Diaspora

Now with a full-color design, the new Fourth Edition of Zill's Advanced Engineering Mathematics provides an in-depth overview of the many mathematical topics necessary for students planning a career in engineering or the sciences. A key strength of this text is Zill's emphasis on differential equations as mathematical models, discussing the constructs and pitfalls of each. The Fourth Edition is comprehensive, yet flexible, to meet the unique needs of various course offerings ranging from ordinary differential equations to vector calculus. Numerous new projects contributed by esteemed mathematicians have been added. New modern applications and engaging projects makes Zill's classic text a must-have text and resource for Engineering Math students! Cover......Page 1 Contents......Page 3 Preface......Page 9 Part 1. Ordinary Differential Equations......Page 15 Chapter 1. Introduction to Differential Equations......Page 16 1.1 Definitions and Termfaology......Page 17 1.2 Initial-Value Problems......Page 26 1.3 Differential Equations as Mathematical Models......Page 32 Chapter 1 in Review......Page 43 Chapter 2. First-Order Differential Equations......Page 46 2.1 Solution Curves Without a Solution......Page 47 2.2 Separable Equations......Page 56 2.3 Linear Equations......Page 64 2.4 Exact Equations......Page 72 2.5 Solutions by Substitutions......Page 78 2.6 A Numerical Method......Page 82 2.7 Linear Models......Page 86 2.8 Nonlinear Models......Page 97 2.9 Modeling with Systems of First-Order DEs......Page 105 Chapter 2 in Review......Page 112 Chapter 3. Higher-Order Differential Equations......Page 117 3.1 Theory of Linear Equations......Page 118 3.2 Reduction of Order......Page 129 3.3 Homogeneous Linear Equations with Constant Coefficients......Page 132 3.4 Undetermined Coefficients......Page 139 3.5 Variation of Parameters......Page 148 3.6 Cauchy-Euler Equations......Page 153 3.7 Nonlinear Equations......Page 159 3.8 Linear Models: Initial-Value Problems......Page 164 3.9 Linear Models: Boundary-Value Problems......Page 179 3.10 Green's Functions......Page 188 3.11 Nonlinear Models......Page 199 3.12 Solving Systems of Linear Equations......Page 208 Chapter 3 in Review......Page 215 Chapter 4. The Laplace Transform......Page 221 4.1 Definition of the Laplace Transform......Page 222 4.2 The Inverse Transform and Transforms of Derivatives......Page 228 4.3 Translation Theorems......Page 236 4.4 Additional Operational Properties......Page 246 4.5 The Dirac Delta Function......Page 256 4.6 Systems of Linear Differential Equations......Page 259 Chapter 4 in Review......Page 265 Chapter 5. Series Solutions of Linear Differential Equations......Page 268 5.1 Solutions about Ordinary Points......Page 269 5.2 Solutions about Singular Points......Page 278 5.3 Special Functions......Page 287 Chapter 3 in Review......Page 300 Chapter 6. Numerical Solutions of Ordinary Differential Equations......Page 302 6.1 Euler Methods and Error Analysis......Page 303 6.2 Runge-Kutta Methods......Page 307 6.3 Multistep Methods......Page 312 6.4 Higher-Order Equations and Systems......Page 314 6.5 Second-Order Boundary-Value Problems......Page 318 Chapter 6 in Review......Page 322 Part 2. Vectors, Matrices, and Vector Calculus......Page 323 Chapter 7. Vectors......Page 324 7 .1 Vectors in 2-Space......Page 325 7 .2 Vectors in 3-Space......Page 330 7 .3 Dot Product......Page 335 7 .4 Cross Product......Page 341 7.5 Lines and Planes in 3-Space......Page 348 7 .6 Vector Spaces......Page 354 7.7 Gram-Schmidt Orthogonalization Process......Page 362 Chapter 7 in Review......Page 367 Chapter 8. Matrices......Page 369 8.1 Matrix Algebra......Page 370 8.2 Systems of Linear Algebraic Equations......Page 378 8.3 Rank of a Matrix......Page 390 8.4 Determinants......Page 395 8.5 Properties of Determinants......Page 401 8.6 Inverse of a Matrix......Page 407 8.7 Cramer's Rule......Page 417 8.8 The Eigenvalue Problem......Page 420 8.9 Powers of Matrices......Page 427 8.10 Orthogonal Matrices......Page 431 8.11 Approximation of Eigenvalues......Page 438 8.12 Diagonalization......Page 445 8.13 LU-Factorization......Page 453 8.14 Cryptography......Page 460 8.15 An Error-Correcting Code......Page 464 8.16 Method of Least Squares......Page 469 8.17 Discrete Compartmental Models......Page 472 Chapter 8 in Review......Page 476 Chapter 9. Vector Calculus......Page 478 9.1 Vector Functions......Page 479 9.2 Motion on a Curve......Page 485 9.3 Curvature and Components of Acceleration......Page 490 9.4 Partial Derivatives......Page 495 9.5 Directional Derivative......Page 500 9.6 Tangent Planes and Normal Lines......Page 506 9.7 Curl and