Engineering Mathematics- II (As per the new syllabus of VTU (B. E. , II Semester)

Cover
......Page 1
Preface......Page 6
Acknowledgement......Page 7
Contents......Page 12
1.2 Radius of Curvature......Page 18
1.2.1 Radius of Curvature in Cartesian Form......Page 19
1.2.2 Radius of Curvature in Parametric Form......Page 20
Worked Out Examples
......Page 21
Exercise 1.1......Page 35
1.2.4 Radius of Curvature in Polar Form......Page 36
Worked Out Examples......Page 38
Exercise 1.2......Page 43
1.3.2 Lagrange’s Mean Value Theorem......Page 44
1.3.3 Cauchy’s Mean Value Theorem......Page 45
1.3.4 Taylor’s Theorem......Page 46
Worked Out Examples......Page 47
Exercise 1.3......Page 67
Additional Problems (From Previous Years VTU Exams.)......Page 69
Objective Questions......Page 74
2.1.1 Indeterminate Form 0/0
......Page 78
Wroked Out Examples......Page 79
Exercise 2.1......Page 85
Worked Out Examples......Page 86
Worked Out Examples......Page 91
Exercise 2.3......Page 94
Worked Out Examples......Page 95
Exercise 2.4......Page 99
2.3.1 Necessary and Sufficient Conditions for Maxima and Minima......Page 100
Worked Out Examples......Page 101
2.4 Lagrange’s Method of Undetermined Multipliers
......Page 106
Working Rules......Page 107
Worked Out Examples......Page 108
Additional Problems (From Previous Years VTU Exams.)
......Page 111
Objective Questions......Page 118
3.3 Double Integrals
......Page 122
Worked Out Examples......Page 123
Exercise 3.1......Page 129
3.3.3 Applications to Area and Volume......Page 130
Type 1. Evaluation over a given region......Page 131
Type 2. Evaluation of a double integral by changing the order of integration......Page 136
Type 3. Evaluation by changing into polars......Page 139
Type 4. Applications of double and triple integrals......Page 141
Exercise 3.2......Page 145
3.4.2 Properties of Beta and Gamma Functions......Page 146
3.4.3 Relationship between Beta and Gamma functions......Page 150
Worked Out Examples......Page 152
Exercise 3.3......Page 173
Additional Problems (From Previous Years VTU Exams)
......Page 176
Objective Questions......Page 179
Worked Out Examples......Page 183
Exercise 4.1......Page 189
4.3.2 Surface integral and Volume integral......Page 190
Worked Out Examples......Page 191
Exercise 4.2......Page 205
4.4.2 Unit Tangent and Unit Normal Vectors......Page 206
4.4.3 The Differential Operators......Page 208
Worked Out Examples......Page 209
4.4.4. Divergence of a Vector......Page 211
Worked Out Examples......Page 212
4.4.5 Curl of a Vector......Page 213
Worked Out Examples......Page 214
Exercise 4.5......Page 215
4.4.7. Particular Coordinate System......Page 216
Worked Out Examples......Page 220
Additional Problems......Page 225
Objective Questions......Page 227
5.2 Linear Differential Equations of Second and Higher Order with Constant Coefficients
......Page 231
Worked Out Examples......Page 232
Exercise 5.1......Page 236
5.4 Inverse Differential Operator And Particular Integral
......Page 237
5.5 Special Forms of X
......Page 238
Worked Out Examples......Page 241
Exercise 5.2......Page 253
Exercise 5.3......Page 258
5.6 Method of Undetermined Coefficients......Page 268
Worked Out Examples......Page 269
Exercise 5.5......Page 279
5.7 Solution of Simultaneous Differential Equations
......Page 281
Worked Out Examples......Page 282
Exercise 5.6......Page 284
Additional Problems (From Previous Years VTU Exams.)
......Page 285
Objective Questions......Page 294
6.1 Methods of Variation of Parameters......Page 297
Worked Out Examples
......Page 298
Exercise 6.1......Page 308
6.2 Solution of Cauchy’s Homogeneous Linear Equation And Lengendre’s Linear Equation
......Page 309
Worked Out Examples......Page 311
Exercise 6.2......Page 323
Worked Out Examples......Page 325
Additional Problems (From Previous Years VTU Exams.)
......Page 327
Objective Questions......Page 335
7.3 Properties of Laplace Transforms
......Page 338
7.3.1 Laplace Transforms of Some Standard Functions......Page 339
Worked Out Examples......Page 342
Exercise 7.1......Page 347
7.3.2 Laplace Transforms of the form eat f (t)......Page 348
Worked Out Examples......Page 349
Exercise 7.2......Page 352
7.3.3 Laplace Transforms of the form t n f (t) Where n is a Positive Integer......Page 353
Worked Out Examples......Page 354
Exercise 7.3......Page 362
7.4 Laplace Transforms of Periodic Functions
......Page 363
Worked Out Examples......Page 364
Exercise 7.3......Page 368
7.5.1 Properties Associated with the Unit Step Function......Page 369
7.5.2 Laplace Transform of the Unit Impulse Function......Page 371
Exercise 7.4......Page 376
Additional Problems (From Previous Years VTU Exams.)......Page 377
Objective Questions......Page 382
8.2 Inverse Laplace Transforms of Some Standard Functions
......Page 386
Worked Out Examples......Page 389
8.3 Inverse Laplace Transforms Using Partial Fractions
......Page 393
Exercise 8.1......Page 399
Worked Out Example
......Page 401
Exercise 8.2......Page 404
8.5 Convolution Theorem......Page 405
Worked out Examples......Page 406
Exercise 8.3......Page 413
8.6 Laplace Transforms of the Derivatives......Page 414
Worked Out Examples......Page 415
Solution of Simultaneous Differential Equations......Page 423
Exercise 8.4......Page 427
8.8 Applications of Laplace Transforms
......Page 428
Worked Out Examples......Page 429
Exercise 8.5......Page 433
Additional Problems (From Previous Years VTU Exams.)......Page 434
Objective Questions......Page 439
Model Question Paper–I
......Page 444
Model Question Paper–Il......Page 458