Engineering mathematics with MATLAB

756 Pages·2018·35.025 MB·English

by  Won Y. Yang, Young K. Choi, Jaekwon Kim, Man Cheol Kim, H. Jin Kim, Taeho Im

ENGINEERING
MATHEMATICS
MATLAB®
with

ENGINEERING
MATHEMATICS
MATLAB®
with
Won Y. Yang
Young K. Choi
Jaekwon Kim
Man Cheol Kim
H. Jin Kim
Taeho Im
CRC Press
Taylor & Francis Group
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Version Date: 20171028
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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