Contents Preface .................................................................................................................... (vii)
CHAPTER - 0 PRELIMINARIES 0.1
Limit............................................................................................................. 1
0.2
One Sided limits ........................................................................................... 2 0.2.1 Closed Interval ................................................................................ 2 0.2.2 Open Interval................................................................................... 3 0.2.3 Absolute Value ................................................................................ 3 0.2.4 Neighbourhood of a Point ............................................................... 3
0.3
The → definition of limit (Cauchy’s Definition) ................................... 4
0.4
One-Sided Limits ......................................................................................... 5 0.4.1 Left-Hand Limit .............................................................................. 5 0.4.2 Right Hand Limit ............................................................................ 5
0.5
Algebra of Limits ......................................................................................... 6
0.6
Limits at Infinity .......................................................................................... 6
0.7
Continuity .................................................................................................. 11
0.8
Derivative of a Function ............................................................................ 14
0.9
Some standard Results ............................................................................... 18 0.9.1 The Derivative of a Constant Function is Zero ............................. 18
0.10 Derivability and Continuity .......................................................................... 19 0.11 Some Standard Rules of Differentiation ....................................................... 20
Contents
0.11.1
Theorem ................................................................................... 20
0.11.2
Differentiation of Sum and Difference .................................... 20
0.11.3
Product Rule............................................................................. 25
0.11.4
Quotient Rule ........................................................................... 30 (xi)
Contents
(x)
0.11.5 0.11.6 0.11.7 0.11.8 0.11.9 0.11.10
Chain Rule ............................................................................... 36 Derivatives of inverse trigonometrical function ...................... 41 Differentiation of functions in parametric form ....................... 42 Implicit differentiation ............................................................. 43 Logarithmic differentiation ...................................................... 46 Differentiation of one function with respect to another function ............................................... 51
0.12 Hyperbolic Functions and their Derivatives ................................................. 52 0.13 Higher Order Derivatives and Successive Differentiation ........................... 53 0.14 Partial Differentiation ................................................................................... 57 0.14.1 Partial derivatives..................................................................... 46 0.14.2 Homogenous functions ............................................................ 59 0.15 Maxima and Minima .................................................................................... 62 0.15.1 Increasing function................................................................... 46 0.15.2 Decreasing function ................................................................. 63 0.15.3 Stationary point ........................................................................ 64 0.15.4 Stationary value ....................................................................... 64 0.15.5 Maximum and minimum values of a function ......................... 64 0.15.6 Points of inflexion .................................................................... 65 0.15.7 First derivative test ................................................................... 65 0.15.8
Second derivative test .............................................................. 65
0.16 Introduction .................................................................................................. 70 0.17 Indefinite Integral ......................................................................................... 70 0.18 Standard Integration Formulae ..................................................................... 71 0.19 Basic Rules of Integration ............................................................................ 72 0.20 Methods of Integration ................................................................................. 79 0.20.1
Integration by Method of Substitution .......................................... 79
0.20.2
Standard forms .............................................................................. 90
0.21 Integration by Parts....................................................................................... 99
Contents
(xi)
0.22 Integrals of the Type ................................................................................. 103 0.23 Integrals of the
a 2 - x 2 dx ..................................................................... 105
0.24 Integrals of the Form
x 2 – a 2 dx, a 2 + x 2 dx .................................. 106
0.25
Definite Integrals ..................................................................................... 107
0.26
Properties of Definite Integral.................................................................. 113
0.27
Integration by Parts .................................................................................. 113
CHAPTER - 1 MATRICES 1.1
Introduction .............................................................................................. 115
1.2
Elementary Row and Column Operations ............................................... 115 1.2.1 Equivalent Matrices ................................................................... 116 1.2.2 Elementary Matrices .................................................................. 116
1.3
Rank of a Matrix ...................................................................................... 117 1.3.1 Rank and Elementary Matrices .................................................. 118
CHAPTER - 2 DIFFERENTIAL CALCULUS - I 2.1
