ENGINEERING MATHEMATICS

A TEXTBOOK OF ENGINEERING MATHEMATICS For B.Sc. (Engg.), B.E., B. Tech., M.E. and Equivalent Professional Examinations By N.P. BALI Formerly Principal S.B. College, Gurgaon Haryana Dr. MANISH GOYAL M.Sc. (Mathematics), Ph.D., CSIR-NET Associate Professor Department of Mathematics Institute of Applied Sciences & Humanities G.L.A. University, Mathura, U.P. LAXMI PUBLICATIONS (P) LTD BANGALORE CHENNAI COCHIN GUWAHATI HYDERABAD JALANDHAR KOLKATA LUCKNOW MUMBAI RANCHI NEW DELHI BOSTON, USA

Copyright 2014 by Laxmi Publications Pvt. Ltd. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher. Published by: LAXMI PUBLICATIONS (P) LTD 113, Golden House, Daryaganj, New Delhi-110002 Phone: 011-43 53 25 00 Fax: 011-43 53 25 28 www.laxmipublications.com info@laxmipublications.com Price: ` 875.00 Only. First Edition : 1996, Sixth Edition : 2004, Seventh Edition : 2007, Reprint : 2008, 2009, 2010, Eighth Edition : 2011, Ninth Edition : 2014 OFFICES Bangalore 080-26 75 69 30 Jalandhar 0181-222 12 72 Chennai 044-24 34 47 26 Kolkata 033-22 27 43 84 Cochin 0484-237 70 04, 405 13 03 Lucknow 0522-220 99 16 Guwahati 0361-254 36 69, 251 38 81 Mumbai 022-24 91 54 15, 24 92 78 69 Hyderabad 040-24 65 23 33 Ranchi 0651-220 44 64 EEM-0539-875-ATB ENGG MATH-BAL C Typeset at: Excellent Graphics, Delhi. Printed at:

CONTENTS 1. Complex Numbers... 1 83 1.1. Real Numbers... 1 1.2. Basic Properties of Real Numbers... 1 1.3. Complex Numbers... 2 1.4. Conjugate Complex Numbers... 2 1.5. Geometrical Representation of Complex Numbers... 2 1.6. Properties of Complex Numbers... 3 1.7. Standard Form of a Complex Number... 3 1.8. Effect of Rotation, in the Anti-clockwise Direction, Through an Angle on the Complex Number... 12 1.9. De Moivre s Theorem... 20 1.10. Roots of a Complex Number... 30 1.11. Exponential Function of a Complex Variable... 53 1.12. Circular Functions of a Complex Variable... 54 1.13. Trigonometrical Formulae for Complex Quantities... 55 1.14. Logarithms of Complex Numbers... 57 1.15. The General Exponential Function... 60 1.16. Hyperbolic Functions... 63 1.17. Formulae of Hyperbolic Functions... 65 1.18. Inverse Hyperbolic Functions... 72 1.19. C + is Method of Summation... 75 2. Theory of Equations and Curve Fitting... 84 138 2.1. Polynomial... 84 2.2. Zero Polynomial... 84 2.3. Equality of Two Polynomials... 84 2.4. Complete and Incomplete Polynomials... 84 2.5. Zero of a Polynomial... 85 2.6. Division Algorithm... 85 2.7. Polynomial Equation... 85 2.8. Root of an Equation... 85 2.9. Synthetic Division... 86 2.10. Fundamental Theorem of Algebra... 88 2.11. Multiplication of Roots... 93 2.12. Diminishing and Increasing the Roots... 94 2.13. Removal of Terms... 96 ( v )

