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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To our parents and families who love and support us and to our teachers and students who enriched our knowledge vii Contents PREFACE XIII Chapter 1: Vectors and Matrices .................................................................................................... 1 1.1 Vectors ..................................................................................................................................... 1 1.1.1 Geometry with Vector .................................................................................................. 1 1.1.2 Dot Product .................................................................................................................. 2 1.1.3 Cross Product ............................................................................................................... 6 1.1.4 Lines and Planes ........................................................................................................... 9 1.1.5 Vector Space .............................................................................................................. 17 1.1.6 Coordinate Systems .................................................................................................... 18 1.1.7 Gram–Schmidt Orthonolization ................................................................................. 29 1.2 Matrices ................................................................................................................................. 32 1.2.1 Matrix Algebra ........................................................................................................... 32 1.2.2 Rank and Row/Column Spaces .................................................................................. 32 1.2.3 Determinant and Trace ............................................................................................... 34 1.2.4 Eigenvalues and Eigenvectors .................................................................................... 35 1.2.5 Inverse of a Matrix ..................................................................................................... 39 1.2.6 Similarity Transformation and Diagonalization ......................................................... 42 1.2.7 Special Matrices ......................................................................................................... 44 1.2.8 Positive Definiteness .................................................................................................. 47 1.2.9 Matrix Inversion Lemma ............................................................................................ 48 1.2.10 LU, Cholesky, QR, and Singular Value Decompositions ........................................... 49 1.2.11 Geometrical Meaning of Eigenvalues/Eigenvectors ................................................... 50 1.3 Systems of Linear Equations ............................................................................................... 55 1.3.1 Nonsingular Case ....................................................................................................... 55 1.3.2 Undetermined Case — Minimum-Norm Solution.................................................... 56 1.3.3 Overdetermined Case — Least-Squares Error Solution ........................................... 58 1.3.4 Gauss(ian) Elimination ............................................................................................... 61 1.3.5 RLS (Recursive Least Squares) Algorithm ................................................................ 65 Problems ....................................................................................................................................... 69 Chapter 2: Vector Calculus ............................................................................................................ 99 2.1 Derivatives ............................................................................................................................. 99 2.2 Vector Functions ................................................................................................................. 104 2.3 Velocity and Acceleration .................................................................................................. 107 2.4 Divergence and Curl........................................................................................................... 113 2.5 Line Integrals and Path Independence ............................................................................. 127 2.5.1 Line Integrals ............................................................................................................ 127 2.5.2 Path Independence .................................................................................................... 132 2.6 Double Integrals ................................................................................................................. 134 2.7 Green's Theorem ................................................................................................................ 138 2.8 Surface Integrals ................................................................................................................. 142 2.9 Stokes' Theorem ................................................................................................................. 149 viii Contents 2.10 Triple Integrals .................................................................................................................... 155 2.11 Divergence Theorem ........................................................................................................... 160 Problems ..................................................................................................................................... 169 Chapter 3: Ordinary Differential Equations ......................................................................191 3.1 First-Order Differential Equations ................................................................................... 192 3.1.1 Separable Equations ................................................................................................. 192 3.1.2 Exact Differential Equations and Integrating Factors ............................................. 195 3.1.3 Linear First-Order Differential Equations .............................................................. 200 3.1.4 Nonlinear First-Order Differential Equations ........................................................... 204 3.1.5 Systems of First-Order Differential Equations ....................................................... 205 3.2 Higher-Order Differential Equations ............................................................................... 213 3.2.1 Undetermined Coefficients ....................................................................................... 213 3.2.2 Variation of Parameters .......................................................................................... 222 3.2.3 Cauchy–Euler Equations ........................................................................................ 225 3.2.4 Systems of Linear Differential Equations................................................................. 229 3.3 Special Second-Order Linear ODEs ................................................................................. 233 3.3.1 Bessel's Equation ...................................................................................................... 233 3.3.2 Legendre's Equation ............................................................................................... 239 3.3.3 Chebyshev's Equation............................................................................................. 241 3.3.4 Hermite's Equation ................................................................................................. 244 3.3.5 Laguerre's Equation ................................................................................................ 246 3.4 Boundary Value Problems ................................................................................................. 