Advanced Engineering Mathematics

Advanced Engineering Mathematics

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The tenth edition of this bestselling text includes examples in more detail and more applied exercises; both changes are aimed at making the material more relevant and accessible to readers. Kreyszig introduces engineers and computer scientists to advanced math topics as they relate to practical problems. It goes into the following topics at great depth differential equations, partial differential equations, Fourier analysis, vector analysis, complex analysis, and linear algebra/differential equations.
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Product details

  • Paperback | 1152 pages
  • 201 x 253 x 35mm | 1,938g
  • John Wiley & Sons Ltd
  • Chichester, United Kingdom
  • English
  • 10th Edition International Student Version
  • 0470646136
  • 9780470646137
  • 78,988

Table of contents

PART A Ordinary Differential Equations (ODEs) 1


CHAPTER 1 First-Order ODEs 2


1.1 Basic Concepts. Modeling 2


1.2 Geometric Meaning of y (x, y). Direction Fields, Euler s Method 9


1.3 Separable ODEs. Modeling 12


1.4 Exact ODEs. Integrating Factors 20


1.5 Linear ODEs. Bernoulli Equation. Population Dynamics 27


1.6 Orthogonal Trajectories. Optional 36


1.7 Existence and Uniqueness of Solutions for Initial Value Problems 38


CHAPTER 2 Second-Order Linear ODEs 46


2.1 Homogeneous Linear ODEs of Second Order 46


2.2 Homogeneous Linear ODEs with Constant Coefficients 53


2.3 Differential Operators. Optional 60


2.4 Modeling of Free Oscillations of a Mass Spring System 62


2.5 Euler Cauchy Equations 71


2.6 Existence and Uniqueness of Solutions. Wronskian 74


2.7 Nonhomogeneous ODEs 79


2.8 Modeling: Forced Oscillations. Resonance 85


2.9 Modeling: Electric Circuits 93


2.10 Solution by Variation of Parameters 99


CHAPTER 3 Higher Order Linear ODEs 105


3.1 Homogeneous Linear ODEs 105


3.2 Homogeneous Linear ODEs with Constant Coefficients 111


3.3 Nonhomogeneous Linear ODEs 116


CHAPTER 4 Systems of ODEs. Phase Plane. Qualitative Methods 124


4.0 For Reference: Basics of Matrices and Vectors 124


4.1 Systems of ODEs as Models in Engineering Applications 130


4.2 Basic Theory of Systems of ODEs. Wronskian 137


4.3 Constant-Coefficient Systems. Phase Plane Method 140


4.4 Criteria for Critical Points. Stability 148


4.5 Qualitative Methods for Nonlinear Systems 152


4.6 Nonhomogeneous Linear Systems of ODEs 160


CHAPTER 5 Series Solutions of ODEs. Special Functions 167


5.1 Power Series Method 167


5.2 Legendre's Equation. Legendre Polynomials Pn(x) 175


5.3 Extended Power Series Method: Frobenius Method 180


5.4 Bessel s Equation. Bessel Functions (x) 187


5.5 Bessel Functions of the Y (x). General Solution 196


CHAPTER 6 Laplace Transforms 203


6.1 Laplace Transform. Linearity. First Shifting Theorem (s-Shifting) 204


6.2 Transforms of Derivatives and Integrals. ODEs 211


6.3 Unit Step Function (Heaviside Function). Second Shifting Theorem (t-Shifting) 217


6.4 Short Impulses. Dirac's Delta Function. Partial Fractions 225


6.5 Convolution. Integral Equations 232


6.6 Differentiation and Integration of Transforms. ODEs with Variable Coefficients 238


6.7 Systems of ODEs 242


6.8 Laplace Transform: General Formulas 248


6.9 Table of Laplace Transforms 249


PART B Linear Algebra. Vector Calculus 255


CHAPTER 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems 256


7.1 Matrices, Vectors: Addition and Scalar Multiplication 257