( vi ) 2.14. Reciprocal Equations... 100 2.15. Sum of the Integral Powers of the Roots and Symmetric Functions... 105 2.16. Symmetric Functions of the Roots... 109 2.17. Descarte s Rule of Signs... 111 2.18. Cardon s Method... 111 2.19. Irreducible Case of Cardon s Solution... 116 2.20. Descarte s Method... 117 2.21. Ferrari s Solution of the Biquadratic... 120 2.22. Curve Fitting... 122 2.23. Graphical Method... 122 2.24. Method of Group Averages... 124 2.25. Equations Involving Three Constants... 126 2.26. Principle of Least Squares... 130 2.27. Method of Moments... 136 3. Matrices... 139 194 3.1. Definitions (Matrices)... 139 3.2. Addition of Matrices... 142 3.3. Multiplication of a Matrix by a Scalar... 142 3.4. Properties of Matrix Addition... 143 3.5. Matrix Multiplication... 144 3.6. Properties of Matrix Multiplication... 146 3.7. Transpose of a Matrix... 149 3.8. Properties of Transpose of a Matrix... 149 3.9. Symmetric Matrix... 150 3.10. Skew-symmetric Matrix (or Anti-symmetric Matrix)... 150 3.11. Every Square Matrix can Uniquely be Expressed as the Sum of a Symmetric Matrix and a Skew-symmetric Matrix... 151 3.12. Orthogonal Matrix... 151 3.13. For any Two Orthogonal Matrices A and B, Show that AB is an Orthogonal Matrix... 151 3.14. Adjoint of a Square Matrix... 152 3.15. Singular and Non-singular Matrices... 153 3.16. Inverse (or Reciprocal) of a Square Matrix... 153 3.17. The Inverse of a Square Matrix, if it Exists, is Unique... 153 3.18. Theorem : The Necessary and Sufficient Condition for a Square Matrix A to Possess Inverse is that A 0 (i.e., A is Non-singular)... 153 3.19. If A is Invertible, Then so is A 1 and (A 1 ) 1 = A... 155 3.20. If A and B be Two Non-singular Square Matrices of the Same Order, then (AB) 1 = B 1 A 1... 155 3.21. If A is a Non-singular Square Matrix, then so is A and (A ) 1 = (A 1 )... 155 3.22. If A and B are Two Non-singular Square Matrices of the Same Order, then adj(ab) = (adj B) (adj A)... 156

( vii ) 3.23. Elementary Transformations (or Operations)... 157 3.24. Elementary Matrices... 158 3.25. The Following Theorems on the Effect of E-operations on Matrices Hold Good... 158 3.26. Inverse of Matrix by E-operations (Gauss-jordan Method)... 159 3.27. Rank of a Matrix... 160 3.28. Solution of a System of Linear Equations... 165 3.29. Vectors... 171 3.30. Linear Dependence and Linear Independence of Vectors... 171 3.31. Linear Transformations... 172 3.32. Orthogonal Transformation... 173 3.33. Complex Matrices... 175 3.34. Characteristic Equation... 178 3.35. Eigen Vectors... 178 3.36. Cayley Hamilton Theorem... 181 3.37. Reduction of a Matrix to Diagonal Form... 184 3.38. Quadratic Forms... 186 3.39. Linear Transformation of a Quadratic Form... 187 3.40. Canonical Form... 187 3.41. Index and Signature of the Quadratic Form... 188 3.42. Definite, Semi-definite and Indefinite Real Quadratic Forms... 188 3.43. Law-of-inertia of Quadratic Form... 188 3.44. Reduction to Canonical Form by Orthogonal Transformation... 191 4. Analytical Solid Geometry... 195 336 4.1. Introduction... 195 4.2. Co-ordinate Axes and Co-ordinate Planes... 195 4.3. Co-ordinates of a Point... 195 4.4. Distance between Two Points... 197 4.5. Section Formula... 198 4.6. Centroid of a Triangle... 201 4.7. Tetrahedron... 201 4.8. Centroid of a Tetrahedron... 202 4.9. Angle between Two Skew (or Non-coplanar) Lines... 203 4.10. Direction Cosines of a Line... 203 4.11. A Useful Result... 203 4.12. Relation between Direction Cosines... 204 4.13. Direction Ratios of a Line... 205 4.14. Direction Ratios of the Line Joining Two Points... 206 4.15. Angle between Two Lines... 206 4.16. Find the Angle between Two Lines whose Direction Ratios are a 1, b 1, c 1 and a 2, b 2, c 2. Deduce the Condition for Perpendicularity and Parallelism of Two Lines... 208