Introduction .............................................................................................. 170
2.2
Higher Order Derivatives ......................................................................... 170
2.3
Derivatives of Higher Order and Leibnitz’s Theorem ............................. 173 2.3.1 Leibnitz’s Theorem ..................................................................... 175
2.4
Partial Differentiation .............................................................................. 180
2.5
Homogeneous Functions .......................................................................... 183
2.6
Euler’s Theorem....................................................................................... 184
2.7
Change of Variables ................................................................................. 190
2.8
Expansions of Functions ......................................................................... 195 2.8.1 Taylor’s Theorem with Lagrange’s form of Remainder ............ 195
Contents
(xii)
2.8.2 Taylor’s Theorem with Cauchy’s form of Remainder ............... 195 2.8.3 Taylor’s Theorem with Generalized form of Remainder ............ 196 2.8.4 Maclaurin’s Theorem with Generalized form of Remainder ...... 196 2.8.5 Taylor’s Theorem for the Function of Two Variables ............. 199
CHAPTER - 3 DIFFERENTIAL CALCULUS – II 3.1
Introduction .............................................................................................. 201
3.2
Properties of Jacobian .............................................................................. 208
3.3
Chain Rule ............................................................................................... 212
3.4
Functional Dependence ............................................................................ 212
3.5
Errors and Approximations ...................................................................... 216 3.5.1 Approximate Calculations ........................................................... 217
CHAPTER - 4 MULTIPLE INTEGRALS 4.1
Introduction .............................................................................................. 246
4.2
Double Integrals ....................................................................................... 246
4.3
Evaluation of Double Integrals in Polar Coordinates .............................. 259
4.4
Triple Integrals ......................................................................................... 264 4.4.1 Evaluation of Triple Integrals ..................................................... 264
4.5
Change of Variables ................................................................................. 271 4.5.1 Double Integrals .......................................................................... 271 4.5.2 Triple Integrals ............................................................................ 272
4.6
Changing the Order of Integration ........................................................... 279
4.7
Application of Multiple Integrals ............................................................. 287
CHAPTER - 5 MULTIPLE INTEGRALS 5.1
Introduction .............................................................................................. 331
5.2
Point Functions ........................................................................................ 331
Contents
5.2.1 5.2.2 5.2.3 5.2.4
(xiii)
Scalar Point Function .................................................................. 331 Scalar Field ................................................................................. 332 Vector Point Function ................................................................. 332 Vector Field................................................................................ 332
5.3
Vector Differential Operator .................................................................. 332 5.3.1 Level surface ............................................................................... 333
5.4
Gradient of a Scalar Function .................................................................. 333 5.4.1 Magnitude of ........................................................................ 333
5.5
Properties of Gradient .............................................................................. 333
5.6
Directional Derivative .............................................................................. 343
5.7
Equation of Tangent Plane and Normal to a Level Surface ..................... 345 5.7.1 Equation of the Tangent Plane .................................................... 345 5.7.2 Equation of the Normal ............................................................... 346
5.8
Divergence ............................................................................................... 355
5.9
Curl .......................................................................................................... 364 5.9.1 Irrotational ................................................................................... 364
CHAPTER - 6 DIFFERENTIAL EQUATIONS 6.1
Introduction .............................................................................................. 420
6.2
Solution of a Differential Equation .......................................................... 421 6.2.1 General solution .......................................................................... 422 6.2.2 Particular solution ....................................................................... 422 6.2.3 Singular solution ......................................................................... 422
6.3
Formation of Differential Equation.......................................................... 422
6.4
First Order and First Degree Differential Equations ................................ 426
6.5
Variables .................................................................................................. 426
6.6
Equations Reducible to the Form in which Variables .............................. 429
6.7
Homogenous Equations ........................................................................... 434 6.7.1 Method of solving homogenous differential equations ............... 434 6.7.2 Equations reducible to homogenous form................................... 438
Contents
(xiv)
6.8
Exact Differential Equations .................................................................... 446
6.9
Integrating Factors ................................................................................... 454 6.9.1 Number of integrating factors ..................................................... 455
6.10
Rules for Finding the Integrating Factors ................................................ 462