9780521679718

Mathematical Methods for Physics and Engineering: A Comprehensive Guide

by
  • ISBN13:

    9780521679718

  • ISBN10:

    0521679710

  • Edition: 3rd
  • Format: Paperback
  • Copyright: 3/13/2006
  • Publisher: Cambridge University Press

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Summary

The third edition of this highly acclaimed undergraduate textbook is suitable for teaching all the mathematics for an undergraduate course in any of the physical sciences. As well as lucid descriptions of all the topics and many worked examples, it contains over 800 exercises. New stand-alone chapters give a systematic account of the 'special functions' of physical science, cover an extended range of practical applications of complex variables, and give an introduction to quantum operators. Further tabulations, of relevance in statistics and numerical integration, have been added. In this edition, half of the exercises are provided with hints and answers and, in a separate manual available to both students and their teachers, complete worked solutions. The remaining exercises have no hints, answers or worked solutions and can be used for unaided homework; full solutions are available to instructors on a password-protected web site, www.cambridge.org/9780521679718.

Table of Contents

Preface to the third edition xx
Preface to the second edition xxiii
Preface to the first edition xxv
Preliminary algebra
1(40)
Simple functions and equations
Polynomial equations: factorisation; properties of roots
1(9)
Trigonometric identities
Single angle; compound angles; double- and half-angle identities
10(5)
Coordinate geometry
15(3)
Partial fractions
Complications and special cases
18(7)
Binomial expansion
25(2)
Properties of binomial coefficients
27(3)
Some particular methods of proof
Proof by induction; proof by contradiction; necessary and sufficient conditions
30(6)
Exercises
36(3)
Hints and answers
39(2)
Preliminary calculus
41(42)
Differentiation
Differentiation from first principles; products; the chain rule; quotients; implicit differentiation; logarithmic differentiation; Leibnitz' theorem; special points of a function; curvature; theorems of differentiation
41(18)
Integration
Integration from first principles: the inverse of differentiation; by inspection; sinusoidal functions; logarithmic integration; using partial fractions; substitution method; integration by parts; reduction formulae; infinite and improper integrals; plane polar coordinates; integral inequalities; applications of integration
59(17)
Exercises
76(5)
Hints and answers
81(2)
Complex numbers and hyperbolic functions
83(32)
The need for complex numbers
83(2)
Manipulation of complex numbers
Addition and subtraction; modulus and argument; multiplication; complex conjugate; division
85(7)
Polar representation of complex numbers
Multiplication and division in polar form
92(3)
de Moivre's theorem
trigonometric identities; finding the nth roots of unity; solving polynomial equations
95(4)
Complex logarithms and complex powers
99(2)
Applications to differentiation and integration
101(1)
Hyperbolic functions
Definitions; hyperbolic-trigonometric analogies; identities of hyperbolic functions; solving hyperbolic equations; inverses of hyperbolic functions; calculus of hyperbolic functions
102(7)
Exercises
109(4)
Hints and answers
113(2)
Series and limits
115(36)
Series
115(1)
Summation of series
Arithmetic series; geometric series; arithmetico-geometric series; the difference method; series involving natural numbers; transformation of series
116(8)
Convergence of infinite series
Absolute and conditional convergence; series containing only real positive terms; alternating series test
124(7)
Operations with series
131(1)
Power series
Convergence of power series; operations with power series
131(5)
Taylor series
Taylor's theorem; approximation errors; standard Maclaurin series
136(5)
Evaluation of limits
141(3)
Exercises
144(5)
Hints and answers
149(2)
Partial differentiation
151(36)
Definition of the partial derivative
