Essential Mathematical Methods for Physicists, ISE

by Weber; Arfken

• ISBN13: 9780120598779
• ISBN10: 0120598779
• Format: Hardcover
• Copyright: 2003-08-08
• Publisher: Elsevier Science

Summary

This new adaptation of Arfken and Weber's bestselling Mathematical Methods for Physicists, Fifth Editionlt;/b>, is the most comprehensive, modern, and accessible text for using mathematics to solve physics problems. Additional explanations and examples make it student-friendly and more adaptable to a course syllabus. KEY FEATURES: This is a more accessible version of Arfken and Weber's blockbuster reference, Mathematical Methods for Physicists, 5th Edition Many more detailed, worked-out examples illustrate how to use and apply mathematical techniques to solve physics problems More frequent and thorough explanations help readers understand, recall, and apply the theory New introductions and review material provide context and extra support for key ideas Many more routine problems reinforce basic concepts and computations

Table of Contents

Preface	xix
Vector Analysis	1	(95)
Elementary Approach	1	(11)
Vectors and Vector Space Summary	9	(3)
Scalar or Dot Product	12	(8)
Free Motion and Other Orbits	14	(6)
Vector or Cross Product	20	(9)
Triple Scalar Product and Triple Vector Product	29	(6)
Triple Scalar Product	29	(2)
Triple Vector Product	31	(4)
Gradient	35	(9)
Partial Derivatives	35	(5)
Gradient as a Vector Operator	40	(2)
A Geometrical Interpretation	42	(2)
Divergence	44	(3)
A Physical Interpretation	45	(2)
Curl, x	47	(6)
Successive Applications of	53	(5)
Vector Integration	58	(10)
Line Integrals	59	(3)
Surface Integrals	62	(3)
Volume Integrals	65	(1)
Integral Definitions of Gradient, Divergence, and Curl	66	(2)
Gauss's Theorem	68	(4)
Green's Theorem	70	(2)
Stokes's Theorem	72	(4)
Potential Theory	76	(6)
Scalar Potential	76	(6)
Gauss's Law and Poisson's Equation	82	(4)
Gauss's Law	82	(2)
Poisson's Equation	84	(2)
Dirac Delta Function	86	(10)
Additional Reading	95	(1)
Vector Analysis in Curved Coordinates and Tensors	96	(63)
Special Coordinate Systems	97	(1)
Rectangular Cartesian Coordinates	97	(1)
Integrals in Cartesian Coordinates	98	(1)
Circular Cylinder Coordinates	98	(15)
Integrals in Cylindrical Coordinates	101	(6)
Gradient	107	(1)
Divergence	108	(2)
Curl	110	(3)
Orthogonal Coordinates	113	(8)
Differential Vector Operators	121	(5)
Gradient	121	(1)
Divergence	122	(2)
Curl	124	(2)
Spherical Polar Coordinates	126	(10)
Integrals in Spherical Polar Coordinates	130	(6)
Tensor Analysis	136	(13)
Rotation of Coordinate Axes	137	(4)
Invariance of the Scalar Product under Rotations	141	(1)
Covariance of Cross Product	142	(1)
Covariance of Gradient	143	(1)
Definition of Tensors of Rank Two	144	(1)
Addition and Subtraction of Tensors	145	(1)
Summation Convention	145	(1)
Symmetry--Antisymmetry	146	(1)
Spinors	147	(2)
Contraction and Direct Product	149	(2)
Contraction	149	(1)
Direct Product	149	(2)
Quotient Rule	151	(2)
Dual Tensors	153	(6)
Levi--Civita Symbol	153	(1)
Dual Tensors	154	(3)
Additional Reading	157	(2)
Determinants and Matrices	159	(70)
Determinants	159	(15)
Linear Equations: Examples	159	(1)
Homogeneous Linear Equations	160	(1)
Inhomogeneous Linear Equations	161	(3)
Laplacian Development by Minors	164	(2)
Antisymmetry	166	(8)
Matrices	174	(19)
Basic Definitions, Equality, and Rank	174	(1)
Matrix Multiplication, Inner Product	175	(3)
Dirac Bra-ket, Transposition	178	(1)
Multiplication (by a Scalar)	178	(1)
Addition	179	(1)
Product Theorem	180	(2)
Direct Product	182	(1)
Diagonal Matrices	182	(2)
Trace	184	(1)
Matrix Inversion	184	(9)
Orthogonal Matrices	193	(13)
Direction Cosines	194	(1)
Applications to Vectors	195	(3)
Orthogonality Conditions: Two-Dimensional Case	198	(2)
Euler Angles	200	(2)
Symmetry Properties and Similarity Transformations	202	(2)
