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Advanced Engineering Mathematics, 7th Edition

Name: Advanced Engineering Mathematics, 7th Edition
Author: Dennis G. Zill
ISBN: 9781284206258

by Dennis G. Zill

December 2020

Intermediate to advanced

1064 pages

49h 13m

English

Jones & Bartlett Learning

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Cover
Title Page
Copyright Page
Contents
Preface
1 Introduction to Differential Equations
1.1 Definitions and Terminology
1.2 Initial-Value Problems
1.3 Differential Equations as Mathematical Models
Chapter 1 in Review

2 First-Order Differential Equations
2.1 Solution Curves Without a Solution
2.1.1 Direction Fields2.1.2 Autonomous First-Order DEs
2.2 Separable Equations
2.3 Linear Equations
2.4 Exact Equations
2.5 Solutions by Substitutions
2.6 A Numerical Method
2.7 Linear Models
2.8 Nonlinear Models
2.9 Modeling with Systems of First-Order DEs
Chapter 2 in Review
3 Higher-Order Differential Equations
3.1 Theory of Linear Equations
3.1.1 Initial-Value and Boundary-Value Problems3.1.2 Homogeneous Equations3.1.3 Nonhomogeneous Equations
3.2 Reduction of Order
3.3 Linear Equations with Constant Coefficients
3.4 Undetermined Coefficients
3.5 Variation of Parameters
3.6 Cauchy–Euler Equations
3.7 Nonlinear Equations
3.8 Linear Models: Initial-Value Problems
3.8.1 Spring/Mass Systems: Free Undamped Motion3.8.2 Spring/Mass Systems: Free Damped Motion3.8.3 Spring/Mass Systems: Driven Motion3.8.4 Series Circuit Analogue
3.9 Linear Models: Boundary-Value Problems
3.10 Green’s Functions
3.10.1 Initial-Value Problems3.10.2 Boundary-Value Problems
3.11 Nonlinear Models
3.12 Solving Systems of Linear DEs
Chapter 3 in Review
4 The Laplace Transform
4.1 Definition of the Laplace Transform
4.2 Inverse Transforms and Transforms of Derivatives
4.2.1 Inverse Transforms4.2.2 Transforms of Derivatives
4.3 Translation Theorems
4.3.1 Translation on the s-axis4.3.2 Translation on the t-axis
4.4 Additional Operational Properties
4.4.1 Derivatives of Transforms4.4.2 Transforms of Integrals4.4.3 Transform of a Periodic Function
4.5 Dirac Delta Function
4.6 Systems of Linear Differential Equations
Chapter 4 in Review
5 Series Solutions of Linear Equations
5.1 Solutions about Ordinary Points
5.1.1 Review of Power Series5.1.2 Power Series Solutions
5.2 Solutions about Singular Points
5.3 Special Functions
5.3.1 Bessel Functions5.3.2 Legendre Functions
Chapter 5 in Review
6 Numerical Solutions of Ordinary Differential Equations
6.1 Euler Methods and Error Analysis
6.2 Runge–Kutta Methods
6.3 Multistep Methods
6.4 Higher-Order Equations and Systems
6.5 Second-Order Boundary-Value Problems
Chapter 6 in Review
7 Vectors
7.1 Vectors in 2-Space
7.2 Vectors in 3-Space
7.3 Dot Product
7.4 Cross Product
7.5 Lines and Planes in 3-Space
7.6 Vector Spaces
7.7 Gram–Schmidt Orthogonalization Process
Chapter 7 in Review
8 Matrices
8.1 Matrix Algebra
8.2 Systems of Linear Equations
8.3 Rank of a Matrix
8.4 Determinants
8.5 Properties of Determinants
8.6 Inverse of a Matrix
8.6.1 Finding the Inverse8.6.2 Using the Inverse to Solve Systems
8.7 Cramer’s Rule
8.8 Eigenvalue Problem
8.9 Powers of Matrices
8.10 Orthogonal Matrices
8.11 Approximation of Eigenvalues
8.12 Diagonalization
8.13 LU-Factorization
8.14 Cryptography
8.15 Error-Correcting Code
8.16 Method of Least Squares
8.17 Discrete Compartmental Models
Chapter 8 in Review
9 Vector Calculus
9.1 Vector Functions
9.2 Motion on a Curve
9.3 Curvature
9.4 Partial Derivatives
9.5 Directional Derivative
9.6 Tangent Planes and Normal Lines
9.7 Curl and Divergence
9.8 Line Integrals
9.9 Independence of Path
9.10 Double Integrals
9.11 Double Integrals in Polar Coordinates
9.12 Green’s Theorem
9.13 Surface Integrals
9.14 Stokes’ Theorem
9.15 Triple Integrals
9.16 Divergence Theorem
9.17 Change of Variables in Multiple Integrals
Chapter 9 in Review
10 Systems of Linear Differential Equations
10.1 Theory of Linear Systems
10.2 Homogeneous Linear Systems
