Page 12 - Advanced engineering mathematics

vi Contents

3.3.1 The First Shifting Theorem 84
3.3.2 The Heaviside Function and Pulses 86
3.3.3 Heaviside’s Formula 93
3.4 Convolution 96
3.5 Impulses and the Delta Function 102
3.6 Solution of Systems 106
3.7 Polynomial Coefficients 112
3.7.1 Differential Equations with Polynomial Coefficients 112
3.7.2 Bessel Functions 114

CHAPTER 4 Series Solutions 121
4.1 Power Series Solutions 121
4.2 Frobenius Solutions 126
CHAPTER 5 Approximation of Solutions 137
5.1 Direction Fields 137
5.2 Euler’s Method 139
5.3 Taylor and Modified Euler Methods 142

PART 2 Vectors, Linear Algebra, and Systems
of Linear Differential Equations 145

CHAPTER 6 Vectors and Vector Spaces 147
6.1 Vectors in the Plane and 3-Space 147
6.2 The Dot Product 154
6.3 The Cross Product 159
6.4 The Vector Space R n 162
6.5 Orthogonalization 175
6.6 Orthogonal Complements and Projections 177
6.7 The Function Space C[a,b] 181

CHAPTER 7 Matrices and Linear Systems 187
7.1 Matrices 187
7.1.1 Matrix Multiplication from Another Perspective 191
7.1.2 Terminology and Special Matrices 192
7.1.3 Random Walks in Crystals 194
7.2 Elementary Row Operations 198
7.3 Reduced Row Echelon Form 203
7.4 Row and Column Spaces 208
7.5 Homogeneous Systems 213
7.6 Nonhomogeneous Systems 220
7.7 Matrix Inverses 226
7.8 Least Squares Vectors and Data Fitting 232
7.9 LU Factorization 237
7.10 Linear Transformations 240

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