( viii ) 4.17. Projection... 216 4.18. To Prove that the Projection of the Join of two Points (x 1, y 1, z 1 ), (x 2, y 2, z 2 ) on a Line whose Direction Cosines are l, m, n is l(x 2 x 1 ) + m(y 2 y 1 ) + n(z 2 z 1 )... 216 4.19. The Plane... 218 4.20. General Equation of First Degree in x, y, z Represents a Plane... 218 4.21. Intercept Form... 219 4.22. Normal Form... 221 4.23. Three Point Form... 223 4.24. (a) Angle between Two Planes... 225 4.24. (b) Perpendicular Distance of a Point from a Plane... 227 4.25. Any Plane Through the Intersection of Two Given Planes... 229 4.26. Planes Bisecting the Angles between Two Planes... 231 4.27. Projection on a Plane... 232 4.28. Theorem... 232 4.29. General Form... 237 4.30. Symmetrical Form... 237 4.31. Reduction of the General Equations to the Symmetrical Form... 241 4.32. Perpendicular Distance Formula... 242 4.33. To Find the Point of Intersection of the Line x x 1 y y1 z z1 l m n with the plane ax + by + cz + d = 0... 248 4.34. The Conditions that the Line x x 1 y y1 z z1 may be Parallel to l m n the Plane ax + by + cz + d = 0 are al + bm + cn = 0 and ax 1 + by 1 + cz 1 + d 0... 249 4.35. The Conditions that the Line x x 1 y y1 z z may Lie in the Plane l m n ax + by + cz + d = 0 are al + bm + cn = 0 and ax 1 + by 1 + cz 1 + d = 0... 249 4.36. The Condition for the Line x x 1 y y1 z z to be Perpendicular l m n to the Plane ax + by + cz + d = 0... 249 4.37. Angle between a Line and a Plane... 253 4.38. Any Plane Through a Given Line... 253 4.39. To Find the Condition that the Two Lines x x 1 y y1 z z1, l1 m1 n1 x x2 y y2 = z z 2 l2 m n 2 2 may Intersect (or May be Coplanar) and to Find the Equation of the Plane in which they Lie... 261 4.40. Shortest Distance between Two Lines... 265

( ix ) 4.41. Magnitude and Equations of Shortest Distance... 265 4.42. Intersection of Three Planes... 275 4.43. Definition (The Sphere)... 281 4.44. Equations of a Sphere in Different Forms... 281 4.45. Touching Spheres... 282 4.46. Four-point Form... 283 4.47. Diameter Form... 284 4.48. Section of a Sphere by a Plane... 289 4.49. Intersection of Two Spheres... 290 4.50. Equations of a Circle... 290 4.51. Any Sphere Through a Given Circle... 294 4.52. Great Circle... 294 4.53. Definition of the Tangent Plane... 298 4.54. Equation of the Tangent Plane at a Point... 298 4.55. Angle of Intersection of Two Spheres... 303 4.56. Condition of Orthogonality of Two Spheres... 304 4.57. Definition (The Cone)... 308 4.58. Equation of the Cone with Vertex at the Origin... 308 4.59. The Direction Cosines (or Direction Ratios) of a Generator of a Cone Satisfy the Equation of the Cone whose Vertex is the Origin... 311 4.60. Quadric Cone Through the Axes... 311 4.61. Right Circular Cone... 312 4.62. To Find the Equation to the Cone whose Vertex is the Point (,, ) and Base the Conic F(x, y) = ax 2 + by 2 + 2hxy + 2fy + 2gx + c = 0, z = 0... 315 4.63. Enveloping Cone... 317 4.64. Angle between Two Lines in which a Plane Through the Vertex Cuts a Cone... 318 4.65. Definitions (The Cylinder)... 323 4.66. To Find the Equation to the Cylinder whose Generators are Parallel to the Line x y z and Intersect the Curve... 324 l m n 4.67. Equation of Right Circular Cylinder... 326 4.68. Enveloping Cylinder... 328 4.69. Definition (The Conicoids)... 330 5. Succesive and Partial Differentiation... 337 426 5.1. Successive Differentiation... 337 5.2. Calculation of n th Order Derivatives... 337 5.3. Use of Partial Fractions... 342 5.4. Leibnitz Theorem... 345 5.5. Determination of the Value of The n th Derivative of a Function at x = 0... 351 5.6. Function of Two Variables... 354

