

Linear Dependence and Independence

132

(5)

Special Matrices and Formulas

137

(5)

Linear Vector Spaces

142

(6)

Eigenvalues and Eigenvectors; Diagonalizing Matrices

148

(14)

Applications of Diagonalization

162

(10)

A Brief Introduction to Groups

172

(7)

General Vector Spaces

179

(5)

Miscellaneous Problems

184

(4)

Partial Differentiation

188

(53)

Introduction and Notation

188

(3)

Power Series in Two Variables

191

(2)

Total Differentials

193

(3)

Approximations using Differentials

196

(3)

Chain Rule or Differentiating a Function of a Function

199

(3)

Implicit Differentiation

202

(1)

More Chain Rule

203

(8)

Application of Partial Differentiation to Maximum and Minimum Problems

211

(3)

Maximum and Minimum Problems with Constraints; Lagrange Multipliers

214

(9)

Endpoint or Boundary Point Problems

223

(5)

Change of Variables

228

(5)

Differentiation of Integrals; Leibniz' Rule

233

(5)

Miscellaneous problems

238

(3)

Multiple Integrals

241

(35)

Introduction

241

(1)

Double and Triple Integrals

242

(7)

Applications of Integration; Single and Multiple Integrals

249

(9)

Change of Variables in Integrals; Jacobians

258

(12)

Surface Integrals

270

(3)

Miscellaneous Problems

273

(3)

Vector Analysis

276

(64)

Introduction

276

(1)

Applications of Vector Multiplication

276

(2)

Triple Products

278

(7)

Differentiation of Vectors

285

(4)

Fields

289

(1)

Directional Derivative; Gradient

290

(6)

Some Other Expressions Involving

296

(3)

Line Integrals

299

(10)

Green's Theorem in the Plane

309

(5)

The Divergence and the Divergence Theorem

314

(10)

The Curl and Stokes' Theorem

324

(12)

Miscellaneous Problems

336

(4)

Fourier Series and Transforms

340

(50)

Introduction

340

(1)

Simple Harmonic Motion and Wave Motion; Periodic Functions

340

(5)

Applications of Fourier Series

345

(2)

Average Value of a Function

347

(3)

Fourier Coefficients

350

(5)

Dirichlet Conditions

355

(3)

Complex Form of Fourier Series

358

(2)

Other Intervals

360

(4)

Even and Odd Functions

364

(8)

An Application to Sound

372

(3)

Parseval's Theorem

375

(3)

Fourier Transforms

378

(8)

Miscellaneous Problems

386

(4)

Ordinary Differential Equations

390

(82)

Introduction

390

(5)

Separable Equations

395

(6)

Linear First-Order Equations

401

(3)

Other Methods for First-Order Equations

404

(4)

Second-Order Linear Equations with Constant Coefficients and Zero Right-Hand Side

408

(9)

Second-Order Linear Equations with Constant Coefficients and Right-Hand Side Not Zero

417

(13)

Other Second-Order Equations

430

(7)

The Laplace Transform

437

(3)

Solution of Differential Equations by Laplace Transforms

440

(4)

Convolution

444

(5)

The Dirac Delta Function

449

(12)

A Brief Introduction to Green Functions

461

(5)

Miscellaneous Problems

466

(6)

Calculus of Variations

472

(24)

Introduction

472

(2)

The Euler Equation

474

(4)

Using the Euler Equation

478

(4)

The Brachistochrone Problem; Cycloids

482

(3)

Several Dependent Variables; Lagrange's Equations

485

(6)

Isoperimetric Problems

491

(2)

Variational Notation

493

(1)

Miscellaneous Problems

494

(2)

Tensor Analysis

496

(41)

Introduction

496

(2)

Cartesian Tensors

498

(4)

Tensor Notation and Operations

502

(3)

Inertia Tensor

505

(3)

Kronecker Delta and Levi-Civita Symbol

508

(6)

Pseudovectors and Pseudotensors

514

(4)

