Engineering Mathematics

The book is designed as a self-contained, comprehensive study mate- rial. Each topic is treated in systematic and logical manner. A large number of Worked Examples and graded exercises with answers form an integral part of the text. It is believed that the book will fulfill the needs of students and teachers for whom it is intended. Constructive criticism, comments and suggestions are most welcome.

1 Differential Calculus – I

1.1 Successive Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 nth derivatives of some standard functions . . . . . . . . . . . . . . . . 2

1.1.2 Leibniz's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Differential Calculus – II

2.1 Polar Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.1.1 Angle between radius vector and tangent . . . . . . . . . . . . . . . . 35

2.1.2 Angle between Polar Curves . . . . . . . . . . . . . . . . . . . . . . . 40

2.1.3 Perpendicular from the pole on the tangent; Pedal Equations . . . . . . 46

2.2 Derivatives of arc-length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.3 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.3.1 Radius of curvature in Cartesian form . . . . . . . . . . . . . . . . . . 65

2.3.2 Radius of curvature in parametric form . . . . . . . . . . . . . . . . . 72

2.3.3 Radius of curvature in pedal form . . . . . . . . . . . . . . . . . . . . 77

2.3.4 Radius of curvature in polar form . . . . . . . . . . . . . . . . . . . . 79

3 Differential Calculus – III

3.1 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.2 Taylor's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.3 Maclaurin's Theorem/Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.4 Indeterminate forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.4.1 Evaluation of limits of the form. . . . . . . . . . . . . . . . . . 111

3.4.2 Evaluation of limits of the form 0 ×∞, ∞−∞ . . . . . . . . . . . . . 121

3.4.3 Evaluation of limits of the form 00, ∞0,1∞

4 Differential Calculus – IV 135–208

4.1 Partial Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

4.1.1 Partial Differentiation of Homogeneous Functions; Euler's Theorem . . 153

4.1.2 Total Differential and Total Derivative . . . . . . . . . . . . . . . . . . 164

4.1.3 Differentiation of Implicit Functions . . . . . . . . . . . . . . . . . . . 168

4.1.4 Partial Differentiation of Composite Functions . . . . . . . . . . . . . 171

4.2 Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

4.3 Extreme Values of f(x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Calculus 209–267

5.1 Vector Function of a Single Variable . . . . . . . . . . . . . . . . . . . . . . . 209

5.2 Scalar and Vector point functions . . . . . . . . . . . . . . . . . . . . . . . . . 224

5.2.1 Gradient of a Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

5.2.2 Some Geometrical Considerations . . . . . . . . . . . . . . . . . . . . 230

5.2.3 Divergence of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . 237

5.2.4 Laplacian of a Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

5.2.5 Curl of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

5.2.6 Some More Vector Identities . . . . . . . . . . . . . . . . . . . . . . . 261

6 Integral Calculus

6.2.2 Reduction Formula for

6.2.3 Reduction Formula for

6.1 Differentiation under the Integral Sign . . . . . . . . . . . . . . . . . . . . . . 268

6.2 Reduction Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

6.2.1 Reduction Formula

6.3 Tracing of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

6.3.1 Cartesian Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

6.3.2 Polar Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

7 Differential Equations

7.1 Some Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

7.2 Equations of First-Order and First-Degree . . . . . . . . . . . . . . . . . . . . 319

7.2.1 Linear Equations; Bernoulli Equation . . . . . . . . . . . . . . . . . . 320

7.2.2 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

7.2.3 Equations Reducible to Exact Equations . . . . . . . . . . . . . . . . . 343

7.3 Orthogonal Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

7.3.1 Orthogonal Trajectories in Cartesian form . . . . . . . . . . . . . . . . 356

7.3.2 Orthogonal Trajectories in Polar form . . . . . . . . . . . . . . . . . . 365

7.4 Some other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

8 Linear Algebra – I

8.1 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

8.2 Elementary Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

8.3 Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

8.4 Homogeneous Systems of Linear Equations . . . . . . . . . . . . . . . . . . . 415

8.5 Non-homogeneous Systems of Linear Equations . . . . . . . . . . . . . . . . . 428

8.5.1 LU-Decomposition Method . . . . . . . . . . . . . . . . . . . . . . . 448

8.5.2 Gauss-Seidel Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

9 Linear Algebra – II

9.1 Eigenvalues and Eigenvectors of a Square Matrix . . . . . . . . . . . . . . . . 466

9.1.1 Rayleigh's Power Method . . . . . . . . . . . . . . . . . . . . . . . . 475

9.2 Reduction of a square matrix to Diagonal form . . . . . . . . . . . . . . . . . 483

9.3 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500

9.4 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

9.4.1 Orthogonal Reduction of a Quadratic form to Canonical form . . . . . 508

Appendix A : Derivatives of some standard functions

Appendix B : Integrals of some standard functions

Syllabus

Index