Engineering Mathematics with MATLAB

Chapter 1: Vectors and Matrices1.1 Vectors   1.1.1 Geometry with Vector   1.1.2 Dot Product   1.1.3 Cross Product   1.1.4 Lines and Planes   1.1.5 Vector Space    1.1.6 Coordinate Systems   1.1.7 Gram-Schmidt Orthonolization1.2 Matrices   1.2.1 Matrix Algebra   1.2.2 Rank and Row/Column Spaces   1.2.3 Determinant and Trace   1.2.4 Eigenvalues and Eigenvectors   1.2.5 Inverse of a Matrix   1.2.6 Similarity Transformation and Diagonalization   1.2.7 Special Matrices   1.2.8 Positive Definiteness   1.2.9 Matrix Inversion Lemma   1.2.10 LU, Cholesky, QR, and Singular Value Decompositions   1.2.11 Physical Meaning of Eigenvalues/Eigenvectors1.3 Systems of Linear Equations   1.3.1 Nonsingular Case   1.3.2 Undetermined Case - Minimum-Norm Solution   1.3.3 Overdetermined Case - Least-Squares Error Solution   1.3.4 Gauss(ian) Elimination   1.3.5 RLS (Recursive Least Squares) AlgorithmProblems Chapter 2: Vector Calculus2.1 Derivatives2.2 Vector Functions2.3 Velocity and Acceleration 2.4 Divergence and Curl2.5 Line Integrals and Path Independence 2.5.1 Line Integrals 2.5.2 Path Independence2.6 Double Integrals2.7 Green's Theorem2.8 Surface Integrals2.9 Stokes' Theorem2.10 Triple Integrals2.11 Divergence TheoremProblemsChapter 3: Ordinary Differential Equation3.1 First-Order Differential Equations 3.1.1 Separable Equations 3.1.2 Exact Differential Equations and Integrating Factors 3.1.3 Linear First-Order Differential Equations 3.1.4 Nonlinear First-Order Differential Equations 3.1.5 Systems of First-Order Differential Equations3.2 Higher-Order Differential Equations 3.2.1 Undetermined Coefficients 3.2.2 Variation of Parameters 3.2.3 Cauchy-Euler Equations 3.2.4 Systems of Linear Differential Equations3.3 Special Second-Order Linear ODEs 3.3.1 Bessel's Equation 3.3.2 Legendre's Equation 3.3.3 Chebyshev's Equation 3.3.4 Hermite's Equation 3.3.5 Laguerre's Equation3.4 Boundary Value ProblemsProblemsChapter 4: Laplace Transform4.1 Definition of the Laplace Transform    4.1.1 Laplace Transform of the Unit Step Function   4.1.2 Laplace Transform of the Unit Impulse Function   4.1.3 Laplace Transform of the Ramp Function   4.1.4 Laplace Transform of the Exponential Function   4.1.5 Laplace Transform of the Complex Exponential Function4.2 Properties of the Laplace Transform   4.2.1 Linearity   4.2.2 Time Differentiation   4.2.3 Time Integration   4.2.4 Time Shifting - Real Translation   4.2.5 Frequency Shifting - Complex Translation   4.2.6 Real Convolution   4.2.7 Partial Differentiation   4.2.8 Complex Differentiation    4.2.9 Initial Value Theorem (IVT)   4.2.10 Final Value Theorem (FVT)4.3 The Inverse Laplace Transform4.4 Using of the Laplace Transform4.5 Transfer Function of a Continuous-Time SystemProblems 300Chapter 5: The Z-transform5.1 Definition of the Z-transform5.2 Properties of the Z-transform   5.2.1 Linearity   5.2.2 Time Shifting - Real Translation   5.2.3 Frequency Shifting - Complex Translation   5.2.4 Time Reversal   5.2.5 Real Convolution   5.2.6 Complex Convolution   5.2.7 Complex Differentiation   5.2.8 Partial Differentiation   5.2.9 Initial Value Theorem   5.2.10 Final Value Theorem5.3 The Inverse Z-transform5.4 Using The Z-transform5.5 Transfer Function of a Discrete-Time System 5.6 Differential Equation and Difference EquationProblemsChapter 6: Fourier Series and Fourier Transform6.1 Continuous-Time Fourier