Advanced Engineering Mathematics / Edition 10

by Erwin Kreyszig

Overview

The content and character of mathematics needed in applications are changing rapidly. Introduces students of engineering, physics, mathematics and computer science to those areas that are vital to address practical problems. The Seventh Edition offers a self-contained treatment of ordinary differential equations, linear algebra, vector calculus, fourier analysis and partial differential equations, complex analysis, numerical methods, optimization and graphs, probability and statistics. New in this edition are: many sections rewritten to increase readability; problems have been revised and more closely related to examples; instructors manual quadrupled in content; improved balance between applications, algorithmic ideas and theory; reorganized differential equations and linear algebra sections; added and improved examples throughout.

Product Details

ISBN-13:	9780470458365
Publisher:	Wiley
Publication date:	12/08/2010
Pages:	1280
Product dimensions:	8.40(w) x 10.10(h) x 1.90(d)

Table of Contents

	Introduction, General Commands	1
Part A.	Ordinary Differential Equations (ODE's)	6
Chapter 1	First-Order ODE's	7
Ex. 1.1	General Solutions	7
Ex. 1.2	Direction Fields	8
Ex. 1.3	Mixing Problems	10
Ex. 1.4	Integrating Factors	11
Ex. 1.5	Bernoulli's Equation	12
Ex. 1.6	RL-Circuit	13
	Problems for Chapter 1	14
Chapter 2	Linear ODE's of Second and Higher Order	17
Ex. 2.1	General Solution. Initial Value Problem	17
Ex. 2.2	Mass-Spring System. Complex Roots. Damped Oscillations	19
Ex. 2.3	The Three Cases of Damping	19
Ex. 2.4	The Three Cases for an Euler-Cauchy Equation	21
Ex. 2.5	Wronskian	22
Ex. 2.6	Nonhomogeneous Linear ODE's	23
Ex. 2.7	Solution by Undetermined Coefficients	24
Ex. 2.8	Solution by Variation of Parameters	25
Ex. 2.9	Forced Vibrations. Resonance. Beats	26
Ex. 2.10	RLC-Circuit	27
	Problems for Chapter 2	28
Chapter 3	Systems of Differential Equations. Phase Plane, Qualitative Methods	31
Ex. 3.1	Solving a System of ODE's by DSolve. Initial Value Problem	31
Ex. 3.2	Use of Matrices in Solving Systems of ODE's	33
Ex. 3.3	Critical Points. Node	34
Ex. 3.4	Proper Node, Saddle Point, Center, Spiral Point	36
Ex. 3.5	Pendulum Equation	37
Ex. 3.6	Nonhomogeneous System	38
Ex. 3.7	Method of Undetermined Coefficients	39
	Problems for Chapter 3	40
Chapter 4	Series Solutions of Differential Equations	42
Ex. 4.1	Power Series Solutions. Plots from Them. Numerical Values	43
Ex. 4.2	Legendre Polynomials	44
Ex. 4.3	Legendre's Differential Equation	45
Ex. 4.4	Orthogonality. Fourier-Legendre Series	47
Ex. 4.5	Frobenius Method	48
Ex. 4.6	Bessel's Equation. Bessel Functions	50
	Problems for Chapter 4	52
Chapter 5	Laplace Transform Method for Solving ODE's	55
Ex. 5.1	Transforms and Inverse Transforms	55
Ex. 5.2	Differential Equations	56
Ex. 5.3	Forced Vibrations. Resonance	58
Ex. 5.4	Unit Step Function (Heaviside Function), Dirac's Delta	59
Ex. 5.5	Solution of Systems by Laplace Transform	61
Ex. 5.6	Formulas on General Properties of the Laplace Transform	62
	Problems for Chapter 5	64
Part B.	Linear Algebra, Vector Calculus	66
Chapter 6	Matrices, Vectors, Determinants. Linear Systems of Equations	66
Ex. 6.1	Matrix Addition, Scalar Multiplication, Matrix Multiplication. Vectors	66
