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Advanced Engineering Mathematics, 10th Edition,9780470458365
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Advanced Engineering Mathematics, 10th Edition



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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.

Table of Contents

Ordinary Differential Equations (ODEs)
First-Order ODEs
Basic Concepts. Modeling
Geometric Meaning of y' = f(x, y). Direction Fields
Separable ODEs. Modeling
Exact ODEs. Integrating Factors
Linear ODEs. Bernoulli Equation. Population Dynamics
Orthogonal Trajectories. Optional
Existence and Uniqueness of Solutions
Chapter 1 Review Questions and Problems
Summary of Chapter 1
Second-Order Linear ODEs
Homogeneous Linear ODEs of Second Order
Homogeneous Linear ODEs with Constant Coefficients
Differential Operators. Optional
Modeling: Free Oscillations. (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 2 Review Questions and Problems
Summary of Chapter 2
Higher Order Linear ODEs
Homogeneous Linear ODEs
Homogeneous Linear ODEs with Constant Coefficients
Nonhomogeneous Linear ODEs
Chapter 3 Review Questions and Problems
Summary of Chapter 3
Systems of ODEs. Phase Plane. Qualitative Methods
Basics of Matrices and Vectors
Systems of ODEs as Models
Basic Theory of Systems of ODEs
Constant-Coefficient Systems. Phase Plane Method
Criteria for Critical Points. Stability
Qualitative Methods for Nonlinear Systems
Nonhomogeneous Linear Systems of ODEs
Chapter 4 Review Questions and Problems
Summary of Chapter 4
Series Solutions of ODEs. Special Functions
Power Series Method
Legendre's Equation. Legendre Polynomials Pn(x)
Frobenius Method
Bessel's Equation. Bessel Functions Jv(x)
Bessel Functions of the Second Kind Yv(x)
Chapter 5 Review Questions and Problems
Summary of Chapter 5
Laplace Transforms
Laplace Transform. Inverse Transform. Linearity. ^-Shifting
Transforms of Derivatives and Integrals. ODEs
Unit Step Function. f-Shifting
Short Impulses. Dirac's Delta Function. Partial Fractions
Convolution. Integral Equations
Differentiation and Integration of Transforms
Systems of ODEs
Laplace Transform: General Formulas
Table of Laplace Transforms
Chapter 6 Review Questions and Problems
Summary of Chapter 6
Linear Algebra. Vector Calculus
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 7 Review Questions and Problems
Summary of Chapter 7
Linear Algebra: Matrix Eigenvalue Problems
Eigenvalues, Eigenvectors
Some Applications of Eigenvalue Problems
Symmetric, Skew-Symmetric, and Orthogonal Matrices
Eigenbases. Diagonalization. Quadratic Forms
Complex Matrices and Forms. Optional
Chapter 8 Review Questions and Problems
Summary of Chapter 8
Vector Differential Calculus. Grad, Div, Curl
Vectors in 2-Space and 3-Space
Inner Product
Vector Product
Vector and Scalar Functions and Fields. 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 9 Review Questions and Problems
Summary of Chapter 9
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
Chapter 10 Review Questions and Problems
Summary of Chapter 10
Fourier Analysis. Partial Differential Equations (PDEs)
Fourier Series, Integrals, and Transforms
Fourier Series
Functions of Any Period p = 2L. Even and Odd Functions. Half-Range Expansions
Forced Oscillations
Approximation by Trigonometric Polynomials
Sturm-Liouville Problems. Orthogonal Functions
Orthogonal Eigenfunction Expansions
Fourier Integral
Fourier Cosine and Sine Transforms
Fourier Transform. Discrete and Fast Fourier Transforms
Tables of Transforms
Chapter 11 Review Questions and Problems
Summary of Chapter 11
Partial Differential Equations (PDEs)
Basic Concepts
Modeling: Vibrating String, Wave Equation
Solution by Separating Variables. Use of Fourier Series
D'Alembert's Solution of the Wave Equation. Characteristics
Introduction to the Heat Equation
Heat Equation: Solution by Fourier Series
Heat Equation: 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
Chapter 12 Review Questions and Problems
Summary of Chapter 12
Complex Analysis
Complex Numbers and Functions
Complex Numbers. Complex Plane
Polar Form of Complex Numbers. Powers and Roots
Derivative. Analytic Function
Cauchy-Riemann Equations. Laplace's Equation
Exponential Function
Trigonometric and Hyperbolic Functions
Logarithm. General Power
Chapter 13 Review Questions and Problems
Summary of Chapter 13
Complex Integration
Line Integral in the Complex Plane
Cauchy's Integral Theorem
Cauchy's Integral Formula
Derivatives of Analytic Functions
Chapter 14 Review Questions and Problems
Summary of Chapter 14
Power Series, Taylor Series
Sequences, Series, Convergence Tests
Power Series
Functions Given by Power Series
Taylor and Maclaurin Series
Uniform Convergence. Optional
Chapter 15 Review Questions and Problems
Summary of Chapter 15
Laurent Series. Residue Integration
Laurent Series
Singularities and Zeros. Infinity
Residue Integration Method
Residue Integration of Real Integrals
Review Questions and Problems
Summary of Chapter 16
Conformal Mapping
Geometry of Analytic Functions: Conformal Mapping
Linear Fractional Transformations
Special Linear Fractional Transformations
Conformal Mapping by Other Functions
Riemann Surfaces. Optional
Chapter 17 Review Questions and Problems
Summary of Chapter 17
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
Chapter 18 Review Questions and Problems
Summary of Chapter 18
Numeric Analysis
Numerics in General
Solution of Equations by Iteration
Spline Interpolation
Numeric Integration and Differentiation
Chapter 19 Review Questions and Problems
Summary of Chapter 19
Numeric Linear Algebra
Linear Systems: Gauss Elimination
Linear Systems: LU-Factorization, 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
Tridiagonalization and QR-Factorization
Chapter 20 Review Questions and Problems
Summary of Chapter 20
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
Chapter 21 Review Questions and Problems
Summary of Chapter 21
Optimization, Graphs
Unconstrained Optimization. Linear Programming
Basic Concepts. Unconstrained Optimization
Linear Programming
Simplex Method
Table of Contents provided by Publisher. All Rights Reserved.

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