Engineering mathematics/ S.S.Sastry

1. Differentiation (Review) 3-11
1.1 Differential Coefficient 3
1.2 Formulae for Differentiation 3
1.2.1 Other Useful Formulae 4
Exercise 1.1 10
2. Successive Differentiation 12-23
2.1 Higher Derivatives of Functions 12
Exercise 2.1 16
2.2 Leibnitz's Theorem 17
Exercise 2.2 22
3. Partial Differentiation 24-40
3.1 Introduction 24
3.1.1 Partial Derivatives 24
3.1.2 Geometrical Interpretation of dzJ^x and dzl^y 26
3.1.3 Partial Derivatives of Higher Orders 27
Exercise 3.1 30
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iv Contents
3.2 Homogeneous Functions 31
3.2.1 Euler's Theorem 32
3.2.2 Deductions from Euler's Theorem 33
Exercise 3.2 38
4. Total Differentiation 41-58
4.1 Total Differential Coefficient 41
4.1.1 Important Deductions 42
Exercise 4.1 51
4.2 Change of Variables 52
4.2.1 Change of Independent Variable to the Dependent
Variable 53
4.2.2 Change to Another Variable 53
4.2.3 Change of Both Dependent and Independent
Variables 54
4.2.4 Change of Two Independent Variables 54
Exercise 4.2 57
5. Expansion of Functions of Several Variables 59-66
5.1 Taylor's Series 59
Exercise 5.1 65
6. Curve Tracing 67-87
6.1 Introduction 67
6.1.1 Definitions 67
6.1.2 Asymptotes of a Curve 69
6.1.3 Tracing of Cartesian Curves 71
6.1.4 Sketches of Some Important Curves 75
Exercise 6.1 76
6.2 Tracing of Polar Curves 76
6.2.1 Sketches of Some Important Polar Curves 81
Exercise 6.2 81
6.3 Tracing of Curves Given in Parametric Form 81
Unit n Differential Calculus H
7. Jacobians 91-101
7.1 Definition 91
7.1.1 Properties of Jacobians 92
7.1.2 Jacobians of Implicit Functions 97
Exercise 7.1 100
8. Approximation of Errors 102-109
8.1 Differentials 102
8.1.1 Definitions 103
Exercise 8.1 108
9. Extrema of Functions of Several Variables 110-124
9.1 Maxima and Minima 110
9.1.1 Criteria for Maxima or Minima 110
Exercise 9.1 117
9.2 Lagrange's Method of Undetermined Multipliers 118
Exercise 9.2 123
Unit III Matrices
10. Algebra of Matrices 127-189
10.1 Introduction 127
10.1.1 Basic Definitions 127
10.1.2 Matrix Operations 129
10.1.3 Determinants 132
10.1.4 Transpose, Adjoint and Inverse of a Matrix 134
10.1.5 Some Special Matrices 138
Exercise 10.1 143
10.2 Elementary Transformations 145
10.2.1 Elementary Matrices 145
10.2.2 Rank of the Matrix 146
Exercise 10.2 152
10.3 Linear Dependence and Linear Independence of
Vectors 153
Exercise 10.3 155
10.4 Solution of a System of Linear Equations 156
10.4.1 Consistency of a Nonhomogenous Linear
System 156
10.4.2 Solution of Homogeneous Systems 160
10.4.3 Solution of AX = 5 when Ais Nonsingular 162
Exercise 10.4 165
10.5 Eigenvalues and Eigenvectors of Matrices 166
10.5.1 Eigenvalues of Some Important Matrices * 167
10.5.2 Properties of Eigenvalues and Eigenvectors 174
10.5.3 Cayley-Hamilton Theorem 177
10.5.4 Diagonalization of a Matrix 179
Contents V
10.5.5 Diagonalization by Orthogonal Transformation 183
Exercise 10.5 186
Unit IV Multiple Integrals
11. Double Integration 193-215
11.1 Double Integral 193
11.1.1 Evaluation of a Double Integral 194
11.1.2 Change of Order of Integration 199