Divergence......Page 509 9.8 Line Integrals......Page 515 9.9 Independence of the Path......Page 523 9.10 Double Integrals......Page 533 9.11 Double Integrals in Polar Coordinates......Page 541 9.12 Green's Theorem......Page 545 9.13 Surface Integrals......Page 551 9.14 Stokes' Theorem......Page 558 9.15 Triple Integrals......Page 563 9.16 Divergence Theorem......Page 573 9.17 Change of Variables in Multiple Integrals......Page 579 Chapter 9 in Review......Page 585 Part 3. Systems of Differential Equations......Page 589 Chapter 10. Systems of Linear Differential Equations......Page 590 10.1 Theory of Linear Systems......Page 591 10.2 Homogeneous Linear Systems......Page 597 10.3 Solution by Diagonalization......Page 610 10.4 Nonhomogeneous Linear Systems......Page 613 10.5 Matrix Exponential......Page 620 Chapter 10 in Review......Page 624 Chapter 11. Systems of Nonlinear Differential Equations......Page 626 11.1 Autonomous Systems......Page 627 11.2 Stability of Linear Systems......Page 633 11.3 Linearization and Local Stability......Page 640 11.4 Autonomous Systems as Mathematical Models......Page 649 11.5 Periodic Solutions, Limit Cycles, and Global Stability......Page 656 Chapter 11 in Review......Page 664 Part 4. Partial Differential Equations......Page 667 Chapter 12. Orthogonal Functions and Fourier Series......Page 668 12.1 Orthogonal Functions......Page 669 12.2 Fourier Series......Page 674 12.3 Fourier Cosine and Sine Series......Page 678 12.4 Complex Fourier Series......Page 685 12.5 Sturm-Liouville Problem......Page 688 12.6 Bessel and Legendre Series......Page 695 Chapter 12 in Review......Page 701 Chapter 13. Boundary-Value Problems in Rectangular Coordinates......Page 702 13.1 Separable Partial Differential Equations......Page 703 13.2 Classical PDEs and Boundary-Value Problems......Page 706 13.3 Heat Equation......Page 711 13.4 Wave Equation......Page 714 13.5 Laplace's Equation......Page 720 13.6 Nonhomogeneous BVPs......Page 725 13. 7 Orthogonal Series Expansions......Page 731 13.8 Fourier Series in Two Variables......Page 735 Chapter 13 in Review......Page 738 Chapter 14. Boundary-Value Problems in Other Coordinate Systems......Page 740 14.1 Problems in Polar Coordinates......Page 741 14.2 Problems in Cylindrical Coordinates......Page 746 14.3 Problems in Spherical Coordinates......Page 753 Chapter 14 in Review......Page 756 Chapter 15. Integral Transform Method......Page 758 15.1 Error Function......Page 759 15.2 Applications of the Laplace Transform......Page 760 15.3 Fourier Integral......Page 768 15.4 Fourier Transforms......Page 773 15.5 Fast Fourier Transform......Page 778 Chapter 15 in Review......Page 787 Chapter 16. Numerical Solutions of Partial Differential Equations......Page 789 16.1 Laplace's Equation......Page 790 16.2 The Heat Equation......Page 795 16.3 The Wave Equation......Page 800 Chapter 16 in Review......Page 803 Part 5. Complex Analysis......Page 805 Chapter 17. Functions of a Complex Variable......Page 806 17.1 Complex Numbers......Page 807 17 .2 Powers and Roots......Page 810 17.3 Sets in the Complex Plane......Page 815 17.4 Functions of a Complex Variable......Page 817 17.5 Cauchy-Riemann Equations......Page 822 17.6 Exponential and Logarithmic Functions......Page 826 17. 7 Trigonometric and Hyperbolic Functions......Page 832 17.8 Inverse Trigonometric and Hyperbolic Functions......Page 836 Chapter 17 in Review......Page 838 Chapter 18. Integration in the Complex Plane......Page 840 18.1 Contour Integrals......Page 841 18.2 Cauchy-Goursat Theorem......Page 846 18.3 Independence of the Path......Page 850 18.4 Cauchy's Integral Formulas......Page 855 Chapter 18 in Review......Page 860 Chapter 19. Series and Residues......Page 862 19.1 Sequences and Series......Page 863 19.2 Taylor Series......Page 867 19.3 Laurent Series......Page 872 19.4 Zeros and Poles......Page 879 19.5 Residues and Residue Theorem......Page 882 19.6 Evaluation of Real Integrals......Page 887 Chapter 19 in Review......Page 894 Chapter 20. Conformal Mappings......Page 896 20.1 Complex Functions as Mappings......Page 897 20.2 Conformal Mappings......Page 901 20.3 Linear Fractional Transformations......Page 907 20.4 Schwarz-Christoffel Transformations......Page 913 20.5 Poisson Integral Formulas......Page 917 20.6 Applications......Page 921 Chapter 20 in Review......Page 927 Appendices......Page 929 Appendix I. Derivative and Integral Formulas......Page 930 Appendix II. Gamma Function......Page 932 Appendix III. Table of Laplace Transforms......Page 934 Appendix IV. Conformal Mappings......Page 937 Answers to Selected Odd-Numbered Problems......Page 943 Index......Page 989 Credits......Page 1017