Relation to Tensors	204	(2)
Hermitian Matrices and Unitary Matrices	206	(5)
Definitions	206	(2)
Pauli Matrices	208	(3)
Diagonalization of Matrices	211	(18)
Moment of Inertia Matrix	211	(1)
Eigenvectors and Eigenvalues	212	(2)
Hermitian Matrices	214	(2)
Anti-Hermitian Matrices	216	(2)
Normal Modes of Vibration	218	(2)
Ill-Conditioned Systems	220	(1)
Functions of Matrices	221	(7)
Additional Reading	228	(1)
Group Theory	229	(28)
Introduction to Group Theory	229	(8)
Definition of Group	230	(4)
Homomorphism and Isomorphism	234	(1)
Matrix Representations: Reducible and Irreducible	234	(3)
Generators of Continuous Groups	237	(6)
Rotation Groups SO(2) and SO(3)	238	(1)
Rotation of Functions and Orbital Angular Momentum	239	(1)
Special Unitary Group SU(2)	240	(3)
Orbital Angular Momentum	243	(5)
Ladder Operator Approach	244	(4)
Homogeneous Lorentz Group	248	(9)
Vector Analysis in Minkowski Space-Time	251	(4)
Additional Reading	255	(2)
Infinite Series	257	(61)
Fundamental Concepts	257	(5)
Addition and Subtraction of Series	260	(2)
Convergence Tests	262	(7)
Comparison Test	262	(1)
Cauchy Root Test	263	(1)
d'Alembert or Cauchy Ratio Test	263	(1)
Cauchy or Maclaurin Integral Test	264	(5)
Alternating Series	269	(5)
Leibniz Criterion	270	(1)
Absolute and Conditional Convergence	271	(3)
Algebra of Series	274	(2)
Multiplication of Series	275	(1)
Series of Functions	276	(5)
Uniform Convergence	276	(2)
Weierstrass M (Majorant) Test	278	(1)
Abel's Test	279	(2)
Taylor's Expansion	281	(10)
Maclaurin Theorem	283	(1)
Binomial Theorem	284	(2)
Taylor Expansion---More Than One Variable	286	(5)
Power Series	291	(5)
Convergence	291	(1)
Uniform and Absolute Convergence	291	(1)
Continuity	292	(1)
Differentiation and Integration	292	(1)
Uniqueness Theorem	292	(1)
Inversion of Power Series	293	(3)
Elliptic Integrals	296	(6)
Definitions	297	(1)
Series Expansion	298	(2)
Limiting Values	300	(2)
Bernoulli Numbers and the Euler-Maclaurin Formula	302	(12)
Bernoulli Functions	305	(1)
Euler-Maclaurin Integration Formula	306	(1)
Improvement of Convergence	307	(2)
Improvement of Convergence by Rational Approximations	309	(5)
Asymptotic Series	314	(4)
Error Function	314	(3)
Additional Reading	317	(1)
Functions of a Complex Variable I	318	(54)
Complex Algebra	319	(12)
Complex Conjugation	321	(4)
Functions of a Complex Variable	325	(6)
Cauchy-Riemann Conditions	331	(6)
Analytic Functions	335	(2)
Cauchy's Integral Theorem	337	(7)
Contour Integrals	337	(2)
Stokes's Theorem Proof of Cauchy's Integral Theorem	339	(2)
Multiply Connected Regions	341	(3)
Cauchy's Integral Formula	344	(6)
Derivatives	346	(1)
Morera's Theorem	346	(4)
Laurent Expansion	350	(10)
Taylor Expansion	350	(1)
Schwarz Reflection Principle	351	(1)
Analytic Continuation	352	(2)
Laurent Series	354	(6)
Mapping	360	(8)
Translation	360	(1)
Rotation	361	(1)
Inversion	361	(2)
Branch Points and Multivalent Functions	363	(5)
Conformal Mapping	368	(4)
Additional Reading	370	(2)
Functions of a Complex Variable II	372	(38)
Singularities	372	(6)
Poles	373	(1)
Branch Points	374	(4)
Calculus of Residues	378	(22)
Residue Theorem	378	(1)
Evaluation of Definite Integrals	379	(5)
Cauchy Principal Value	384	(6)
Pole Expansion of Meromorphic Functions	390	(2)
Product Expansion of Entire Functions	392	(8)
Method of Steepest Descents	400	(10)
Analytic Landscape	400	(2)
Saddle Point Method	402	(7)
Additional Reading	409	(1)
Differential Equations	410	(72)
Introduction	410	(1)
First-Order ODEs	411	(13)
Separable Variables	411	(2)
Exact Differential Equations	413	(1)
Linear First-Order ODEs	414	(4)
ODEs of Special Type	418	(6)
Second-Order ODEs	424	(15)
Inhomogeneous Linear ODEs and Particular Solutions	430	(1)
Inhomogeneous Euler ODE	430	(1)
Inhomogeneous ODE with Constant Coefficients	431	(3)