7.2 Matrix Multiplication 263


7.3 Linear Systems of Equations. Gauss Elimination 272


7.4 Linear Independence. Rank of a Matrix. Vector Space 282


7.5 Solutions of Linear Systems: Existence, Uniqueness 288


7.6 For Reference: Second- and Third-Order Determinants 291


7.7 Determinants. Cramer s Rule 293


7.8 Inverse of a Matrix. Gauss Jordan Elimination 301


7.9 Vector Spaces, Inner Product Spaces. Linear Transformations. Optional 309


CHAPTER 8 Linear Algebra: Matrix Eigenvalue Problems 322


8.1 The Matrix Eigenvalue Problem. Determining Eigenvalues and Eigenvectors 323


8.2 Some Applications of Eigenvalue Problems 329


8.3 Symmetric, Skew-Symmetric, and Orthogonal Matrices 334


8.4 Eigenbases. Diagonalization. Quadratic Forms 339


8.5 Complex Matrices and Forms. Optional 346


CHAPTER 9 Vector Differential Calculus. Grad, Div, Curl 354


9.1 Vectors in 2-Space and 3-Space 354


9.2 Inner Product (Dot Product) 361


9.3 Vector Product (Cross Product) 368


9.4 Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives 375


9.5 Curves. Arc Length. Curvature. Torsion 381


9.6 Calculus Review: Functions of Several Variables. Optional 392


9.7 Gradient of a Scalar Field. Directional Derivative 395


9.8 Divergence of a Vector Field 403


9.9 Curl of a Vector Field 406


CHAPTER 10 Vector Integral Calculus. Integral Theorems 413


10.1 Line Integrals 413


10.2 Path Independence of Line Integrals 419


10.3 Calculus Review: Double Integrals. Optional 426


10.4 Green s Theorem in the Plane 433


10.5 Surfaces for Surface Integrals 439


10.6 Surface Integrals 443


10.7 Triple Integrals. Divergence Theorem of Gauss 452


10.8 Further Applications of the Divergence Theorem 458


10.9 Stokes s Theorem 463


PART C Fourier Analysis. Partial Differential Equations (PDEs) 473


CHAPTER 11 Fourier Analysis 474


11.1 Fourier Series 474


11.2 Arbitrary Period. Even and Odd Functions. Half-Range Expansions 483


11.3 Forced Oscillations 492


11.4 Approximation by Trigonometric Polynomials 495


11.5 Sturm Liouville Problems. Orthogonal Functions 498


11.6 Orthogonal Series. Generalized Fourier Series 504


11.7 Fourier Integral 510


11.8 Fourier Cosine and Sine Transforms 518


11.9 Fourier Transform. Discrete and Fast Fourier Transforms 522


11.10 Tables of Transforms 534


CHAPTER 12 Partial Differential Equations (PDEs) 540


12.1 Basic Concepts of PDEs 540


12.2 Modeling: Vibrating String, Wave Equation 543


12.3 Solution by Separating Variables. Use of Fourier Series 545


12.4 D Alembert s Solution of the Wave Equation. Characteristics 553


12.5 Modeling: Heat Flow from a Body in Space. Heat Equation 557


12.6 Heat Equation: Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem 558


12.7 Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms 568


12.8 Modeling: Membrane, Two-Dimensional Wave Equation 575


12.9 Rectangular Membrane. Double Fourier Series 577


12.10 Laplacian in Polar Coordinates. Circular Membrane. Fourier Bessel Series 585


12.11 Laplace s Equation in Cylindrical and Spherical Coordinates. Potential 593


12.12 Solution of PDEs by Laplace Transforms 600


PART D Complex Analysis 607


CHAPTER 13 Complex Numbers and Functions. Complex Differentiation 608


13.1 Complex Numbers and Their Geometric Representation 608


13.2 Polar Form of Complex Numbers. Powers and Roots 613


13.3 Derivative. Analytic Function 619


13.4 Cauchy Riemann Equations. Laplace s Equation 625


13.5 Exponential Function 630