151(2)
The total differential and total derivative
153(2)
Exact and inexact differentials
155(2)
Useful theorems of partial differentiation
157(1)
The chain rule
157(1)
Change of variables
158(2)
Taylor's theorem for many-variable functions
160(2)
Stationary values of many-variable functions
162(5)
Stationary values under constraints
167(6)
Envelopes
173(3)
Thermodynamic relations
176(2)
Differentiation of integrals
178(1)
Exercises
179(6)
Hints and answers
185(2)
Multiple integrals
187(25)
Double integrals
187(3)
Triple integrals
190(1)
Applications of multiple integrals
Areas and volumes; masses, centres of mass and centroids; Pappus' theorems: moments of inertia; mean values of functions
191(8)
Change of variables in multiple integrals
Change of variables in double integrals; evaluation of the integral I = ∞-∞ e-x2 change of variables in triple integrals; general properties of Jacobians
199(8)
Exercises
207(4)
Hints and answers
211(1)
Vector algebra
212(29)
Scalars and vectors
212(1)
Addition and subtraction of vectors
213(1)
Multiplication by a scalar
214(3)
Basis vectors and components
217(1)
Magnitude of a vector
218(1)
Multiplication of vectors
Scalar product; vector product; scalar triple product; vector triple product
219(7)
Equations of lines, planes and spheres
226(3)
Using vectors to find distances
Point to line; point to plane; line to line; line to plane
229(4)
Reciprocal vectors
233(1)
Exercises
234(6)
Hints and answers
240(1)
Matrices and vector spaces
241(75)
Vector spaces
Basis vectors; inner product; some useful inequalities
242(5)
Linear operators
247(2)
Matrices
249(1)
Basic matrix algebra
Matrix addition; multiplication by a scalar; matrix multiplication
250(5)
Functions of matrices
255(1)
The transpose of a matrix
255(1)
The complex and Hermitian conjugates of a matrix
256(2)
The trace of a matrix
258(1)
The determinant of a matrix
Properties of determinants
259(4)
The inverse of a matrix
263(4)
The rank of a matrix
267(1)
Special types of square matrix
Diagonal; triangular; symmetric and antisymmetric; orthogonal; Hermitian and anti-Hermitian; unitary; normal
268(4)
Eigenvectors and eigenvalues
Of a normal matrix; of Hermitian and anti-Hermitian matrices; of a unitary matrix; of a general square matrix
272(8)
Determination of eigenvalues and eigenvectors
Degenerate eigenvalues
280(2)
Change of basis and similarity transformations
282(3)
Diagonalisation of matrices
285(3)
Quadratic and Hermitian forms
Stationary properties of the eigenvectors; quadratic surfaces
288(4)
Simultaneous linear equations
Range; null space; N simultaneous linear equations in N unknowns; singular value decomposition
292(15)
Exercises
307(7)
Hints and answers
314(2)
Normal modes
316(18)
Typical oscillatory systems
31(291)
Symmetry and normal modes
322(5)
Rayleigh-Ritz method
327(2)
Exercises
329(3)
Hints and answers
332(2)
Vector calculus
334(43)
Differentiation of vectors
Composite vector expressions; differential of a vector
334(5)
Integration of vectors
339(1)
Space curves
340(4)
Vector functions of several arguments
344(1)
Surfaces
345(2)
Scalar and vector fields
347(1)
Vector operators
Gradient of a scalar field; divergence of a vector field; curl of a vector field
347(7)
Vector operator formulae
Vector operators acting on sums and products; combinations of grad, div and curl
354(3)
Cylindrical and spherical polar coordinates
357(7)
General curvilinear coordinates
364(5)
Exercises
369(6)
Hints and answers
375(2)
Line, surface and volume integrals
377(38)
Line integrals
Evaluating line integrals; physical examples; line integrals with respect to a scalar
377(6)
Connectivity of regions
383(1)
Green's theorem in a plane
384(3)
Conservative fields and potentials
387(2)
Surface integrals
Evaluating surface integrals; vector areas of surfaces; physical examples
389(7)
Volume integrals
Volumes of three-dimensional regions
396(2)
Integral forms for grad, div and curl
398(3)
Divergence theorem and related theorems
Green's theorems; other related integral theorems; physical applications