( x ) 5.7. Continuity... 354 5.8. Partial Derivatives of First Order... 355 5.9. Partial Derivatives of Higher Order... 356 5.10. Homogeneous Functions... 363 5.11. Euler s Theorem on Homogeneous Functions... 364 5.12. If u is a Homogeneous Function of Degree n in x and y,... 364 5.13. Deductions From Euler s Theorem... 365 5.14. Composite Functions... 372 5.15. Differentiation of Composite Functions... 373 5.16. Taylor s Theorem for a Function of Two Variables... 380 5.17. Jacobians... 385 5.18. Definitions... 385 5.19. Properties of Jacobians (Chain Rules)... 385 5.20. Theorem... 386 5.21. Jacobian of Implicit Functions... 387 5.22. Functional Relationship... 388 5.23. Approximation of Errors... 397 5.24. Maxima and Minima of Functions of Two Variables... 403 5.25. Conditions for F(x, y) to be Maximum or Minimum... 404 5.26. Rule to Find The Extreme Values of a Function z = f(x, y)... 404 5.27. Conditions for f(x, y, z) to be Maximum or Minimum... 405 5.28. Lagrange s Method of Undetermined Multipliers... 408 5.29. Geometrical Meaning of Partial Derivatives... 417 5.30. Tangent Plane and Normal to a Surface... 418 5.31. Differentiation under Integral Sign... 420 6. Multiple Integrals... 427 475 6.1. Double Integrals... 427 6.2. Evaluation of Double Integrals... 428 6.3. Evaluation of Double Integrals in Polar Co-ordinates... 434 6.4. Change of Order of Integration... 437 6.5. Triple Integrals... 440 6.6. Change of Variables... 442 6.7. Area by Double Integration... 449 6.8. Volume as a Double Integral... 449 6.9. Volume as a Triple Integral... 455 6.10. Volumes of Solids of Revolution... 457 6.11. Calculation of Mass... 458 6.12. Centre of Gravity (c.g.)... 460 6.13. Centre of Pressure... 463 6.14. Moment of Inertia... 466

( xi ) 6.15. Product of Inertia... 467 6.16. Principal Axes... 467 7. Vector Calculus... 476 532 7.1. Vector Functions... 476 7.2. Derivative of a Vector Function with respect to a Scalar... 476 7.3. General Rules for Differentiation... 477 7.4. Derivative of a Constant Vector... 479 7.5. Derivative of a Vector Function in terms of its Components... 479 7.6. If d F F () t has a Constant Magnitude, then F. = 0... 480 dt 7.7. If F F () t has a Constant Direction, then F d = 0... 480 dt 7.8. Geometrical Interpretation of dr dt... 480 7.9. Velocity and Acceleration... 481 7.10. Scalar and Vector Fields... 487 7.11. Gradient of a Scalar Field... 487 7.12. Geometrical Interpretation of Gradient... 487 7.13. Directional Derivative... 488 7.14. Properties of Gradient... 488 7.15. Divergence of a Vector Point Function... 493 7.16. Curl of a Vector Point Function... 493 7.17. Physical Interpretation of Divergence... 494 7.18. Physical Interpretation of Curl... 495 7.19. Properties of Divergence and Curl... 496 7.20. Repeated Operations by... 498 7.21. Integration of Vector Functions... 504 7.22. Line Integrals... 506 7.23. Circulation... 507 7.24. Work Done by a Force... 507 7.25. Surface Integrals... 510 7.26. Volume Integrals... 511 7.27. Gauss Divergence Theorem (Relation between Surface and Volume Integrals)... 516 7.28. Green s Theorem in the Plane... 523 7.29. Stoke s Theorem (Relation between Line and Surface Integrals)... 526 8. Curvilinear Co-ordinates... 533 548 8.1. Definitions... 533 8.2. Unit Vectors in Curvilinear System... 533 8.3. Arc Length and Volume Element... 535

