Ramamurti SHANKAR - Basic Training in Mathematics: A Fitness Program for Science Students


CONTENTS


1 DIFFERENTIAL CALCULUS OF ONE VARIABLE 1

 1.1. Introduction 1

 1.2. Differential Calculus 1

 1.3. Exponential and Log Functions 5

 1.4. Trigonometric Functions 19

 1.5. Plotting Functions 23

 1.6. Miscellaneous Problems on Differential Calculus 25

 1.7. Differentials 29

 1.8. Summary 30


2 INTEGRAL CALCULUS 33

 2.1. Basics of Integration 33

 2.2. Some Tricks of the Trade 44

 2.3. Summary 49


3 CALCULUS OF MANY VARIABLES 51

 3.1. Differential Calculus of Many Variables 51

  3.1.1. Lagrange multipliers 55

 3.2. Integral Calculus of Many Variables 61

  3.2.1. Solid angle 68

 3.3. Summary 72


4 INFINITE SERIES 75

 4.1. Introduction 75

 4.2. Tests for Convergence 77

 4.3. Power Series in x 80

 4.4. Summary 87


5 COMPLEX NUMBERS 89

 5.1. Introduction 89

 5.2. Complex Numbers in Cartesian Form 90

 5.3. Polar Form of Complex Numbers 94

 5.4. An Application 98

 5.5. Summary 104


6 FUNCTIONS OF A COMPLEX VARIABLE 107

 6.1. Analytic Functions 107

  6.1.1. Singularities of analytic functions Ill

  6.1.2. Derivatives of analytic functions 114

 6.2. Analytic Functions Defined by Power Series 116

 6.3. Calculus of Analytic Functions 126

 6.4. The Residue Theorem 132

 6.5. Taylor Series for Analytic Functions 139

 6.6. Summary 144


7 VECTOR CALCULUS 149

 7.1. Review of Vectors Analysis 149

 7.2. Time Derivatives of Vectors 156

 7.3. Scalar and Vector Fields 158

 7.4. Line and Surface Integrals 159

 7.5. Scalar Field and the Gradient 167

  7.5.1. Gradient in other coordinates 171

 7.6. Curl of a Vector Field 172

  7.6.1. Curl in other coordinate systems 181

 7.7. The Divergence of a Vector Field 182

  7.7.1. Divergence in other coordinate systems 186

 7.8. Differential Operators 187

 7.9. Summary of Integral Theorems 188

 7.10. Higher Derivatives 189

  7.10.1. Digression on vector identities 190

 7.11. Applications from Electrodynamics 193

 7.12. Summary 202


8 MATRICES AND DETERMINANTS 205

 8.1. Introduction 205

 8.2. Matrix Inverses 211

 8.3. Determinants 214

 8.4. Transformations on Matrices and Special Matrices 219

  8.4.1. Action on row vectors 223

 8.5. Summary 227


9 LINEAR VECTOR SPACES 229

 9.1. Linear Vector Spaces: Basics 229

 9.2. Inner Product Spaces 237

 9.3. Linear Operators 247

 9.4. Some Advanced Topics 253

 9.5. The Eigenvalue Problem 255

  9.5.1. Degeneracy 263

 9.6. Applications of Eigenvalue Theory 266

  9.6.1. Normal modes of vibration: two coupled masses 267

  9.6.2. Quadratic forms 274

 9.7. Function Spaces 277

  9.7.1. Generation of orthonormal bases 279

  9.7.2. Fourier integrals 287

  9.7.3. Normal modes: vibrating string 289

 9.8. Some Terminology 294

 9.9. Tensors: An Introduction 294

 9.10. Summary 300


10 DIFFERENTIAL EQUATIONS 305

 10.1. Introduction 305

 10.2. ODEs with Constant Coefficients 307

 10.3. ODEs with Variable Coefficients: First Order 315

 10.4. ODEs with Variable Coefficients: Second Order and Homogeneous 318

  10.4.1. Frobenius method for singular coefficients 324

 10.5. Partial Differential Equations 329

  10.5.1. The wave equation in one and two space dimensions .... 329

  10.5.2. The heat equation in one and two dimensions 336

 10.6. Green's Function Method 344

 10.7. Summary 347


ANSWERS 351

INDEX 359


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