A Textbook of Engineering Mathematics by NP Bali and Dr Manish Goyal

16 Pages·2015·1.32 MB·English

A TEXTBOOK OF
ENGINEERING
MATHEMATICS
For
B.Sc. (Engg.), B.E., B. Tech., M.E. and
Equivalent Professional Examinations
By
N.P. BALI Dr. MANISH GOYAL
Formerly Principal M.Sc. (Mathematics), Ph.D., CSIR-NET
S.B. College, Gurgaon Associate Professor
Haryana Department of Mathematics
Institute of Applied Sciences & Humanities
G.L.A. University, Mathura,
U.P.
LLLLLAAAAAXXXXXMMMMMIIIII PPPPPUUUUUBBBBBLLLLLIIIIICCCCCAAAAATTTTTIIIIIOOOOONNNNNSSSSS (((((PPPPP))))) LLLLLTTTTTDDDDD
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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 (cid:68)
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 | (cid:122) 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(cid:99) and (A(cid:99))–1 = (A–1)(cid:99)............ 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 , b , c
1 1 1
and a , b , c . Deduce the Condition for Perpendicularity and Parallelism
2 2 2
of Two Lines....................................................................................................... 208
( viii )
4.17. Projection........................................................................................................... 216
4.18. To Prove that the Projection of the Join of two Points (x , y , z ), (x , y , z )
1 1 1 2 2 2
on a Line whose Direction Cosines are l, m, n is l(x – x ) + m(y – y )
2 1 2 1
+ n(z – z ) .......................................................................................................... 216
2 1
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
x(cid:16) x y(cid:16) y z(cid:16) z
4.33. To Find the Point of Intersection of the Line 1 (cid:32) 1 (cid:32) 1
l m n
with the plane ax + by + cz + d = 0................................................................... 248
x(cid:16) x y(cid:16) y z(cid:16) z
4.34. The Conditions that the Line 1 (cid:32) 1 (cid:32) 1 may be Parallel to
l m n
the Plane ax + by + cz + d = 0 are al + bm + cn = 0 and
ax + by + cz + d (cid:122) 0......................................................................................... 249
1 1 1
x(cid:16) x y(cid:16) y z(cid:16) z
4.35. The Conditions that the Line 1 (cid:32) 1 (cid:32) 1 may Lie in the Plane
l m n
ax + by + cz + d = 0 are al + bm + cn = 0 and ax + by + cz + d = 0............... 249
1 1 1
x(cid:16) x y(cid:16) y z(cid:16) z
4.36. The Condition for the Line 1 (cid:32) 1 (cid:32) 1 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
x(cid:16) x y(cid:16) y z(cid:16) z
4.39. To Find the Condition that the Two Lines 1 (cid:32) 1 (cid:32) 1,
l m n
1 1 1
x(cid:16) x2 (cid:32) y(cid:16) y2 = z(cid:16) z2 may Intersect (or May be Coplanar)
l2 m2 n2
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 ((cid:68), (cid:69), (cid:74)) and
Base the Conic F(x, y) = ax2 + by2 + 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
x y z
to the Line (cid:32) (cid:32) 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 nth Order Derivatives ................................................................ 337
5.3. Use of Partial Fractions.................................................................................... 342
5.4. Leibnitz Theorem .............................................................................................. 345
5.5. Determination of the Value of The nth 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