Mathematical methods for physics and engineering - Riley, Angielskie techniczne
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Contents
Preface to the second edition
xix
Preface to the first edition
xxi
1
Preliminary algebra
1
1.1 Simple functions and equations
1
Polynomial equations; factorisation; properties of roots
1.2 Trigonometric identities
10
Single angle; compound-angles; double- and half-angle identities
1.3 Coordinate geometry
15
1.4 Partial fractions
18
Complications and special cases; complex roots; repeated roots
1.5 Binomial expansion
25
1.6 Properties of binomial coecients
27
1.7 Some particular methods of proof
30
Methods of proof; by induction; by contradiction; necessary and sucient
conditions
1.8 Exercises
36
1.9 Hints and answers
39
2
Preliminary calculus
42
2.1 Differentiation
42
Differentiation from first principles; products; the chain rule; quotients;
implicit differentiation; logarithmic differentiation; Leibniz’ theorem; special
points of a function; theorems of differentiation
v
CONTENTS
2.2 Integration
60
Integration from first principles; the inverse of differentiation; integration
by inspection; sinusoidal functions; logarithmic integration; integration
using partial fractions; substitution method; integration by parts; reduction
formulae; infinite and improper integrals; plane polar coordinates; integral
inequalities; applications of integration
2.3 Exercises
77
2.4 Hints and answers
82
3
Complex numbers and hyperbolic functions
86
3.1 The need for complex numbers
86
3.2 Manipulation of complex numbers
88
Addition and subtraction; modulus and argument; multiplication; complex
conjugate; division
3.3 Polar representation of complex numbers
95
Multiplication and division in polar form
3.4 de Moivre’s theorem
98
trigonometric identities; finding the
n
th roots of unity; solving polynomial
equations
3.5 Complex logarithms and complex powers
102
3.6 Applications to differentiation and integration
104
3.7 Hyperbolic functions
105
Definitions; hyperbolic–trigonometric analogies; identities of hyperbolic
functions; solving hyperbolic equations; inverses of hyperbolic functions;
calculus of hyperbolic functions
3.8 Exercises
112
3.9 Hints and answers
116
4
Series and limits
118
4.1 Series
118
4.2 Summation of series
119
Arithmetic series; geometric series; arithmetico-geometric series; the
difference method; series involving natural numbers; transformation of series
4.3 Convergence of infinite series
127
Absolute and conditional convergence; convergence of a series containing
only real positive terms; alternating series test
4.4 Operations with series
134
4.5 Power series
134
Convergence of power series; operations with power series
4.6 Taylor series
139
Taylor’s theorem; approximation errors in Taylor series; standard Maclaurin
series
vi
CONTENTS
4.7 Evaluation of limits
144
4.8 Exercises
147
4.9 Hints and answers
152
5
Partial differentiation
154
5.1 Definition of the partial derivative
154
5.2 The total differential and total derivative
156
5.3 Exact and inexact differentials
158
5.4 Useful theorems of partial differentiation
160
5.5 The chain rule
160
5.6 Change of variables
161
5.7 Taylor’s theorem for many-variable functions
163
5.8 Stationary values of many-variable functions
165
5.9 Stationary values under constraints
170
5.10 Envelopes
176
5.11 Thermodynamic relations
179
5.12 Differentiation of integrals
181
5.13 Exercises
182
5.14 Hints and answers
188
6
Multiple integrals
190
6.1 Double integrals
190
6.2 Triple integrals
193
6.3 Applications of multiple integrals
194
Areas and volumes; masses, centres of mass and centroids; Pappus’
theorems; moments of inertia; mean values of functions
6.4 Change of variables in multiple integrals
202
Change of variables in double integrals; evaluation of the integral
I
=
e
−
x
2
dx
; change of variables in triple integrals; general properties of
Jacobians
6.5 Exercises
∞
−∞
210
6.6 Hints and answers
214
7
Vector algebra
216
7.1 Scalars and vectors
216
7.2 Addition and subtraction of vectors
217
7.3 Multiplication by a scalar
218
7.4 Basis vectors and components
221
7.5 Magnitude of a vector
222
7.6 Multiplication of vectors
223
Scalar product; vector product; scalar triple product; vector triple product
vii
CONTENTS
7.7 Equations of lines, planes and spheres
230
Equation of a line; equation of a plane
7.8 Using vectors to find distances
233
Point to line; point to plane; line to line; line to plane
7.9 Reciprocal vectors
237
7.10 Exercises
238
7.11 Hints and answers
244
8
Matrices and vector spaces
246
8.1 Vector spaces
247
Basis vectors; the inner product; some useful inequalities
8.2 Linear operators
252
Properties of linear operators
8.3 Matrices
254
Matrix addition and multiplication by a scalar; multiplication of matrices
8.4 Basic matrix algebra
255
8.5 Functions of matrices
260
8.6 The transpose of a matrix
260
8.7 The complex and Hermitian conjugates of a matrix
261
8.8 The trace of a matrix
263
8.9 The determinant of a matrix
264
Properties of determinants
8.10 The inverse of a matrix
268
8.11 The rank of a matrix
272
8.12 Special types of square matrix
273
Diagonal; symmetric and antisymmetric; orthogonal; Hermitian; unitary;
normal
8.13 Eigenvectors and eigenvalues
277
Of a normal matrix; of Hermitian and anti-Hermitian matrices; of a unitary
matrix; of a general square matrix
8.14 Determination of eigenvalues and eigenvectors
285
Degenerate eigenvalues
8.15 Change of basis and similarity transformations
288
8.16 Diagonalisation of matrices
290
8.17 Quadratic and Hermitian forms
293
The stationary properties of the eigenvectors; quadratic surfaces
8.18 Simultaneous linear equations
297
N
simultaneous linear equations in
N
unknowns
8.19 Exercises
312
8.20 Hints and answers
319
viii
CONTENTS
9
Normal modes
322
9.1 Typical oscillatory systems
323
9.2 Symmetry and normal modes
328
9.3 Rayleigh–Ritz method
333
9.4 Exercises
335
9.5 Hints and answers
338
10 Vector calculus
340
10.1 Differentiation of vectors
340
Composite vector expressions; differential of a vector
10.2 Integration of vectors
345
10.3 Space curves
346
10.4 Vector functions of several arguments
350
10.5 Surfaces
351
10.6 Scalar and vector fields
353
10.7 Vector operators
353
Gradient of a scalar field; divergence of a vector field; curl of a vector field
10.8 Vector operator formulae
360
Vector operators acting on sums and products; combinations of grad, div
and curl
10.9 Cylindrical and spherical polar coordinates
363
Cylindrical polar coordinates; spherical polar coordinates
10.10 General curvilinear coordinates
370
10.11 Exercises
375
10.12 Hints and answers
381
11 Line, surface and volume integrals
383
11.1 Line integrals
383
Evaluating line integrals; physical examples of line integrals; line integrals
with respect to a scalar
11.2 Connectivity of regions
389
11.3 Green’s theorem in a plane
390
11.4 Conservative fields and potentials
393
11.5 Surface integrals
395
Evaluating surface integrals; vector areas of surfaces; physical examples of
surface integrals
11.6 Volume integrals
402
Volumes of three-dimensional regions
11.7 Integral forms for grad, div and curl
404
11.8 Divergence theorem and related theorems
407
Green’s theorems; other related integral theorems; physical applications of
the divergence theorem
ix
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