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11.1.3 Double Integral in Polar Coordinates 205
Exercise 11.1 208
11.2 Applications of Double Integrals 209
11.2.1 Moments and Centroid of Area 210
11.2.2 Moments of Inertia 210
11.2.3 Volume of a Solid of Revolution 212
11.2.4 Volume of a Solid as a Double Integral 212
Exercise 11.2 214
12. Triple Integrals 216-231
12.1 Volume as a Triple Integral 216
12.1.1 Cylindrical Coordinates 220
12.1.2 Spherical Polar Coordinates 221
12.1.3 Applications of Triple Integrals 222
12.1.4 Change of Variables 225
Exercise 12.1 229
13. Beta and Gamma Functions Ihl-IAl
13.1 Introduction 252
13.1.1 A Formula for r(n) 233
13.1.2 Transformations of Gamma Function 235
13.1.3 Transformations of Beta Function 236
13.1.4 Relation between Beta and Ganrmia Functions 237
13.1.5 Deductions 239
Exercise 13.1 246
Unit V Vector Calculus '
14. Vector Differential Calculus 251-278
14.1 Introduction 251
14,1.1 Scalar and Vector Fields 251
•14.1.2 Derivatives of Vectors 251
14.1.3 The Directional Derivative 255
Exercise 14.1 261
14.2 Divergence and Curl of a Vector Point Function 263
14.2.1 Divergence of a Vector Point Function 263
14.2.2 The Curl of a Vector 267
14.2.3 Second-Order Expressions 272
Exercise 14.2 275
15. Vector Integration 279-304
15.1 Introduction 279
15.1.1 Line Integrals 279
15.1.2 Surface and Volume Integrals 284
. Exercise 15.1 287
15.2 Integral Theorems 289
15.2.1 Green's Theorem for a Plane 289
15.2.2 Stoke's Theorem 292
15.2.3 Divergence Theorem of Gauss 296
Exercise 15.2 301
UPTU Examination Paper 305-307
PART II (SECOND SEMESTER)
Unit I Differential Equations
1. Differential Equations (Revision) 311-324
1.1 Introduction 311
1.1.1 Variables Separable 311
1.1.2 Exact Differential Equations 313
1.1.3 Homogeneous Equations 314
1.1.4 First-Order Linear Differential Equations 316
1.1.5 Bernoulli's Differential Equation 318
1.2 Differential Equations of First Order and Higher
Degree 319
Exercise 1.1 322
2. Linear Differential Equations 325-370
2.1 Differential Equations of Higher Order 325
2.1.1 Homogeneous Linear Equations 327
2.1.2 Nonhomogeneous Linear Differential Equations 331
2.1.3 Method of Variation of Parameters 332
2.1.4 The Operator Method for Particular Integrals 336
Exercise 2.1 344
2.2 Simultaneous Linear Differential Equations 345
Exercise 2.2 350
2.3 Solution of Certain Types of Differential Equations 351
Exercise 2.3 358
2.4 Linear Equations with Variable Coefficients 359
2.4.1 Change of Independent Variable 359
2.4.2 Method of Variation of Parameters 363
Exercise 2.4 367
3. Applications of Differential Equations 371-402
3.1 Introduction 371
3.1.1 Electrical Circuits 371
3.1.2 Motion under Gravity 380
3.1.3 Elastic Strings 384
Contents Vii
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3.1.4 Vibration of a Particle
3.1.5 Elastic Curves 391
3.1.6 Motion of a Particle in a Plane 397
Exercise 3.1 399
Unit II Series Solutions and Special Functions
4. Solution in Series of Differential Equations 405-420
4.1 Ordinary Points of a Differential Equation 405
4.1.1 Solution of Differential Equations with Ordinary
Points 405
4.2 Singular Points of a Differential Equation 408