( xii ) 8.4. Gradient in Orthogonal Curvilinear Co-ordinates... 538 8.5. Divergence in Orthogonal Curvilinear Co-ordinates... 538 8.6. Curl in Orthogonal Curvilinear Co-ordinates... 539 8.7. Laplacian in Terms Of Orthogonal Curvilinear Co-ordinates... 540 8.8. Special Curvilinear Co-ordinate Systems... 540 8.9. Some More Special Curvilinear Co-ordinate Systems... 547 9. Infinite Series... 549 597 9.1. Sequence... 549 9.2. Real Sequence... 549 9.3. Range of a Sequence... 549 9.4. Constant Sequence... 549 9.5. Bounded and Unbounded Sequences... 549 9.6. Convergent, Divergent and Oscillating Sequences... 550 9.7. Monotonic Sequences... 551 9.8. Limit of a Sequence... 551 9.9. Every Convergent Sequence is Bounded... 551 9.10. Convergence of Monotonic Sequences... 552 9.11. Infinite Series... 554 9.12. Series of Positive Terms... 554 9.13. Alternating Series... 554 9.14. Partial Sums... 554 9.15. Behaviour of an Infinite Series... 555 9.16. Absolute Convergence of a Series... 591 9.17. Every Absolutely Convergent Series is Convergent... 593 9.18. Uniform Convergence of Series of Functions... 595 10. Fourier Series... 598 648 10.1. Periodic Functions... 598 10.2. Fourier Series... 598 10.3. Euler s Formulae... 601 10.4. Dirichlet s Conditions... 602 10.5. Fourier Series for Discontinuous Functions... 614 10.6. Change of Interval... 619 10.7. Half Range Series... 625 10.8. Fourier Series of Different Waveforms... 637 10.9. Parseval s Identity... 639 10.10. Root Mean Square Value (r.m.s. Value)... 640 10.11. Complex Form of Fourier Series... 642 10.12. Practical Harmonic Analysis... 644

( xiii ) 11. Differential Equations of First Order... 649 682 11.1. Definitions (Differential Equations)... 649 11.2. Geometrical Meaning of a Differential Equation of the First Order and First Degree... 650 11.3. Formation of a Differential Equation... 651 11.4. Solution of Differential Equations of the First Order and First Degree... 654 11.5. Variables Separable Form... 654 11.6. Homogeneous Equations... 656 11.7. Equations Reducible to Homogeneous Form... 659 11.8. Linear Differential Equations... 661 11.9. Equations Reducible to the Linear Form (Bernoulli s Equation)... 664 11.10. Exact Differential Equations... 667 11.11. Theorem... 667 11.12. Equations Reducible to Exact Equations... 670 11.13. Differential Equations of the First Order and Higher Degree... 675 11.14. Equations Solvable for p... 675 11.15. Equations Solvable for y... 678 11.16. Equations Solvable for x... 679 11.17. Clairaut s Equation... 681 12. Applications of Differential Equations of First Order... 683 707 12.1. Introduction... 683 12.2. Geometrical Applications... 683 12.3. Orthogonal Trajectories... 687 12.4. Working Rule to Find the Equation of Orthogonal Trajectories... 688 12.5. Physical Applications... 691 12.6. Application to Electric Circuits... 699 12.7. Conduction of Heat... 700 12.8. Rate of Growth or Decay... 701 12.9. Newton s Law of Cooling... 702 12.10. Chemical Reactions and Solutions... 703 13. Linear Differential Equations... 708 744 13.1. Definitions (Linear Differential Equations)... 708 13.2. The Operator D... 708 13.3. Theorems... 709 13.4. Auxiliary Equation (A.E.)... 709 13.5. Rules for Finding the Complementary Function... 710 13.6. 1 The Inverse Operator f ( D )... 713 13.7. Rules for Finding the Particular Integral... 714 13.8. Method of Variation of Parameters to Find P.I.... 727

