Advanced Engineering Mathematics International

by Zill, Dennis G.

• ISBN13: 9780763713577
• ISBN10: 0763713570
• Edition: 2nd
• Format: Hardcover
• Copyright: 1999-12-01
• Publisher: Jones & Bartlett Learning

Summary

* Text is divided into six modules: Ordinary Differential Equations; Vectors, Matrices, and Vector Calculus; Systems of Differential Equations; Fourier Series and Boundary-Value Problems; Numberical Analysis; Complex Analysis.* Topics are presented in a succinct and easy-to-read manner.* Numerous illustrations help students visualize problems.

Table of Contents

Part 1 Ordinary Differential Equations	1	(562)
Introduction to Differential Equations	5	(30)
Definitions and Terminology	6	(9)
Initial-Value Problems	15	(6)
Differential Equations as Mathematical Models	21	(14)
Review Exercises	33	(2)
First-Order Differential Equations	35	(66)
Solution Curves Without the Solution	36	(8)
Separable Variables	44	(7)
Linear Equations	51	(8)
Exact Equations	59	(5)
Solutions by Substitutions	64	(4)
A Numerical Solution	68	(5)
Linear Models	73	(9)
Nonlinear Models	82	(9)
Systems: Linear and Nonlinear Models	91	(10)
Review Exercises	98	(3)
Higher-Order Differential Equations	101	(88)
Preliminary Theory: Linear Equations	102	(12)
Initial-Value and Boundary-Value Problems	102	(2)
Homogeneous Equations	104	(6)
Nonhomogeneous Equations	110	(4)
Reduction of Order	114	(3)
Homogeneous Linear Equations with Constant Coefficients	117	(6)
Undetermined Coefficients	123	(9)
Variation of Parameters	132	(4)
Cauchy-Euler Equation	136	(6)
Nonlinear Equations	142	(5)
Linear Models: Initial-Value Problems	147	(16)
Spring/Mass Systems: Free Undamped Motion	147	(4)
Spring/Mass Systems: Free Damped Motion	151	(3)
Spring/Mass Systems: Driven Motion	154	(3)
Series Circuit Analogue	157	(6)
Linear Models: Boundary-Value Problems	163	(7)
Nonlinear Models	170	(10)
Solving Systems of Linear Equations	180	(9)
Review Exercises	186	(3)
The Laplace Transform	189	(46)
Definition of the Laplace Transform	190	(5)
The Inverse Transform and Transforms of Derivatives	195	(9)
Translation Theorems	204	(11)
Translation on the s-axis	204	(3)
Translation on the t-axis	207	(8)
Additional Operational Properties	215	(9)
Dirac Delta Function	224	(3)
Solving Systems of Linear Equations	227	(8)
Review Exercises	232	(3)
Series Solutions of Linear Equations	235	(32)
Solutions about Ordinary Points	236	(10)
Review of Power Series	236	(2)
Power Series Solutions	238	(8)
Solutions about Singular Points	246	(10)
Two Special Equations	256	(11)
Review Exercises	290
Numerical Solutions of Ordinary Differential Equations	267	(30)
Euler Methods and Error Analysis	268	(5)
Runge-Kutta Methods	273	(5)
Methods	278	(3)
Higher-Order Equations and Systems	281	(5)
Second-Order Boundary-Value Problems	286	(11)
Review Exercises	290	(7)
Part 2 Vectors, Matrices, and Vector Calculus
Vectors	297	(44)
Vectors in 2-Space	298	(6)
Vectors in 3-Space	304	(5)
The Dot Product	309	(8)
The Cross Product	317	(6)
Lines and Planes in 3-Space	323	(8)
Vector Spaces	331	(10)
Review Exercises	338	(3)
Matrices	341	(106)
Matrix Algebra	342	(9)
Systems of Linear Algebraic Equations	351	(12)
Rank of a Matrix	363	(5)
Determinants	368	(6)
Properties of Determinants	374	(7)
Inverse of a Matrix	381	(10)
Finding the Inverse
Using the Inverse to Solve Systems	387	(4)
Cramer's Rule	391	(4)
The Eigenvalue Problem	395	(5)
Powers of Matrices	400	(4)
Orthogonal Matrices	404	(7)
Approximation of Eigenvalues	411	(7)
Diagonalization	418	(9)
Cryptography	427	(3)
An Error-Correcting Code	430	(6)
Method of Least Squares	436	(3)
Discrete Compartmental Models	439	(8)
Review Exercises	444	(3)
Vector Calculus	447	(116)
Vector Functions	448	(6)
Motion on a Curve	454	(5)
Curvature and Components of Acceleration	459	(5)
Functions of Several Variables	464	(6)
The Directional Derivative	470	(7)
Planes and Normal Lines	477	(3)
Divergence and Curl	480	(6)
Line Integrals	486	(9)
Line Integrals Independent of the Path	495	(7)