4.2.1 Roots Distinct and do not Differ by an
Integer 408
4.2.2 The Roots C| and C2 are Equal 412
4.2.3 Roots Distinct and Differ by an Integer 416
Exercise 4.1 419
5. Bessel Functions 421-440
5.1 Introduction 421
5.2 Bessel's Differential Equation and Bessel Functions 421
5.3 Recurrence Formulae for y„(jc) 426
5.4 Generating Function for Jn{x) 430
5.5 Integral Relations Involving Bessel Functions 435
5.6 Bessel Functions of the Second Kind 435
Exercise 5.1 439
6. Legendre Polynomials 441-460
6.1 Legendre's Equation and Its Solutions 441
6.1.1 Solution of Legendre's Equation 441
6.2 Rodrigue's Formula 444
6.3 Generating Function 447
6.4 Orthogonal Properties 448
6.5 Recurrence Formulae 452
6.6 Legendre's Polynomials as Integrals 454
Exercise 6.1 458
Unit III Laplace Transforms
7. Laplace Transforms 463-509
7.1 Introduction 463
7.1.1 Definition and Conditions for Existence 463
7.1.2 Laplace Transforms of Some Elenientary
Functions 465
7.1.3 Table of Laplace Transforms 470
7.1.4 Properties of Laplace Transforms 470
7.1.5 Transforms of Derivatives 484
Exercise 7.1 486
7.2 The Inverse Laplace Transform 487
1.2.1 Definition and Uniqueness 487
7.2.2 Properties of Inverse Transforms 489
7.2.3 Use of Partial Fractions 496
Exercise 7.2 499
7.3 Applications to Differential Equations 500
7.3.1 Linear Differential Equations with Constant
Coefficients 500
7.3.2 Linear Differential Equations with Variable
Coefficients 503
7.3.3 Simultaneous Differential Equations 504
Exercise 7.3 506
Unit IV Fourier Series and Partial Differential Equations
8. Fourier Series 513-556
9.
8.1 Introduction 513
8.1.1 Definitions and Derivations 513
8.1.2 Odd and Even Functions 523
Exercise 8.1 530
8.2 Half-Range Series from 0 to ;r 531
Exercise 8.2 535
8.3 Change of Scale 536
8.3.1 Parseval's Equalities 542
Exercise 8.3 548
8.4 Numerical Harmonic Analysis 550
Exercise 8.4 552
Contents ix
Partial Differential Equations 557-586
9.1 Introduction 557
9.1.1 Formation of Partial Differential Equations 559
9.1.2 Solution of a Partial Differential Equation 562
Exercise 9.1 564
9.2 Partial Differential Equations of the First Order 565
9.2.1 Solutions of Some Standard Types 565
9.2.2 Lagrange's Linear Equation 570
Exercise 9.2 575
9.3 Nonlinear First-Order Partial Differential Equations 576
9.3.1 Some Standard Types 576
9.3.2 Charpit's Method 580
Exercise 9.3 585
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10. Partial Differential Equations of the Second Order 587-608
10.1 Introduction 587
10.1.1 Classification of Second-Order Linear
Equations 587
10.1.2 Homogeneous Linear Partial Differential
Equations 589
10.1.3 Linear Non-homogeneous Equations 591
Exercise 10.1 592
10.2 Evaluation of Particular Integrals 593
10.2.1 Method of Separation of Variables 603
Exercise 10.2 606
Unit V Applications of Partial Differential Equations
11. Initial Boundary-Value Problems 611-639
11.1 Introduction 611
11.1.1 One-Dimensional Wave Equation (Vibrations
of a String) 611
11.1.2 One-Dimensional Heat Flow 620
11.1.3 Two-Dimensional Heat Flow 626
11.1.4 Laplace's Equation in Polar Coordinates 632
11.1.5 Equations of Transmission Lines 634