Page 13 - Advanced engineering mathematics

Contents vii

CHAPTER 8 Determinants 247
8.1 Definition of the Determinant 247
8.2 Evaluation of Determinants I 252
8.3 Evaluation of Determinants II 255
8.4 A Determinant Formula for A −1 259
8.5 Cramer’s Rule 260
8.6 The Matrix Tree Theorem 262

CHAPTER 9 Eigenvalues, Diagonalization, and Special Matrices 267
9.1 Eigenvalues and Eigenvectors 267
9.2 Diagonalization 277
9.3 Some Special Types of Matrices 284
9.3.1 Orthogonal Matrices 284
9.3.2 Unitary Matrices 286
9.3.3 Hermitian and Skew-Hermitian Matrices 288
9.3.4 Quadratic Forms 290

CHAPTER 10 Systems of Linear Differential Equations 295
10.1 Linear Systems 295
10.1.1 The Homogeneous System X = AX. 296

10.1.2 The Nonhomogeneous System 301
10.2 Solution of X = AX for Constant A 302

10.2.1 Solution When A Has a Complex Eigenvalue 306
10.2.2 Solution When A Does Not Have n Linearly Independent Eigenvectors 308
10.3 Solution of X = AX + G 312

10.3.1 Variation of Parameters 312
10.3.2 Solution by Diagonalizing A 314
10.4 Exponential Matrix Solutions 316
10.5 Applications and Illustrations of Techniques 319
10.6 Phase Portraits 329
10.6.1 Classification by Eigenvalues 329
10.6.2 Predator/Prey and Competing Species Models 338

PART 3 Vector Analysis 343

CHAPTER 11 Vector Differential Calculus 345
11.1 Vector Functions of One Variable 345
11.2 Velocity and Curvature 349
11.3 Vector Fields and Streamlines 354
11.4 The Gradient Field 356
11.4.1 Level Surfaces, Tangent Planes, and Normal Lines 359
11.5 Divergence and Curl 362
11.5.1 A Physical Interpretation of Divergence 364
11.5.2 A Physical Interpretation of Curl 365

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