ADVANCED ENGINEERING MATHEMATICS WITH MATLAB® is written for engineers and engineering students who are interested in applying MATLAB® to solve practical engineering problems. The book emphasizes mathematical principles, not computations, with MATLAB® employed as a tool for analysis that shows how engineering problems are defined and solved. The book features complete MATLAB® integration throughout, abundant examples which show real practical applications, and end-of-chapter problems that reinforce techniques.

### Table of Contents

1. INTRODUCTION

Introduction to MATLAB® / MATLAB® Commands for Display and Plotting / Creating MATLAB® Programs / MATLAB® Programming Language / Problem Solving and Programming (Optional) / Reinforcement Exercises and Exploration Problems / Annotated Bibliography / Answers

2. NUMBERS AND VECTORS

Properties of Real Numbers / MATLAB® Computer Numbers (Optional) / Complex Numbers / Vectors in Two Dimensions and Three Dimensions / Vectors in Higher Dimensions / MATLAB® Vectors / Complex Vectors / Vector Spaces / Vector Spaces of Functions / Reinforcement Exercises and Exploration Problems / Annotated Bibliography / Answers

3. MATRICES

Basic Properties of Matrices / MATLAB® Matrix Operations / Square and Symmetric Matrices / Determinants and Matrix Inverses / Orthogonal and Triangular Matrices / Systems of Linear Equations / MATLAB® Matrix Functions / Linear Transformations / Reinforcement Exercises and Exploration Problems / Annotated Bibliography / Answers

4. EIGENVALUES AND EIGENVECTORS

General Discussion of Eigenvalues / Eigenvalues and Eigenvectors / Matrix Eigenvalue Theorems / Complex Vectors and Matrices / MATLAB® Commands for Eigenvectors / Matrix Calculus / Similar and Diagonalizable Matrices / Special Matrices and Their Eigenvalues (Optional) / Applications to Differential Equations / Reinforcement Exercises and Exploration Problems / Annotated Bibliography / Answers

5. LINEAR DIFFERENTIAL EQUATIONS

Classification of Differential Equations / Linear Differential Equations / Higher Order Differential Equations / Second Order Differential Equations / Particular Solutions of Differential Equations / Systems of Differential Equations / MATLAB® Solutions of Systems of Differential Equations / Homogeneous Systems with Repeated Eigenvalues / Nonhomogeneous Systems of Differential Equations / Transforming Differential Equations / Reinforcement Exercises and Exploration Problems / Annotated Bibliography / Answers

6. ADVANCED DIFFERENTIAL EQUATIONS

Functions and Differential Equations / Sequences and Series / Taylor Series / Numerical Methods for Differential Equations / Stiff Differential Equations / Vector Equations / Boundary Value Problems / Equations with Variable Coefficients / Bessel and Legendre Equations / Reinforcement Exercises and Exploration Problems / Annotated Bibliography / Answers

7. APPROXIMATION OF FUNCTIONS

Polynomial Interpolation / Interpolation by Spline Functions / Least-Squares Curve Fitting / Orthogonal Functions / Reinforcement Exercises and Exploration Problems / Annotated Bibliography / Answers

8. FOURIER ANALYSIS

Fourier Series / Properties of Fourier Series / Fourier Transforms / Reinforcement Exercises and Exploration Problems / Annotated Bibliography / Answers

9. LAPLACE TRANSFORMS

Definition and Properties of the Laplace Transform / Computation of Inverse Laplace Transforms / MATLAB® and Laplace Transforms / Applications to Differential Equations / Applications of Laplace Transforms to Linear Systems / Relationship of Fourier and Laplace Transforms / Summary of Laplace Transform Properties / Reinforcement Exercises and Exploration Problems / Annotated Bibliography / Answers

10. DISCRETE SYSTEMS

Introduction to the Sequences and Discrete Functions / Linear Difference Equations / Approximation to Differential Equations / Smoothing and Digital Filters / Introduction to z-Transforms / MATLAB® Commands for Discrete Systems / z-Transform Solution of Difference Equations / Applications of z-transforms to Linear Discrete Systems / Frequency Response and Fourier Analysis / Summary of z-transform Properties / Reinforcement Exercises and Exploration Problems / Annotated Bibliography / Answers

11. THE DISCRETE FOURIER TRANSFORM AND THE FFT

Frequency Analysis of Signals / Discrete and Fast Fourier Transforms / MATLAB® Fourier Commands / Practical Signal Analysis / Practical Signal Sampling and DFT Errors / Analysis of DFT for Computation (Optional) / Reinforcement Exercises and Exploration Problems / Annotated Bibliography / Answers

12. ADVANCED CALCULUS

Functions of Several Variables / Derivatives of a Multivariate Function / Differentials and Linear Approximation / Two-Dimensional Taylor Series / MATLAB® Two-Dimensional Interpolation / MATLAB® Differentiation / Extrema of Real-Valued Functions / Constrained Extrema and Lagrange Multipliers / Reinforcement Exercises and Exploration Problems / Annotated Bibliography / Answers

13. VECTOR DIFFERENTIAL OPERATORS

Vector and Scalar Fields / MATLAB® Commands for Vector Differential Calculus / Directional Derivatives and the Gradient / The Divergence / The Curl / The Laplacian and Laplace''s Equation / Vector Field Theory / Physical Application and Interpretation / Curvilinear Coordinates / Reinforcement Exercises and Exploration Problems / Annotated Bibliography / Answers

14. VECTOR INTEGRAL CALCULUS

Integration / Applications of Single Integrals / Double and Triple Integrals / Change of Variables in Double Integrals / Change of Variables in Triple Integrals / Applications of Multiple Integrals / MATLAB® Commands for Integration / Line Integrals / Surface Integrals / Theorems of Vector Integral Calculus / Applications of Vector Field Theory / Reinforcement Exercises and Exploration Problems / Annotated Bibliography / Answers

15. PARTIAL DIFFERENTIAL EQUATIONS

Introduction to Partial Differential Equations / Laplace''s Equation / The Heat Equation / The Wave Equation / Reinforcement Exercises and Exploration Problems / Annotated Bibliography / Answers / INDEX / INDEX OF MATLAB® COMMANDS