Review of Double Integrals	502	(9)
Double Integrals in Polar Coordinates	511	(5)
Green's Theorem	516	(5)
Surface Integrals	521	(8)
Stokes' Theorem	529	(5)
Review of Triple Integrals	534	(12)
Divergence Theorem	546	(6)
Change of Variables in Multiple Integrals	552	(11)
Review Exercises	558	(5)
Part 3 Systems of Differential Equations	563	(84)
Systems of Linear Differential Equations	567	(34)
Preliminary Theory	568	(7)
Homogeneous Linear Systems	575	(13)
Distinct Real Eigenvalues	576	(3)
Repeated Eigenvalues	579	(4)
Complex Eigenvalues	583	(5)
Solution by Diagonalization	588	(2)
Nonhomogeneous Linear Systems	590	(5)
Matrix Exponential	595	(6)
Review Exercises	599	(2)
Systems of Nonlinear Differential Equations	601	(46)
Autonomous Systems, Critical Points, and Periodic Solutions	602	(6)
Stability of Linear Systems	608	(9)
Linearization and Local Stability	617	(9)
Modeling Using Autonomous Systems	626	(9)
Periodic Solutions, Limit Cycles, and Global Stability	635	(12)
Review Exercises	644	(3)
Part 4 Fourier Series and Partial Differential Equations	647	(138)
Orthogonal Functions and Fourier Series	651	(36)
Orthogonal Functions	652	(4)
Fourier Series	656	(6)
Fourier Cosine and Sine Series	662	(7)
Complex Fourier Series and Frequency Spectrum	669	(3)
Sturm-Liouville Problem	672	(6)
Bessel and Legendre Series	678	(9)
Fourier-Bessel Series	678	(4)
Fourier-Legendre Series	682	(3)
Review Exercises	685	(2)
Boundary-Value Problems in Rectangular Coordinates	687	(32)
Separable Partial Differential Equations	688	(4)
Classical Equations and Boundary-Value Problems	692	(5)
Heat Equation	697	(3)
Wave Equation	700	(4)
Laplace's Equation	704	(5)
Nonhomogeneous Equations and Boundary Conditions	709	(2)
Orthogonal Series Expansions	711	(4)
Fourier Series in Two Variables	715	(4)
Review Exercises	717	(2)
Boundary-Value Problems in Other Coordinate Systems	719	(16)
Problems Involving Laplace's Equation in Polar Coordinates	720	(4)
Problems in Polar and Cylindrical Coordinates: Bessel Functions	724	(6)
Problems in Spherical Coordinates: Legendre Polynomials	730	(5)
Review Exercises	733	(2)
Integral Transform Method	735	(32)
Error Function	736	(1)
Applications of the Laplace Transform	737	(8)
Fourier Integral	745	(5)
Fourier Transforms	750	(6)
Fast Fourier Transform	756	(11)
Review Exercises	765	(2)
Numerical Solutions of Partial Differential Equations	767	(18)
Elliptic Equations	768	(5)
Parabolic Equations	773	(6)
Hyperbolic Equations	779	(6)
Review Exercises	783	(2)
Part 5 Complex Analysis	785	(4)
Functions of a Complex Variable	789	(1)
Complex Numbers	790	(3)
Form of Complex Numbers; Powers and Roots	793	(5)
Sets of Points in the Complex Plane	798	(2)
Functions of a Complex Variable; Analyticity	800	(6)
Cauchy-Riemann Equations	806	(5)
Exponential and Logarithmic Functions	811	(7)
Trigonometric and Hyperbolic Functions	818	(4)
Inverse Trigonometric and Hyperbolic Functions	822	(2)
Review Exercises	824	(3)
Integration in the Complex Plane	827	(1)
Contour Integrals	828	(5)
Cauchy-Goursat Theorem	833	(5)
Independence of Path	838	(7)
Cauchy's Integral Formula	845	(4)
Review Exercises	849	(2)
Series and Residues	851	(1)
Sequences and Series	852	(5)
Taylor Series	857	(6)
Laurent Series	863	(3)
Zeros and Poles	870	(3)
Residues and Residue Theorem	873	(6)
Evaluation of Real Integrals	879	(7)
Review Exercises	886	(3)
Conformal Mappings and Applications	889	(1)
Complex Functions as Mappings	890	(4)
Conformal Mapping and the Dirichlet Problem	894	(8)
Linear Fractional Transformations	902	(6)
Schwarz-Christoffel Transformations	908	(5)
Poisson Integral Formulas	913	(4)
Applications	917	(7)
Review Exercises	924
Appendices	A-1
Appendix I Some Derivative and Integral Formulas	A-3
Appendix II Gamma Function	A-5
Appendix II Exercises	A-6
Appendix III Table of Laplace Transforms	A-7
Appendix IV Conformal Mappings	A-11
Appendix V Some Basic Programs for Numerical Methods	A-19
Selected Answers for Odd-Numbered Problems	A-25