Ex. 6.2	Special Matrices	69
Ex. 6.3	Changing and Composing Matrices, Accessing Entries. Submatrices	70
Ex. 6.4	Solution of a Linear System	73
Ex. 6.5	Linear Systems: A Further Case	74
Ex. 6.6	Gauss Elimination; Back Substitution	75
	Problems for Chapter 6	77
Chapter 7	Matrix Eigenvalue Problems	79
Ex. 7.1	Eigenvalues, Eigenvectors, Accessing Spectrum	79
Ex. 7.2	Real Matrices with Complex Eigenvalues	81
Ex. 7.3	Orthogonal Matrices and Transformations	81
Ex. 7.4	Complex Matrices	83
Ex. 7.5	Similarity of Matrices. Diagonalization	84
	Problems for Chapter 7	86
Chapter 8	Vectors in R[superscript 2] and R[superscript 3]. Dot and Cross Products. Grad, Div, Curl	88
Ex. 8.1	Vectors, Addition, Scalar Multiplication	89
Ex. 8.2	Inner Product. Cross Product	89
Ex. 8.3	Differentiation of Vectors. Curves and their Properties	91
Ex. 8.4	Gradient. Directional Derivative. Potential	92
Ex. 8.5	Divergence, Laplacian, Curl	94
	Problems for Chapter 8	95
Chapter 9	Vector Integral Calculus. Integral Theorems	98
Ex. 9.1	Line Integrals	98
Ex. 9.2	Independence of Path	99
Ex. 9.3	Double Integrals. Moments of Inertia	100
Ex. 9.4	Green's Theorem in the Plane	101
Ex. 9.5	Surface Integrals. Flux	103
Ex. 9.6	Divergence Theorem of Gauss	104
Ex. 9.7	Stokes's Theorem	106
	Problems for Chapter 9	107
Part C.	Fourier Analysis and Partial Differential Equations	110
Chapter 10	Fourier Series, Integrals, and Transforms	110
Ex. 10.1	Functions of Period 2[pi]. Even Functions. Gibbs Phenomenon	111
Ex. 10.2	Functions of Arbitrary Period. Odd Functions	112
Ex. 10.3	Half-Range Expansions	114
Ex. 10.4	Rectifier	117
Ex. 10.5	Trigonometric Approximation. Minimum Square Error	117
Ex. 10.6	Fourier Integral, Fourier Transform	118
	Problems for Chapter 10	119
Chapter 11	Partial Differential Equations (PDE's)	122
Ex. 11.1	Wave Equation. Separation of Variables. Animation	122
Ex. 11.2	One-Dimensional Heat Equation	124
Ex. 11.3	Heat Equation, Laplace Equation	125
Ex. 11.4	Rectangular Membrane. Double Fourier Series	127
Ex. 11.5	Laplacian. Circular Membrane. Bessel Equation	128
	Problems for Chapter 11	131
Part D.	Complex Analysis	133
Chapter 12	Complex Numbers and Functions. Conformal Mapping	133
Ex. 12.1	Complex Numbers. Polar Form. Plotting	133
Ex. 12.2	Equations. Roots. Sets in the Complex Plane	136
Ex. 12.3	Cauchy-Riemann Equations. Harmonic Functions	138
Ex. 12.4	Conformal Mapping	140
Ex. 12.5	Exponential, Trigonometric, and Hyperbolic Functions	143
Ex. 12.6	Complex Logarithm	145
	Problems for Chapter 12	147
Chapter 13	Complex Integration	149
Ex. 13.1	Indefinite Integration of Analytic Functions	149
Ex. 13.2	Integration: Use of Path. Path Dependence	149
Ex. 13.3	Contour Integration by Cauchy's Integral Theorem and Formula	151
	Problems for Chapter 13	153
Chapter 14	Power Series, Taylor Series	154
Ex. 14.1	Sequences and their Plots	154
Ex. 14.2	Convergence Tests for Complex Series	155
Ex. 14.3	Power Series. Radius of Convergence	156
Ex. 14.4	Taylor Series	156
Ex. 14.5	Uniform Convergence	158
	Problems for Chapter 14	160
Chapter 15	Laurent Series. Residue Integration	162
Ex. 15.1	Laurent Series	162
Ex. 15.2	Singularities and Zeros	163
Ex. 15.3	Residue Integration	164
Ex. 15.4	Real Integrals of Rational Functions of cos and sin	166