10.2.1 Distinct Real Eigenvalues10.2.2 Repeated Eigenvalues10.2.3 Complex Eigenvalues
10.3 Solution by Diagonalization
10.4 Nonhomogeneous Linear Systems
10.4.1 Undetermined Coefficients10.4.2 Variation of Parameters10.4.3 Diagonalization
10.5 Matrix Exponential
Chapter 10 in Review
11 Systems of Nonlinear Differential Equations
11.1 Autonomous Systems
11.2 Stability of Linear Systems
11.3 Linearization and Local Stability
11.4 Autonomous Systems as Mathematical Models
11.5 Periodic Solutions, Limit Cycles, and Global Stability
Chapter 11 in Review
12 Fourier Series
12.1 Orthogonal Functions
12.2 Fourier Series
12.3 Fourier Cosine and Sine Series
12.4 Complex Fourier Series
12.5 Sturm–Liouville Problem
12.6 Bessel and Legendre Series
12.6.1 Fourier–Bessel Series12.6.2 Fourier–Legendre Series
Chapter 12 in Review
13 Boundary-Value Problems in Rectangular Coordinates
13.1 Separable Partial Differential Equations
13.2 Classical PDEs and Boundary-Value Problems
13.3 Heat Equation
13.4 Wave Equation
13.5 Laplace’s Equation
13.6 Nonhomogeneous Boundary-Value Problems
13.7 Orthogonal Series Expansions
13.8 Higher-Dimensional Problems
Chapter 13 in Review
14 Boundary-Value Problems in Other Coordinate Systems
14.1 Polar Coordinates
14.2 Cylindrical Coordinates
14.3 Spherical Coordinates
Chapter 14 in Review
15 Integral Transforms
15.1 Error Function
15.2 Laplace Transform
15.3 Fourier Integral
15.4 Fourier Transforms
15.5 Finite Fourier Transforms
15.6 Fast Fourier Transform
Chapter 15 in Review
16 Numerical Solutions of Partial Differential Equations
16.1 Laplace’s Equation
16.2 Heat Equation
16.3 Wave Equation
Chapter 16 in Review
17 Functions of a Complex Variable
17.1 Complex Numbers
17.2 Powers and Roots
17.3 Sets in the Complex Plane
17.4 Functions of a Complex Variable
17.5 Cauchy–Riemann Equations
17.6 Exponential and Logarithmic Functions
17.7 Trigonometric and Hyperbolic Functions
17.8 Inverse Trigonometric and Hyperbolic Functions
Chapter 17 in Review
18 Integration in the Complex Plane
18.1 Contour Integrals
18.2 Cauchy–Goursat Theorem
18.3 Independence of Path
18.4 Cauchy’s Integral Formulas
Chapter 18 in Review
19 Series and Residues
19.1 Sequences and Series
19.2 Taylor Series
19.3 Laurent Series
19.4 Zeros and Poles
19.5 Residue Theorem
19.6 Evaluation of Real Integrals
Chapter 19 in Review
20 Conformal Mappings
20.1 Complex Functions as Mappings
20.2 Conformal Mappings
20.3 Linear Fractional Transformations
20.4 Schwarz–Christoffel Transformations
20.5 Poisson Integral Formulas
20.6 Applications
Chapter 20 in Review
Appendices
A Integral-Defined Functions
B Derivative and Integral Formulas
C Laplace Transforms
D Conformal Mappings
Answers to Selected Odd-Numbered Problems
Index

Content preview from Advanced Engineering Mathematics, 7th Edition

7.2 Vectors in 3-Space

INTRODUCTION

In the plane, or 2-space, one way of describing the position of a point P is to assign to it coordinates relative to two mutually orthogonal, or perpendicular, axes called the x- and y-axes. If P is the point of intersection of the line x = a (perpendicular to the x-axis) and the line y = b (perpendicular to the y-axis), then the ordered pair (a, b) is said to be the rectangular or Cartesian coordinates of the point. See FIGURE 7.2.1. In this section we extend the notions of Cartesian coordinates and vectors to three dimensions.

A point P is marked on quadrant 1 of a coordinate plane. A vertical line from P intersects the x axis at a distance a from the origin. A horizontal line from P intersects the y axis at a distance b from the origin. The coordinates of P are (a, b). — FIGURE 7.2.1 Rectangular coordinates in 2-space

Rectangular Coordinate System ...

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Advanced Engineering Mathematics, 10th Edition

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Advanced Engineering Mathematics, 7th Edition

by Dennis G. Zill

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