401(5)
Stokes' theorem and related theorems
Related integral theorems; physical applications
406(3)
Exercises
409(5)
Hints and answers
414(1)
Fourier series
415(18)
The Dirichlet conditions
415(2)
The Fourier coefficients
417(2)
Symmetry considerations
419(1)
Discontinuous functions
420(2)
Non-periodic functions
422(2)
Integration and differentiation
424(1)
Complex Fourier series
424(2)
Parseval's theorem
426(1)
Exercises
427(4)
Hints and answers
431(2)
Integral transforms
433(35)
Fourier transforms
The uncertainty principle; Fraunhofer diffraction; the Dirac o-function; relation of the o-function to Fourier transforms; properties of Fourier transforms; odd and even functions; convolution and deconvolution; correlation functions and energy spectra; Parseval's theorem; Fourier transforms in higher dimensions
433(20)
Laplace transforms
Laplace transforms of derivatives and integrals; other properties of Laplace transforms
453(6)
Concluding remarks
459(1)
Exercises
460(6)
Hints and answers
466(2)
First-order ordinary differential equations
468(22)
General form of solution
469(1)
First-degree first-order equations
Separable-variable equations; exact equations; inexact equations, integrating factors; linear equations; homogeneous equations; isobaric equations; Bernoulli's equation; miscellaneous equations
470(10)
Higher-degree first-order equations
Equations soluble for p; for x; for y; Clairaut's equation
480(4)
Exercises
484(4)
Hints and answers
488(2)
Higher-order ordinary differential equations
490(41)
Linear equations with constant coefficients
Finding the complementary function yc(x); finding the particular integral yp(x); constructing the general solution yc(x) + yp(x); linear recurrence relations; Laplace transform method
492(11)
Linear equations with variable coefficients
The Legendre and Euler linear equations; exact equations; partially known complementary function; variation of parameters; Green's functions; canonical form for second-order equations
503(15)
General ordinary differential equations
Dependent variable absent: independent variable absent; non-linear exact equations: isobaric or homogeneous equations: equations homogeneous in x or y alone; equations having y = Aex as a solution
518(5)
Exercises
523(6)
Hints and answers
529(2)
Series solutions of ordinary differential equations
531(23)
Second-order linear ordinary differential equations
Ordinary and singular points
531(4)
Series solutions about an ordinary point
535(3)
Series solutions about a regular singular point
Distinct roots not differing by an integer; repeated root of the indicial equation; distinct roots differing by an integer
538(6)
Obtaining a second solution
The Wronskian method; the derivative method; series form of the second solution
544(4)
Polynomial solutions
548(2)
Exercises
550(3)
Hints and answers
553(1)
Eigenfunction methods for differential equations
554(23)
Sets of functions
Some useful inequalities
556(3)
Adjoint, self-adjoint and Hermitian operators
559(2)
Properties of Hermitian operators
Reality of the eigenvalues; orthogonality of the eigenfunctions; construction of real eigenfunctions
561(3)
Sturm-Liouville equations
Valid boundary conditions; putting an equation into Sturm-Liouville form
564(5)
Superposition of eigenfunctions: Green's functions
569(3)
A useful generalisation
572(1)
Exercises
573(3)
Hints and answers
576(1)
Special functions
577(71)
Legendre functions
General solution for integer f; properties of Legendre polynomials
577(10)
Associated Legendre functions
587(6)
Spherical harmonics
593(2)
Chebyshev functions
595(7)
Bessel functions
General solution for non-integer v; general solution for integer v; properties of Bessel functions
602(12)
Spherical Bessel functions
614(2)
Laguerre functions
616(5)
Associated Laguerre functions
621(3)
Hermite functions
624(4)
Hypergeometric functions
628(5)
Confluent hypergeometric functions
633(2)
The gamma function and related functions
635(5)
Exercises
640(6)
Hints and answers
646(2)
Quantum operators
648(27)
Operator formalism
Commutators