Ex. 15.5	Improper Real Integrals of Rational Functions	167
	Problems for Chapter 15	169
Chapter 16	Complex Analysis in Potential Theory	171
Ex. 16.1	Complex Potential. Related Plots	171
Ex. 16.2	Use of Conformal Mapping	172
Ex. 16.3	Fluid Flow	173
Ex. 16.4	Series Representation of Potential	175
Ex. 16.5	Mean Value Theorem for Analytic Functions	176
	Problems for Chapter 16	177
Part E.	Numerical Methods	179
Chapter 17	Numerical Methods in General	179
Ex. 17.1	Loss of Significant Digits. Quadratic Equation	179
Ex. 17.2	Fixed-Point Iteration	180
Ex. 17.3	Solving Equations by Newton's Method	182
Ex. 17.4	Solving Equations by the Secant Method	183
Ex. 17.5	Solving Equations by the Bisection Method. Module	183
Ex. 17.6	Polynomial Interpolation	185
Ex. 17.7	Spline Interpolation	186
Ex. 17.8	Numerical Integration	188
	Problems for Chapter 17	189
Chapter 18	Numerical Linear Algebra	191
Ex. 18.1	Gauss Elimination. Pivoting	191
Ex. 18.2	Doolittle LU-Factorization	193
Ex. 18.3	Cholesky Factorization	195
Ex. 18.4	Gauss-Jordan Elimination. Matrix Inversion	196
Ex. 18.5	Gauss-Seidel Iteration for Linear Systems	198
Ex. 18.6	Vector and Matrix Norms. Condition Numbers	199
Ex. 18.7	Fitting Data by Least Squares	202
Ex. 18.8	Approximation of Eigenvalues: Collatz Method	203
Ex. 18.9	Approximation of Eigenvalues: Power Method	204
Ex. 18.10	Approximation of Eigenvalues: QR-Factorization	205
	Problems for Chapter 18	210
Chapter 19	Numerical Methods for Differential Equations	213
Ex. 19.1	Euler Method	213
Ex. 19.2	Improved Euler Method	214
Ex. 19.3	Classical Runge-Kutta Method (RK). Module	215
Ex. 19.4	Adams-Moulton Multistep Method	217
Ex. 19.5	Classical Runge-Kutta Method for Systems (RKS)	219
Ex. 19.6	Classical Runge-Kutta-Nystroem Method (RKN)	220
Ex. 19.7	Laplace Equation. Boundary Value Problem	221
Ex. 19.8	Heat Equation. Crank-Nicolson Method	225
	Problems for Chapter 19	228
Part F.	Optimization, Graphs	230
Chapter 20	Unconstrained Optimization. Linear Programming	230
Ex. 20.1	Method of Steepest Descent	230
Ex. 20.2	Simplex Method of Constrained Optimization	232
	Problems for Chapter 20	234
Chapter 21	No examples, no problems
Part G.	Probability and Statistics	235
Chapter 22	Data Analysis. Probability Theory	235
Ex. 22.1	Data Analysis: Mean, Variance, Standard Deviation	235
Ex. 22.2	Data Analysis: Histograms	236
Ex. 22.3	Discrete Probability Distributions	237
Ex. 22.4	Normal Distribution	241
	Problems for Chapter 22	243
Chapter 23	Mathematical Statistics	245
Ex. 23.1	Random Numbers	245
Ex. 23.2	Confidence Interval for the Mean of the Normal Distribution With Known Variance	246
Ex. 23.3	Confidence Interval for the Mean of the Normal Distribution With Unknown Variance. t-Distribution	247
Ex. 23.4	Confidence Interval for the Variance of the Normal Distribution. x[superscript 2]-Distribution	248
Ex. 23.5	Test for the Mean of the Normal Distribution	249
Ex. 23.6	Test for the Mean: Power Function	249
Ex. 23.7	Test for the Variance of the Normal Distribution	251
Ex. 23.8	Comparison of Means	252
Ex. 23.9	Comparison of Variances. F-Distribution	253
Ex. 23.10	Chi-Square Test for Goodness of Fit	253
Ex. 23.11	Regression	254
	Problems for Chapter 23	256
Appendix 1	References	A1
Appendix 2	Answers to Odd-Numbered Problems	A2
	Index	J1