Linear Independence of Solutions	434	(5)
Singular Points	439	(2)
Series Solutions---Frobenius's Method	441	(13)
Expansion about xo	445	(1)
Symmetry of ODE and Solutions	445	(1)
Limitations of Series Approach---Bessel's Equation	446	(2)
Regular and Irregular Singularities	448	(2)
Fuchs's Theorem	450	(1)
Summary	450	(4)
A Second Solution	454	(10)
Series Form of the Second Solution	456	(8)
Numerical Solutions	464	(6)
First-Order Differential Equations	464	(1)
Taylor Series Solution	464	(2)
Runge--Kutta Method	466	(1)
Predictor--Corrector Methods	467	(1)
Second-Order ODEs	468	(2)
Introduction to Partial Differential Equations	470	(1)
Separation of Variables	470	(12)
Cartesian Coordinates	471	(3)
Circular Cylindrical Coordinates	474	(2)
Spherical Polar Coordinates	476	(4)
Additional Reading	480	(2)
Sturm--Liouville Theory---Orthogonal Functions	482	(41)
Self-Adjoint ODEs	483	(13)
Eigenfunctions and Eigenvalues	485	(5)
Boundary Conditions	490	(1)
Hermitian Operators	490	(2)
Hermitian Operators in Quantum Mechanics	492	(4)
Hermitian Operators	496	(7)
Real Eigenvalues	496	(2)
Orthogonal Eigenfunctions	498	(1)
Expansion in Orthogonal Eigenfunctions	499	(2)
Degeneracy	501	(2)
Gram-Schmidt Orthogonalization	503	(7)
Orthogonal Polynomials	507	(3)
Completeness of Eigenfunctions	510	(13)
Bessel's Inequality	512	(1)
Schwarz Inequality	513	(2)
Summary of Vector Spaces---Completeness	515	(3)
Expansion (Fourier) Coefficients	518	(4)
Additional Reading	522	(1)
The Gamma Function (Factorial Function)	523	(29)
Definitions and Simple Properties	523	(12)
Infinite Limit (Euler)	523	(1)
Definite Integral (Euler)	524	(2)
Infinite Product (Weierstrass)	526	(2)
Factorial Notation	528	(2)
Double Factorial Notation	530	(1)
Integral Representation	531	(4)
Digamma and Polygamma Functions	535	(5)
Digamma Function	535	(1)
Polygamma Function	536	(1)
Maclaurin Expansion, Computation	537	(1)
Series Summation	537	(3)
Stirling's Series	540	(5)
Derivation from Euler--Maclaurin Integration Formula	540	(1)
Stirling's Series	541	(1)
Numerical Computation	542	(3)
The Incomplete Gamma Functions and Related Functions	545	(7)
Exponential Integral	546	(2)
Error Integrals	548	(2)
Additional Reading	550	(2)
Legendre Polynomials and Spherical Harmonics	552	(37)
Introduction	552	(11)
Physical Basis: Electrostatics	552	(1)
Generating Function	553	(3)
Power Series	556	(2)
Linear Electric Multipoles	558	(1)
Vector Expansion	559	(4)
Recurrence Relations and Special Properties	563	(5)
Recurrence Relations	563	(1)
Differential Equations	564	(2)
Upper and Lower Bounds for Pn (cos θ)	566	(2)
Orthogonality	568	(11)
Expansion of Functions, Legendre Series	569	(10)
Alternate Definitions of Legendre Polynomials	579	(2)
Rodrigues's Formula	579	(2)
Associated Legendre Functions	581	(8)
Spherical Harmonics	584	(4)
Additional Reading	588	(1)
Bessel Functions	589	(49)
Bessel Functions of the First Kind, Jv(x)	589	(22)
Generating Function for Integral Order	590	(3)
Applications of Recurrence Relations	593	(1)
Bessel's Differential Equation	594	(1)
Integral Representations	595	(4)
Orthogonality	599	(1)
Normalization	600	(1)
Bessel Series	600	(8)
Bessel Functions of Nonintegral Order	608	(3)
Neumann Functions, Bessel Functions of the Second Kind	611	(6)
Definition and Series Form	611	(2)
Other Forms	613	(1)
Recurrence Relations	613	(1)
Wronskian Formulas	613	(4)
Asymptotic Expansions	617	(7)
Expansion of an Integral Representation	618	(4)
Numerical Evaluation	622	(2)
Spherical Bessel Functions	624	(14)
Definitions	624	(3)
Limiting Values	627	(2)
Recurrence Relations	629	(1)
Numerical Computation	630	(7)
Additional Reading	637	(1)
Hermite and Laguerre Polynomials	638	(25)
Hermite Polynomials	638	(12)
Quantum Mechanical Simple Harmonic Oscillator	638	(1)
Raising and Lowering Operators	639	(4)
