A Textbook of Engineering Mathematics-I, 2nd Edition

Preface to the Second Revised Edition
......Page 6
Some Useful Formulae......Page 8
Contents......Page 10
1.1 nth Derivative of Some Elementary Functions......Page 14
Exercise 1.1......Page 19
1.2 Leibnitz's Theorem......Page 20
Exercise 1.2......Page 26
Exercise 1.3......Page 32
1.3 Function of Two Variables......Page 33
1.4 Partial Differential Coefficients......Page 34
Exercise 1.4......Page 46
1.5 Homogeneous Function......Page 48
1.6 Euler's Theorem on Homogeneous Functions......Page 49
Exercise 1.5......Page 60
1.7 Total Differential Coefficient......Page 61
Exercise 1.6......Page 75
Curve Tracing......Page 76
1.8 Procedure for Tracing Curves in Cartesian Form......Page 77
Exercise 1.7......Page 84
1.9 Polar Curves......Page 86
Exercise 1.8......Page 90
1.10 Parametric Curves......Page 91
Exercise 1.9......Page 93
1.11 Taylor's Theorem for Functions of Two Variables......Page 94
Exercise 1.10......Page 102
Objective Type Questions......Page 103
Answers to Objective Type Questions......Page 107
2.1 Jacobian......Page 108
Exercise 2.1......Page 122
2.2 Approximation of Errors......Page 124
Exercise 2.2......Page 132
2.3 Extrema of Function of Several Variables......Page 134
Exercise 2.3......Page 147
2.4 Lagrange's Method of Undetermined Multipliers......Page 148
Exercise 2.4......Page 158
Objective Type Questions......Page 160
Answers to Objective Type Questions......Page 163
3.1 Definition of Matrix......Page 164
3.2 Types of Matrices......Page 165
3.3 Operations on Matrices......Page 168
3.6 Properties of Conjugate Matrices......Page 169
3.9 Inverse of a Matrix (Reciprocal)......Page 176
Exercise 3.1......Page 178
3.10 Elementary Row and Column Transformations......Page 180
3.11 Method of Finding Inverse of a Non-Singular Matrix by Elementary Transformations......Page 181
Exercise 3.2......Page 187
3.12 Rank of a Matrix......Page 188
Exercise 3.3......Page 199
3.13 System of Linear Equations (Non-Homogeneous)......Page 201
3.14 System of Homogeneous Equations......Page 210
3.15 Gaussian Elimination Method......Page 213
Exercise 3.4......Page 219
3.16 Linear Dependence of Vectors
......Page 223
3.17 Eigen Values and Eigen Vectors......Page 227
Exercise 3.6......Page 243
3.18 Cayley-Hamilton Theorem......Page 245
Exercise 3.7......Page 251
3.19 Diagonalization of a Matrix......Page 252
3.20 Application of Matrices to Engineering Problems......Page 262
Exercise 3.8......Page 266
Objective Type Questions......Page 268
Answers to Objective Type Questions......Page 270
4.2 Double Integrals......Page 271
4.4 Double Integration for Polar Curves......Page 272
Exercise 4.1......Page 279
4.5 Change of the Order of Integration......Page 281
4.6 Change of Variables in a Multiple Integral
......Page 287
Exercise 4.2......Page 294
4.7 Beta and Gamma Functions......Page 296
4.8 Transformations of Gamma Function
......Page 298
4.9 Transformations of Beta Function
......Page 299
4.11 Some Important Deductions......Page 300
4.12 Duplication Formula......Page 302
4.13 Evaluate the Integrals......Page 307
Exercise 4.3......Page 312
4.14 Application to area (Double Integrals)......Page 313
Exercise 4.4......Page 324
4.15 Triple Integrals......Page 325
Exercise 4.5......Page 327
4.16 Application to Volume (Triple Integrals)......Page 328
Exercise 4.6......Page 335
4.17 Drichlet's* Theorem
......Page 336
Exercise 4.7......Page 342
Objective Type Questions......Page 343
Answers to Objective Type Questions......Page 345
5.2 Vector Differentiation......Page 346
5.3 Some Results on Differentiation......Page 347
Exercise 5.1......Page 349
5.6 Gradient or Slope of Scalar Point Function
......Page 350
5.8 Directional Derivative......Page 351
5.9 Properties of Gradient......Page 352
Exercise 5.2......Page 363
5.10 Divergence of a Vector Point Function
......Page 364
5.11 Physical Interpretation of Divergence......Page 365
5.13 Physical Meaning of a Curl......Page 366
5.14 Vector Identities......Page 367
Exercise 5.3......Page 376
5.16 Line Integral......Page 378
5.17 Surface Integral......Page 379
5.18 Volume Integral......Page 380
Exercise 5.4......Page 387
5.19 Green's Theorem......Page 389
Exercise 5.5......Page 397
5.20 Stoke's Theorem......Page 399
5.21 Cartesian Representation of Stoke's Theorem......Page 401
Exercise 5.6......Page 412
5.22 Gauss's Divergence Theorem......Page 413
5.23 Cartesian Representation of Gauss's Theorem......Page 414
Exercise 5.7......Page 426
Objective Type Questions......Page 427
Answers to Objective Type Questions
......Page 431
Unsolved Question Papers (2004-2009)......Page 432
Index......Page 446