13.9. Cauchy s Homogeneous Linear Equation... 728 13.10. Legendre s Linear Equation... 730 13.11. Simultaneous Linear Equations with Constant Co-efficients... 733 13.12. Total Differential Equations... 738 13.13. Method for Solving Pdx + Qdy + Rdz = 0... 738 13.14. Solution of Simultaneous Equations of the Form dx dy dz... 741 P Q R 14. Applications of Linear Differential Equations... 745 778 14.1. Introduction... 745 14.2. Simple Harmonic Motion (S.H.M.)... 745 14.3. Mechanical and Electrical Oscillatory Circuits... 748 14.4. Simple Pendulum... 765 14.5. Gain or Loss of Beats... 766 14.6. Deflection of Beams... 769 14.7. Boundary Conditions... 770 14.8. Applications of Simultaneous Linear Differential Equations... 774 15. Special Functions and Series Solution of Differential Equations... 779 861 15.1. Gamma Function... 779 15.2. Reduction Formula for (n)... 779 1 15.3. Value of ( )... 780 2 15.4. Beta Function... 781 15.5. Symmetry of Beta Function i.e., B(m, n) = B(n, m)... 781 15.6. Relation between Beta and Gamma Functions... 781 15.7. To Evaluate z /2 p 0 sin q x. cos x dx; p > 1; q > 1... 782 15.8. Elliptic Integrals... 789 15.9. Applications of Elliptic Integrals... 790 15.10. Error Function... 792 15.11. Series Solution of Differential Equations... 795 15.12. Definitions... 795 15.13. Power Series Solution, When x = 0 is an Ordinary Point of the Equation 2 ( xiv ) d y 2 dx + P(x) dy + Q(x) y = 0... 796 dx 15.14. Frobenius Method : Series Solution When x = 0 is a Regular Singular Point of the Differential Equation... 803 15.15. Legendre s Differential Equation... 818 15.16. Legendre s Function of First kind P n (x)... 819 15.17. Legendre s Function of Second kind Q n (x)... 820

( xv ) 15.18. Solution of Legendre s equation... 820 15.19. Generating Function for P n (x)... 820 15.20. Rodrigue s Formula... 823 15.21. Recurrence Relations... 826 15.22. Beltrami s Result... 828 15.23. Orthogonality of Legendre Polynomials... 828 15.24. Laplace s Integral of First Kind... 830 15.25. Laplace s Integral of Second Kind... 830 15.26. Cristoffel s Expansion Formula... 831 15.27. Cristoffel s Summation Formula... 832 15.28. Expansion of a Function in a Series of Legendre Polynomials (Fourier-Legendre Series)... 832 15.29. Bessel s Differential Equation... 838 15.30. Solution of Bessel s Equation... 838 15.31. Series Representation of Bessel functions... 842 15.32. Recurrence Relations for J n (x)... 843 15.33. Generating Function for J n (x)... 850 15.34. Integral Form of Bessel Function... 851 15.35. Equations Reducible to Bessel s Equation... 854 15.36. Modified Bessel s Equation... 856 15.37. Ber and Bei Functions... 857 15.38. Orthogonality of Bessel Functions... 858 15.39. Fourier-bessel Expansion of F(x)... 859 16. Partial Differential Equations... 862 900 16.1. Introduction... 862 16.2. Formation of Partial Differential Equations... 862 16.3. Definitions... 866 16.4. Equations Solvable by Direct Integration... 868 16.5. Linear Partial Differential Equations of the First Order... 870 16.6. Lagrange s Linear Equation... 870 16.7. Working Method... 871 16.8. Non-linear Equations of the First Order... 876 16.9. (a) Equations of the Form f(p, q) = 0... 876 16.9. (b) Equations of the Form z = px + qy + f(p, q)... 878 16.9. (c) Equations of the Form f (z, p, q) = 0... 878 16.9. (d) Equations of the Form f 1 (x, p) = f 2 (y, q)... 880 16.10. Charpit s Method... 882 16.11. Homogeneous Linear Equations with Constant Co-efficients... 884 16.12. Rules for Finding the C.F.... 885 16.13. Rules for Finding the P.I... 887

A Textbook of Engineering Mathematics by NP Bali and Dr Manish Goyal 40% OFF Publisher : Laxmi Publications ISBN : 9788131808320 Author : NP Bali and Dr Manish Goyal Type the URL : http://www.kopykitab.com/product/3179 Get this ebook