648(8)
Physical examples of operators
Uncertainty principle: angular momentum; creation and annihilation operators
656(15)
Exercises
671(3)
Hints and answers
674(1)
Partial differential equations: general and particular solutions
675(38)
Important partial differential equations
The wave equation: the diffusion equation: Laplace's equation; Poisson's equation; Schrodinger's equation
676(4)
General form of solution
680(1)
General and particular solutions
First-order equations; inhomogeneous equations and problems; second-order equations
681(12)
The wave equation
693(2)
The diffusion equation
695(4)
Characteristics and the existence of solutions
First-order equations; second-order equations
699(6)
Uniqueness of solutions
705(2)
Exercises
707(4)
Hints and answers
711(2)
Partial differential equations: separation of variables and other methods
713(62)
Separation of variables: the general method
713(4)
Superposition of separated solutions
717(8)
Separation of variables in polar coordinates
Laplace's equation in polar coordinates; spherical harmonics; other equations in polar coordinates; solution by expansion; separation of variables for inhomogeneous equations
725(22)
Integral transform methods
747(4)
Inhomogeneous problems - Green's functions
Similarities to Green's functions for ordinary differential equations: general boundary-value problems; Dirichlet problems; Neumann problems
751(16)
Exercises
767(6)
Hints and answers
773(2)
Calculus of variations
775(28)
The Euler-Lagrange equation
776(1)
Special cases
F does not contain y explicitly; F does not contain x explicitly
777(4)
Some extensions
Several dependent variables; several independent variables; higher-order derivatives; variable end-points
781(4)
Constrained variation
785(2)
Physical variational principles
Fermat's principle in optics; Hamilton's principle in mechanics
787(3)
General eigenvalue problems
790(2)
Estimation of eigenvalues and eigenfunctions
792(3)
Adjustment of parameters
795(2)
Exercises
797(4)
Hints and answers
801(2)
Integral equations
803(21)
Obtaining an integral equation from a differential equation
803(1)
Types of integral equation
804(1)
Operator notation and the existence of solutions
805(1)
Closed-form solutions
Separable kernels; integral transform methods; differentiation
806(7)
Neumann series
813(2)
Fredholm theory
815(1)
Schmidt-Hilbert theory
816(3)
Exercises
819(4)
Hints and answers
823(1)
Complex variables
824(47)
Functions of a complex variable
825(2)
The Cauchy-Riemann relations
827(3)
Power series in a complex variable
830(2)
Some elementary functions
832(3)
Multivalued functions and branch cuts
835(2)
Singularities and zeros of complex functions
837(2)
Conformal transformations
839(6)
Complex integrals
845(4)
Cauchy's theorem
849(2)
Cauchy's integral formula
851(2)
Taylor and Laurent series
853(5)
Residue theorem
858(3)
Definite integrals using contour integration
861(6)
Exercises
867(3)
Hints and answers
870(1)
Applications of complex variables
871(56)
Complex potentials
871(5)
Applications of conformal transformations
876(3)
Location of zeros
879(3)
Summation of series
882(2)
Inverse Laplace transform
884(4)
Stokes' equation and Airy integrals
888(7)
WKB methods
895(10)
Approximations to integrals
Level lines and saddle points; steepest descents; stationary phase
905(15)
Exercises
920(5)
Hints and answers
925(2)
Tensors
927(57)
Some notation
928(1)
Change of basis
929(1)
Cartesian tensors
930(2)
First- and zero-order Cartesian tensors
932(3)
Second- and higher-order Cartesian tensors
935(3)
The algebra of tensors
938(1)
The quotient law
939(2)
The tensors δij and εijk
941(3)
Isotropic tensors
944(2)
Improper rotations and pseudotensors
946(3)
Dual tensors
949(1)
Physical applications of tensors
950(4)
Integral theorems for tensors
954(1)
Non-Cartesian coordinates
955(2)
The metric tensor
957(3)
General coordinate transformations and tensors
960(3)
Relative tensors
963(2)
Derivatives of basis vectors and Christofiel symbols
965(3)
Covariant differentiation
968(3)
Vector operators in tensor form