248 Problems ..................................................................................................................................... 257 Chapter 4: The Laplace Transform ....................................................................................277 4.1 Definition of the Laplace Transform ................................................................................ 277 4.1.1 Laplace Transform of the Unit Step Function .......................................................... 277 4.1.2 Laplace Transform of the Unit Impulse Function .................................................. 278 4.1.3 Laplace Transform of the Ramp Function .............................................................. 280 4.1.4 Laplace Transform of the Exponential Function ...................................................... 281 4.1.5 Laplace Transform of the Complex Exponential Function .................................... 281 4.2 Properties of the Laplace Transform ................................................................................ 281 4.2.1 Linearity ................................................................................................................... 281 4.2.2 Time Differentiation ................................................................................................. 281 4.2.3 Time Integration ....................................................................................................... 282 4.2.4 Time Shifting — Real Translation ......................................................................... 282 4.2.5 Frequency Shifting — Complex Translation .......................................................... 282 4.2.6 Real Convolution ...................................................................................................... 282 4.2.7 Partial Differentiation ............................................................................................... 283 4.2.8 Complex Differentiation ........................................................................................... 283 4.2.9 Initial Value Theorem (IVT) .................................................................................... 284 4.2.10 Final Value Theorem (FVT) .................................................................................... 284 4.3 The Inverse Laplace Transform ........................................................................................ 287 4.4 Using the Laplace Transform ............................................................................................ 289 4.5 Transfer Function of a Continuous-Time System ........................................................... 295 Problems ..................................................................................................................................... 300 Contents ix Chapter 5: The Z-transform .................................................................................................... . 309 5.1 Definition of the Z-transform ............................................................................................ 309 5.2 Properties of the Z-transform ........................................................................................... 314 5.2.1 Linearity ................................................................................................................... 314 5.2.2 Time Shifting — Real Translation ......................................................................... 314 5.2.3 Frequency Shifting — Complex Translation .......................................................... 315 5.2.4 Time Reversal........................................................................................................... 315 5.2.5 Real Convolution ...................................................................................................... 316 5.2.6 Complex Convolution .............................................................................................. 316 5.2.7 Complex Differentiation ........................................................................................... 317 5.2.8 Partial Differentiation ............................................................................................... 317 5.2.9 Initial Value Theorem .............................................................................................. 317 5.2.10 Final Value Theorem ................................................................................................ 318 5.3 The Inverse Z-transform .................................................................................................... 318 5.4 Using the Z-transform ........................................................................................................ 322 5.5 Transfer Function of a Discrete-Time System ................................................................. 324 5.6 Differential Equation and Difference Equation ............................................................... 327 Problems ..................................................................................................................................... 329 Chapter 6: Fourier Series and Fourier Transform ............................................................337 6.1 Continuous-Time Fourier Series (CTFS) ......................................................................... 337 6.1.1 Definition and Convergence Conditions .................................................................. 337 6.1.2 Examples of CTFS ................................................................................................... 340 6.2 Continuous-Time Fourier Transform (CTFT) ................................................................ 348 6.2.1 Definition and Convergence Conditions .................................................................. 348 6.2.2 (Generalized) CTFT of Periodic Signals .................................................................. 351 6.2.3 Examples of CTFT ................................................................................................... 352 6.2.4 Properties of CTFT ................................................................................................... 357 6.3 Discrete-Time Fourier Transform (DTFT) ...................................................................... 363 6.3.1 Definition and Convergence Conditions .................................................................. 363 6.3.2 Examples of DTFT ................................................................................................... 364 6.3.3 DTFT of Periodic Sequences .................................................................................... 367 6.3.4 Properties of DTFT .................................................................................................. 369 6.4 Discrete Fourier Transform (DFT) ................................................................................... 373 6.5 Fast Fourier Transform (FFT) .......................................................................................... 377 6.5.1 Decimation-in-Time (DIT) FFT ............................................................................... 377 6.5.2 Decimation-in-Frequency (DIF) FFT ....................................................................... 380 6.5.3 Computation of IDFT Using FFT Algorithm ........................................................... 382 6.5.4 Interpretation of DFT Results ................................................................................... 382 6.6 Fourier–Bessel/Legendre/Chebyshev/Cosine/Sine Series ................................................ 385 6.6.1 Fourier–Bessel Series ............................................................................................... 385 6.6.2 Fourier–Legendre Series .......................................................................................... 388 6.6.3 Fourier–Chebyshev Series ........................................................................................ 389 6.6.4 Fourier–Cosine/Sine Series ...................................................................................... 391 Problems ..................................................................................................................................... 395