Series (CTFS)   6.1.1 Definition and Convergence Conditions   6.1.2 Examples of CTFS6.2 Continuous-Time Fourier Transform (CTFT)   6.2.1 Definition and Convergence Conditions   6.2.2 (Generalized) CTFT of Periodic Signals   6.2.3 Examples of CTFT   6.2.4 Properties of CTFT6.3 Discrete-Time Fourier Transform (DTFT)   6.3.1 Definition and Convergence Conditions   6.3.2 Examples of DTFT   6.3.3 DTFT of Periodic Sequences   6.3.4 Properties of DTFT6.4 Discrete Fourier Transform (DFT)6.5 Fast Fourier Transform (FFT)   6.5.1 Decimation-in-Time (DIT) FFT   6.5.2 Decimation-in-Frequency (DIF) FFT   6.5.3 Computation of IDFT Using FFT Algorithm   6.5.4 Interpretation of DFT Results6.6 Fourier-Bessel/Legendre/Chebyshev/Cosine/Sine Series   6.6.1 Fourier-Bessel Series   6.6.2 Fourier-Legendre Series   6.6.3 Fourier-Chebyshev Series   6.6.4 Fourier-Cosine/Sine SeriesProblemsChapter 7: Partial Differential Equation7.1 Elliptic PDE7.2 Parabolic PDE   7.2.1 The Explicit Forward Euler Method   7.2.2 The Implicit Forward Euler Method    7.2.3 The Crank-Nicholson Method   7.2.4 Using the MATLAB Function 'pdepe()'   7.2.5 Two-Dimensional Parabolic PDEs7.3 Hyperbolic PDES   7.3.1 The Explict Central Difference Method   7.3.2 Tw-Dimensional Hyperbolic PDEs7.4 PDES in Other Coordinate Systems    7.4.1 PDEs in Polar/Cylindrical Coordinates   7.4.2 PDEs in Spherical Coordinates7.5 Laplace/Fourier Transforms for Solving PDES    7.5.1 Using the Laplace Transform for PDEs   7.5.2 Using the Fourier Transform for PDEsProblemsChapter 8: Complex Analysis 5098.1 Functions of a Complex Variable   8.1.1 Complex Numbers and their Powers/Roots   8.1.2 Functions of a Complex Variable   8.1.3 Cauchy-Riemann Equations   8.1.4 Exponential and Logarithmic Functions   8.1.5 Trigonometric and Hyperbolic Functions   8.1.6 Inverse Trigonometric/Hyperbolic Functions 8.2 Conformal Mapping   8.2.1 Conformal Mappings   8.2.2 Linear Fractional Transformations8.3 Integration of Complex Functions   8.3.1 Line Integrals and Contour Integrals   8.3.2 Cauchy-Goursat Theorem   8.3.3 Cauchy's Integral Formula8.4 Series and Residues   8.4.1 Sequences and Series    8.4.2 Taylor Series   8.4.3 Laurent Series   8.4.4 Residues and Residue Theorem   8.4.5 Real Integrals Using Residue TheoremProblemsChapter 9: Optimization9.1 Unconstrained Optimization   9.1.1 Golden Search Method   9.1.2 Quadratic Approximation Method    9.1.3 Nelder-Mead Method   9.1.4 Steepest Descent Method   9.1.5 Newton Method9.2 Constrained Optimization    9.2.1 Lagrange Multiplier Method   9.2.2 Penalty Function Method9.3 MATLAB Built-in Functions for Optimization   9.3.1 Unconstrained Optimization   9.3.2 Constrained Optimization   9.3.3 Linear Programming (LP)   9.3.4 Mixed Integer Linear Programing (MILP)ProblemsChapter 10: Probability10.1 Probability    10.1.1 Definition of Probability    10.1.2 Permutations and Combinations     10.1.3 Joint Probability, Conditional Probability, and Bayes' Rule10.2 Random Variables    10.2.1 Random Variables and Probability Distribution/Density Function    10.2.2 Joint Probability Density Function    10.2.3 Conditional Probability Density Function    10.2.4 Independence    10.2.5 Function of a Random Variable    10.2.6 Expectation, Variance, and Correlation    10.2.7 Conditional Expectation    10.2.8 Central Limit Theorem - Normal Convergence Theorem10.3 ML Estimator and MAP Estimator 653Problems