10th Edition

Erwin Kreyszig

© 2011

© 2011

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Students get endless practice and immediate feedback with over 900 algorithmic end-of-section and end-of-chapter homework questions and 160 correlated review exercises.

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- Mathematica Manual
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- 1060 algorithmic End-of-Section and End-of-Chapter homework questions

Content organised by Learning Objectives.

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- Gradable Reading Assignment Questions (embedded with online text)
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Course Materialstohelp you personalise lessons and optimise your time, including:

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**PART A Ordinary Differential Equations (ODEs) ****CHAPTER 1 First-Order ODEs **

- Basic Concepts. Modeling
- Geometric Meaning of y ƒ(x, y). Direction Fields, Euler’s Method
- Separable ODEs. Modeling
- Exact ODEs. Integrating Factors
- Linear ODEs. Bernoulli Equation. Population Dynamics
- Orthogonal Trajectories. Optional
- Existence and Uniqueness of Solutions for Initial Value Problems

**CHAPTER 2 Second-Order Linear ODEs **

- Homogeneous Linear ODEs of Second Order
- Homogeneous Linear ODEs with Constant Coefficients
- Differential Operators. Optional
- Modeling of Free Oscillations of a Mass–Spring System
- Euler–Cauchy Equations
- Existence and Uniqueness of Solutions. Wronskian
- Nonhomogeneous ODEs
- Modeling: Forced Oscillations. Resonance
- Modeling: Electric Circuits
- Solution by Variation of Parameters

**CHAPTER 3 Higher Order Linear ODEs **

- Homogeneous Linear ODEs
- Homogeneous Linear ODEs with Constant Coefficients
- Nonhomogeneous Linear ODEs

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

- For Reference: Basics of Matrices and Vectors
- Systems of ODEs as Models in Engineering Applications
- Basic Theory of Systems of ODEs. Wronskian
- Constant-Coefficient Systems. Phase Plane Method
- Criteria for Critical Points. Stability
- Qualitative Methods for Nonlinear Systems
- Nonhomogeneous Linear Systems of ODEs

**CHAPTER 5 Series Solutions of ODEs. Special Functions**

- Power Series Method
- Legendre's Equation. Legendre Polynomials Pn(x)
- Extended Power Series Method: Frobenius Method
- Bessel’s Equation. Bessel Functions (x)
- Bessel Functions of the Y (x). General Solution

**CHAPTER 6 Laplace Transforms **

- Laplace Transform. Linearity. First Shifting Theorem (s-Shifting)
- Transforms of Derivatives and Integrals. ODEs
- Unit Step Function (Heaviside Function). Second Shifting Theorem (t-Shifting)
- Short Impulses. Dirac's Delta Function. Partial Fractions
- Convolution. Integral Equations
- Differentiation and Integration of Transforms. ODEs with Variable Coefficients
- Systems of ODEs
- Laplace Transform: General Formulas
- Table of Laplace Transforms

**PART B Linear Algebra. Vector Calculus **

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

- Matrices, Vectors: Addition and Scalar Multiplication
- Matrix Multiplication
- Linear Systems of Equations. Gauss Elimination
- Linear Independence. Rank of a Matrix. Vector Space
- Solutions of Linear Systems: Existence, Uniqueness
- For Reference: Second- and Third-Order Determinants
- Determinants. Cramer’s Rule
- Inverse of a Matrix. Gauss–Jordan Elimination
- Vector Spaces, Inner Product Spaces. Linear Transformations. Optional

**CHAPTER 8 Linear Algebra: Matrix Eigenvalue Problems **

- The Matrix Eigenvalue Problem. Determining Eigenvalues and Eigenvectors
- Some Applications of Eigenvalue Problems
- Symmetric, Skew-Symmetric, and Orthogonal Matrices
- Eigenbases. Diagonalisation. Quadratic Forms
- Complex Matrices and Forms. Optional

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

- Vectors in 2-Space and 3-Space
- Inner Product (Dot Product)
- Vector Product (Cross Product)
- Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives
- Curves. Arc Length. Curvature. Torsion
- Calculus Review: Functions of Several Variables. Optional
- Gradient of a Scalar Field. Directional Derivative
- Divergence of a Vector Field
- Curl of a Vector Field

**CHAPTER 10 Vector Integral Calculus. Integral Theorems **

- Line Integrals
- Path Independence of Line Integrals
- Calculus Review: Double Integrals. Optional
- Green’s Theorem in the Plane
- Surfaces for Surface Integrals
- Surface Integrals
- Triple Integrals. Divergence Theorem of Gauss
- Further Applications of the Divergence Theorem
- Stokes’s Theorem