Recurrence Relations and Generating Function	643	(2)
Alternate Representations	645	(1)
Orthogonality	646	(4)
Laguerre Functions	650	(13)
Differential Equation---Laguerre Polynomials	650	(5)
Associated Laguerre Polynomials	655	(7)
Additional Reading	662	(1)
Fourier Series	663	(26)
General Properties	663	(8)
Completeness	664	(2)
Behavior of Discontinuities	666	(5)
Advantages and Uses of Fourier Series	671	(6)
Periodic Functions	671	(3)
Change of Interval	674	(3)
Complex Fourier Series	677	(6)
Abel's Theorem	678	(5)
Properties of Fourier Series	683	(6)
Convergence	683	(1)
Integration	684	(1)
Differentiation	685	(3)
Additional Reading	688	(1)
Integral Transforms	689	(67)
Introduction and Definitions	689	(1)
Linearity	689	(1)
Fourier Transform	690	(4)
Laplace Transform	693	(1)
Development of the Inverse Fourier Transform	694	(4)
Inverse Fourier Transform---Exponential Form	695	(1)
Dirac Delta Function Derivation	696	(2)
Fourier Transforms---Inversion Theorem	698	(8)
Exponential Transform	698	(1)
Cosine Transform	699	(1)
Sine Transform	700	(6)
Fourier Transform of Derivatives	706	(6)
Convolution Theorem	712	(6)
Parseval's Relation	715	(3)
Momentum Representation	718	(6)
Laplace Transforms	724	(6)
Definition	724	(2)
Inverse Transform	726	(1)
Partial Fraction Expansion	726	(3)
Numerical Inversion	729	(1)
Laplace Transform of Derivatives	730	(4)
Dirac Delta Function	732	(2)
Other Properties	734	(8)
Substitution	734	(1)
RLC Analog	735	(1)
Translation	736	(1)
Derivative of a Transform	737	(1)
Integration of Transforms	738	(1)
Limits of Integration---Unit Step Function	738	(4)
Convolution or Faltungs Theorem	742	(4)
Inverse Laplace Transform	746	(10)
Bromwich Integral	746	(6)
Summary: Inversion of Laplace Transform	752	(2)
Additional Reading	754	(2)
Partial Differential Equations	756	(26)
Examples of Partial Differential Equations and Boundary Conditions	756	(4)
Boundary Conditions	758	(2)
Heat Flow or Diffusion PDE	760	(9)
Alternate Solutions	763	(6)
Inhomogeneous PDE---Green's Function	769	(13)
Additional Reading	780	(2)
Probability	782	(44)
Definitions, Simple Properties	782	(7)
Counting of Permutations and Combinations	787	(2)
Random Variables	789	(13)
Binomial Distribution	802	(2)
Poisson Distribution	804	(3)
Gauss's Normal Distribution	807	(5)
Statistics	812	(14)
Error Propagation	812	(3)
Fitting Curves to Data	815	(2)
The x2 Distribution	817	(4)
The Student t Distribution	821	(4)
Additional Reading	825	(1)
Calculus of Variations	826	(41)
Uses of the Calculus of Variations	826	(1)
A Dependent and an Independent Variable	827	(10)
Concept of Variation	827	(10)
Several Dependent Variables	837	(8)
Transversality Condition	839	(2)
Hamilton's Principle	841	(4)
Several Independent Variables	845	(2)
Several Dependent and Independent Variables	847	(1)
Relation to Physics	848	(1)
Lagrangian Multipliers: Variation with Constraints	848	(13)
Variation with Constraints	850	(3)
Lagrangian Equations	853	(8)
Rayleigh-Ritz Variational Technique	861	(6)
Ground-State Eigenfunction	862	(4)
Additional Reading	866	(1)
Nonlinear Methods and Chaos	867	(38)
Introduction	867	(2)
The Logistic Map	869	(5)
Sensitivity to Initial Conditions and Parameters	874	(4)
Lyapunov Exponents	874	(1)
Fractals	875	(3)
Nonlinear Differential Equations	878	(27)
Bernoulli and Riccati Equations	879	(2)
Fixed and Movable Singularities, Special Solutions	881	(1)
Autonomous Differential Equations	882	(3)
Local and Global Behavior in Higher Dimensions	885	(11)
Dissipation in Dynamical Systems	896	(2)
Bifurcations in Dynamical Systems	898	(2)
Chaos in Dynamical Systems	900	(1)
Routes to Chaos in Dynamical Systems	901	(2)
Additional Reading	903	(2)
APPENDIX 1 REAL ZEROS OF A FUNCTION	905	(6)
Bisection Method	905	(2)
Three Warnings	907	(1)
Additional Reading	908	(1)
General References	908	(3)
Index	911