13.6 Trigonometric and Hyperbolic Functions. Euler's Formula 633


13.7 Logarithm. General Power. Principal Value 636


CHAPTER 14 Complex Integration 643


14.1 Line Integral in the Complex Plane 643


14.2 Cauchy's Integral Theorem 652


14.3 Cauchy's Integral Formula 660


14.4 Derivatives of Analytic Functions 664


CHAPTER 15 Power Series, Taylor Series 671


15.1 Sequences, Series, Convergence Tests 671


15.2 Power Series 680


15.3 Functions Given by Power Series 685


15.4 Taylor and Maclaurin Series 690


15.5 Uniform Convergence. Optional 698


CHAPTER 16 Laurent Series. Residue Integration 708


16.1 Laurent Series 708


16.2 Singularities and Zeros. Infinity 714


16.3 Residue Integration Method 719


16.4 Residue Integration of Real Integrals 725


CHAPTER 17 Conformal Mapping 735


17.1 Geometry of Analytic Functions: Conformal Mapping 736


17.2 Linear Fractional Transformations (Moebius Transformations) 741


17.3 Special Linear Fractional Transformations 745


17.4 Conformal Mapping by Other Functions 749


17.5 Riemann Surfaces. Optional 753


CHAPTER 18 Complex Analysis and Potential Theory 756


18.1 Electrostatic Fields 757


18.2 Use of Conformal Mapping. Modeling 761


18.3 Heat Problems 765


18.4 Fluid Flow 768


18.5 Poisson's Integral Formula for Potentials 774


18.6 General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet Problem 778


PART E Numeric Analysis 785


Software 786


CHAPTER 19 Numerics in General 788


19.1 Introduction 788


19.2 Solution of Equations by Iteration 795


19.3 Interpolation 805


19.4 Spline Interpolation 817


19.5 Numeric Integration and Differentiation 824


CHAPTER 20 Numeric Linear Algebra 841


20.1 Linear Systems: Gauss Elimination 841


20.2 Linear Systems: LU-Factorization, Matrix Inversion 849


20.3 Linear Systems: Solution by Iteration 855


20.4 Linear Systems: Ill-Conditioning, Norms 861


20.5 Least Squares Method 869


20.6 Matrix Eigenvalue Problems: Introduction 873


20.7 Inclusion of Matrix Eigenvalues 876


20.8 Power Method for Eigenvalues 882


20.9 Tridiagonalization and QR-Factorization 885


CHAPTER 21 Numerics for ODEs and PDEs 897


21.1 Methods for First-Order ODEs 898


21.2 Multistep Methods 908


21.3 Methods for Systems and Higher Order ODEs 912


21.4 Methods for Elliptic PDEs 919


21.5 Neumann and Mixed Problems. Irregular Boundary 928


21.6 Methods for Parabolic PDEs 933


21.7 Method for Hyperbolic PDEs 939


PART F Optimization, Graphs 947


CHAPTER 22 Unconstrained Optimization. Linear Programming 948


22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent 949


22.2 Linear Programming 952


22.3 Simplex Method 956


22.4 Simplex Method: Difficulties 960


CHAPTER 23 Graphs. Combinatorial Optimization 967


23.1 Graphs and Digraphs 967


23.2 Shortest Path Problems. Complexity 972


23.3 Bellman's Principle. Dijkstra s Algorithm 977


23.4 Shortest Spanning Trees: Greedy Algorithm 980


23.5 Shortest Spanning Trees: Prim s Algorithm 984


23.6 Flows in Networks 987


23.7 Maximum Flow: Ford Fulkerson Algorithm 993


23.8 Bipartite Graphs. Assignment Problems 996


APPENDIX 1 References A1


APPENDIX 2 Answers to Selected Problems A4


APPENDIX 3 Auxiliary Material A51


A3.1 Formulas for Special Functions A51


A3.2 Partial Derivatives A57


A3.3 Sequences and Series A60


A3.4 Grad, Div, Curl, 2 in Curvilinear Coordinates A62


APPENDIX 4 Additional Proofs A65


APPENDIX 5 Tables A85


INDEX I1


PHOTO CREDITS P1
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