971(4)
Absolute derivatives along curves
975(1)
Geodesies
976(1)
Exercises
977(5)
Hints and answers
982(2)
Numerical methods
984(57)
Algebraic and transcendental equations
Rearrangement of the equation: linear interpolation: binary chopping; Newton-Raphson method
985(7)
Convergence of iteration schemes
992(2)
Simultaneous linear equations
Gaussian elimination; Gauss-Seidel iteration; tridiagonal matrices
994(6)
Numerical integration
Trapezium rule; Simpson's rule; Gaussian integration; Monte Carlo methods
1000(19)
Finite differences
1019(1)
Differential equations
Difference equations; Taylor series solutions; prediction and correction; Runge-Kutta methods; isoclines
1020(8)
Higher-order equations
1028(2)
Partial differential equations
1030(3)
Exercises
1033(6)
Hints and answers
1039(2)
Group theory
1041(35)
Groups
Definition of a group; examples of groups
1041(8)
Finite groups
1049(3)
Non-Abelian groups
1052(4)
Permutation groups
1056(3)
Mappings between groups
1059(2)
Subgroups
1061(2)
Subdividing a group
Equivalence relations and classes; congruence and cosets: conjugates and classes
1063(7)
Exercises
1070(4)
Hints and answers
1074(2)
Representation theory
1076(43)
Dipole moments of molecules
1077(1)
Choosing an appropriate formalism
1078(6)
Equivalent representations
1084(2)
Reducibility of a representation
1086(4)
The orthogonality theorem for irreducible representations
1090(2)
Characters
Orthogonality property of characters
1092(3)
Counting irreps using characters
Summation rules for irreps
1095(5)
Construction of a character table
1100(2)
Group nomenclature
1102(1)
Product representations
1103(2)
Physical applications of group theory
Bonding in molecules; matrix elements in quantum mechanics; degeneracy of normal modes; breaking of degeneracies
1105(8)
Exercises
1113(4)
Hints and answers
1117(2)
Probability
1119(102)
Venn diagrams
1119(5)
Probability
Axioms and theorems; conditional probability; Bayes' theorem
1124(9)
Permutations and combinations
1133(6)
Random variables and distributions
Discrete random variables; continuous random variables
1139(4)
Properties of distributions
Mean; mode and median; variance and standard deviation; moments; central moments
1143(7)
Functions of random variables
1150(7)
Generating functions
Probability generating functions; moment generating functions; characteristic functions; cumulant generating functions
1157(11)
Important discrete distributions
Binomial; geometric; negative binomial; hypergeometric; Poisson
1168(11)
Important continuous distributions
Gaussian; log-normal; exponential; gamma; chi-squared; Cauchy; Breit-Wigner; uniform
1179(16)
The central limit theorem
1195(1)
Joint distributions
Discrete bivariate; continuous bivariate; marginal and conditional distributions
1196(3)
Properties of joint distributions
Means; variances; covariance and correlation
1199(6)
Generating functions for joint distributions
1205(1)
Transformation of variables in joint distributions
1206(1)
Important joint distributions
Multinominal; multivariate Gaussian
1207(4)
Exercises
1211(8)
Hints and answers
1219(2)
Statistics
1221(84)
Experiments, samples and populations
1221(1)
Sample statistics
Averages: variance and standard deviation; moments; covariance and correlation
1222(7)
Estimators and sampling distributions
Consistency, bias and efficiency; Fisher's inequality; standard errors; confidence limits
1229(14)
Some basic estimators
Mean; variance; standard deviation; moments; covariance and correlation
1243(12)
Maximum-likelihood method
ML estimator; transformation invariance and bias; efficiency; errors and confidence limits; Bayesian interpretation; large-N behaviour; extended ML method
1255(16)
The method of least squares
Linear least squares; non-linear least squares
1271(6)
Hypothesis testing
Simple and composite hypotheses; statistical tests; Neyman-Pearson; generalised likelihood-ratio; Student's t; Fisher's F; goodness of fit
1277(21)
Exercises
1298(5)
Hints and answers
1303(2)
Index 1305

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