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

**CHAPTER 11 Fourier Analysis **

- Fourier Series
- Arbitrary Period. Even and Odd Functions. Half-Range Expansions
- Forced Oscillations
- Approximation by Trigonometric Polynomials
- Sturm–Liouville Problems. Orthogonal Functions
- Orthogonal Series. Generalised Fourier Series
- Fourier Integral
- Fourier Cosine and Sine Transforms
- Fourier Transform. Discrete and Fast Fourier Transforms
- Tables of Transforms

**CHAPTER 12 Partial Differential Equations (PDEs) **

- Basic Concepts of PDEs
- Modeling: Vibrating String, Wave Equation
- Solution by Separating Variables. Use of Fourier Series
- D’Alembert’s Solution of the Wave Equation. Characteristics
- Modeling: Heat Flow from a Body in Space. Heat Equation
- Heat Equation: Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem
- Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms
- Modeling: Membrane, Two-Dimensional Wave Equation
- Rectangular Membrane. Double Fourier Series
- Laplacian in Polar Coordinates. Circular Membrane. Fourier–Bessel Series
- Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential
- Solution of PDEs by Laplace Transforms

**PART D Complex Analysis**

**CHAPTER 13 Complex Numbers and Functions. Complex Differentiation **

- Complex Numbers and Their Geometric Representation
- Polar Form of Complex Numbers. Powers and Roots
- Derivative. Analytic Function
- Cauchy–Riemann Equations. Laplace’s Equation
- Exponential Function
- Trigonometric and Hyperbolic Functions. Euler's Formula
- Logarithm. General Power. Principal Value

**CHAPTER 14 Complex Integration **

- Line Integral in the Complex Plane
- Cauchy's Integral Theorem
- Cauchy's Integral Formula
- Derivatives of Analytic Functions

**CHAPTER 15 Power Series, Taylor Series **

- Sequences, Series, Convergence Tests
- Power Series
- Functions Given by Power Series
- Taylor and Maclaurin Series
- Uniform Convergence. Optional

**CHAPTER 16 Laurent Series. Residue Integration**

- Laurent Series
- Singularities and Zeros. Infinity
- Residue Integration Method
- Residue Integration of Real Integrals

**CHAPTER 17 Conformal Mapping **

- Geometry of Analytic Functions: Conformal Mapping
- Linear Fractional Transformations (Möbius Transformations)
- Special Linear Fractional Transformations
- Conformal Mapping by Other Functions
- Riemann Surfaces. Optional

**CHAPTER 18 Complex Analysis and Potential Theory **

- Electrostatic Fields
- Use of Conformal Mapping. Modeling
- Heat Problems
- Fluid Flow
- Poisson's Integral Formula for Potentials
- General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet Problem

**PART E Numeric Analysis **

- Software

**CHAPTER 19 Numerics in General**

- Introduction
- Solution of Equations by Iteration
- Interpolation
- Spline Interpolation
- Numeric Integration and Differentiation

**CHAPTER 20 Numeric Linear Algebra **

- Linear Systems: Gauss Elimination
- Linear Systems: LU-Factorisation, Matrix Inversion
- Linear Systems: Solution by Iteration
- Linear Systems: Ill-Conditioning, Norms
- Least Squares Method
- Matrix Eigenvalue Problems: Introduction
- Inclusion of Matrix Eigenvalues
- Power Method for Eigenvalues
- Tridiagonalisation and QR-Factorisation

**CHAPTER 21 Numerics for ODEs and PDEs **

- Methods for First-Order ODEs
- Multistep Methods
- Methods for Systems and Higher Order ODEs
- Methods for Elliptic PDEs
- Neumann and Mixed Problems. Irregular Boundary
- Methods for Parabolic PDEs
- Method for Hyperbolic PDEs

**PART F Optimisation, Graphs **

**CHAPTER 22 Unconstrained Optimisation. Linear Programming **

- Basic Concepts. Unconstrained Optimisation: Method of Steepest Descent
- Linear Programming
- Simplex Method
- Simplex Method: Difficulties

**CHAPTER 23 Graphs. Combinatorial Optimisation **

- Graphs and Digraphs
- Shortest Path Problems. Complexity
- Bellman's Principle. Dijkstra’s Algorithm
- Shortest Spanning Trees: Greedy Algorithm
- Shortest Spanning Trees: Prim’s Algorithm
- Flows in Networks
- Maximum Flow: Ford–Fulkerson Algorithm
- Bipartite Graphs. Assignment Problems

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