More About Applications

518

(3)

Curvilinear Coordinates

521

(4)

Vector Operators in Orthogonal Curvilinear Coordinates

525

(4)

Non-Cartesian Tensors

529

(6)

Miscellaneous Problems

535

(2)

Special Functions

537

(25)

Introduction

537

(1)

The Factorial Function

538

(1)

Definition of the Gamma Function; Recursion Relation

538

(2)

The Gamma Function of Negative Numbers

540

(1)

Some Important Formulas Involving Gamma Functions

541

(1)

Beta Functions

542

(1)

Beta Functions in Terms of Gamma Functions

543

(2)

The Simple Pendulum

545

(2)

The Error Function

547

(2)

Asymptotic Series

549

(3)

Stirling's Formula

552

(2)

Elliptic Integrals and Functions

554

(6)

Miscellaneous Problems

560

(2)

Series Solutions of Differential Equations; Legendre, Bessel, Hermite, and Laguerre Functions

562

(57)

Introduction

562

(2)

Legendre's Equation

564

(3)

Leibniz' Rule for Differentiating Products

567

(1)

Rodrigues' Formula

568

(1)

Generating Function for Legendre Polynomials

569

(6)

Complete Sets of Orthogonal Functions

575

(2)

Orthogonality of the Legendre Polynomials

577

(1)

Normalization of the Legendre Polynomials

578

(2)

Legendre Series

580

(3)

The Associated Legendre Functions

583

(2)

Generalized Power Series or the Method of Frobenius

585

(2)

Bessel's Equation

587

(3)

The Second Solution of Bessel's Equation

590

(1)

Graphs and Zeros of Bessel Functions

591

(1)

Recursion Relations

592

(1)

Differential Equations with Bessel Function Solutions

593

(2)

Other Kinds of Bessel Functions

595

(3)

The Lengthening Pendulum

598

(3)

Orthogonality of Bessel Functions

601

(3)

Approximate Formulas for Bessel Functions

604

(1)

Series Solutions; Fuchs's Theorem

605

(2)

Hermite Functions; Laguerre Functions; Ladder Operators

607

(8)

Miscellaneous Problems

615

(4)

Partial Differential Equations

619

(47)

Introduction

619

(2)

Laplace's Equation; Steady-State Temperature in a Rectangular Plate

621

(7)

The Diffusion or Heat Flow Equation; the Schrodinger Equation

628

(5)

The Wave Equation; the Vibrating String

633

(5)

Steady-state Temperature in a Cylinder

638

(6)

Vibration of a Circular Membrane

644

(3)

Steady-state Temperature in a Sphere

647

(5)

Poisson's Equation

652

(7)

Integral Transform Solutions of Partial Differential Equations

659

(4)

Miscellaneous Problems

663

(3)

Functions of a Complex Variable

666

(56)

Introduction

666

(1)

Analytic Functions

667

(7)

Contour Integrals

674

(4)

Laurent Series

678

(4)

The Residue Theorem

682

(1)

Methods of Finding Residues

683

(4)

Evaluation of Definite Integrals by Use of the Residue Theorem

687

(15)

The Point at Infinity; Residues at Infinity

702

(3)

Mapping

705

(5)

Some Applications of Conformal Mapping

710

(8)

Miscellaneous Problems

718

(4)

Probability and Statistics

722

(57)

Introduction

722

(2)

Sample Space

724

(5)

Probability Theorems

729

(7)

Methods of Counting

736

(8)

Random Variables

744

(6)

Continuous Distributions

750

(6)

Binomial Distribution

756

(5)

The Normal or Gaussian Distribution

761

(6)

The Poisson Distribution

767

(3)

Statistics and Experimental Measurements

770

(6)

Miscellaneous Problems

776

(3)

References

779

(2)

Answers to Selected Problems

781

(30)

Index

811

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Mathematical Methods in the Physical Sciences, 3rd Edition

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Table of Contents

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