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Sequences And Series -- Successive, Differentiation, Mean Value Theorems And Expansion Of Functions -- Curvature -- Asymptotes And Curve Tracing -- Functions Of Seven Variables -- Tangents And Normals -- Beta And Gamma Functions -- Reduction Formulas -- Quadrature And Rectification -- Centre Of Gravity And Moment Of Inertia -- Volumes And Surface Of Solids Of Revolution -- Multiple Integrals -- Vector Calculus -- Three-dimensional Geometry -- Logic -- Elements Of Fuzzy Logic -- Graphs. Babu Ram. Cover......Page 1 Contents......Page 8 Preface to the Revised Edition......Page 12 Symbols and Basic Formulae......Page 14 1.2 Convergence of Sequences......Page 18 1.3 The Upper and Lower Limits of a Sequence......Page 20 1.4 Cauchy’s Principle of Convergence......Page 21 1.5 Monotonic Sequence......Page 23 1.6 Theorems on Limits......Page 25 1.7 Subsequences......Page 28 1.8 Series......Page 29 1.9 Comparison Tests......Page 32 1.10 D’alemberi’s Ratio Test......Page 37 1.11 Cauchy’s Root Test......Page 42 1.12 Raabe’s Test......Page 44 1.13 Logarithmic Test......Page 48 1.14 De Morgan–Berirand Test......Page 50 1.15 Gauss’s Test......Page 51 1.16 Cauchy’s Integral Test......Page 53 1.17 Cauchy’s Condensation Test......Page 55 1.18 Kummer’s Test......Page 57 1.19 Alternating Series......Page 58 1.20 Absolute Convergence of a Series......Page 60 1.21 Convergence of the Series of the Type......Page 65 1.22 Derangement of Series......Page 67 1.23 Nature of Non-Absolutely Convergent Series......Page 68 1.24 Effect of Derangement of Non-Absolutely Convergent Series......Page 69 1.25 Uniform Convergence......Page 71 1.26 Uniform Convergence of a Series of Functions......Page 73 1.27 Properties of Uniformly Convergent Series......Page 74 1.28 Power Series......Page 75 Exercises......Page 76 2.1 Successive Differentiation......Page 80 2.2 Leibnitz’s Theorem and its Applications......Page 84 2.3 General Theorems......Page 88 2.4 Taylor’s Infinite Series and Power Series Expansion......Page 95 2.6 Expansion of Functions......Page 96 2.7 Indeterminate Forms......Page 106 Exercises......Page 115 3.1 Radius of Curvature of Intrinsic Curves......Page 120 3.2 Radius of Curvature for Cartesian Curves......Page 121 3.3 Radius of Curvature for Parametric Curves......Page 125 3.4 Radius of Curvature for Pedal Curves......Page 127 3.5 Radius of Curvature for Polar Curves......Page 128 3.6 Radius of Curvature at the Origin......Page 131 3.7 Center of Curvature......Page 133 3.9 Equation of the Circle of Curvature......Page 134 3.10 Chords of Curvature Parallel to the Coordinate Axes......Page 137 3.11 Chord of Curvature in Polar Coordinates......Page 138 3.12 Miscellaneous Examples......Page 140 Exercises......Page 145 4.1 Determination of Asymptotes When the Equation of the Curve in Cartesian form is Givens......Page 148 4.2 The Asymptotes of the General Rational Algebraic Curve......Page 149 4.3 Asymptotes Parallel to Coordinate Axes......Page 150 4.4 Working Rule for Finding Asymptotes of Rational Algebraic Curve......Page 151 4.5 Intersection of a Curve and its Asymptotes......Page 154 4.6 Asymptotes by Expansion......Page 156 4.7 Asymptotes of the Polar Curves......Page 157 4.9 Concavity, Convexity and Singular Points......Page 159 4.10 Curve Tracing (Cartesian Equations)......Page 163 4.11 Curve Tracing (Polar Equations)......Page 168 4.12 Curve Tracing (Parametric Equations)......Page 170 Exercises......Page 171 Chapter 5: Functions of Several Variables......Page 176 5.3 The Differential Coefficients......Page 177 5.5 Higher-Order Partial Derivatives......Page 178 5.6 Envelopes and Evolutes......Page 183 5.7 Homogeneous Functions and Euler’s Theorem......Page 185 5.8 Differentiation of Composite Functions......Page 190 5.9 Transformation from Cartesian to Polar Coordinates and Vice Versa......Page 194 5.10 Taylor’s Theorem for Functions of Several Variables......Page 196 5.11 Approximation of Errors......Page 199 5.12 General Formula for Errors......Page 200 5.13 Tangent Plane and Normal to a Surface......Page 203 5.15 Properties of Jacobian......Page 205 5.16 Necessary and Sufficient Conditions for Jacobian to Vanish......Page 208 5.17 Differentiation Under the Integral Sign......Page 209 5.18 Miscellaneous Examples......Page 212 5.19 Extreme Values......Page 216 5.20 Lagrange’s Method of Undetermined Multipliers......Page 223 Exercises......Page 228 6.3 Equation of the Normal at a Point of a Curve......Page 232 6.4 Lengths of Tangent, Normal, Sub-Tangent and Subnormal at any Point of a Curve......Page 236 Exercises......Page 237 7.2 Properties of Beta Function......Page 238 7.4 Properties of Gamma Function......Page 241 7.5 Relation Between Beta and Gamma Functions......Page 242 7.6 Dirichlet’s and Liouville’s Theorems......Page 247 7.7 Miscellaneous Examples......Page 249 Exercises......Page 250 8.1 Reduction Formulas for sinn x dx and cosn x dx......Page 252 8.2 Reduction Formula for sinm x cosn x dx 8.3......Page 254 8.3 Reduction Formulas for tann x dx and secn x dx 8.5......Page 256 8.4 Reduction Formulas for xn sinmx dx and xn cosmx dx......Page 257 8.5 Reduction Formulas for x n eaxdx and xm (log x)n dx......Page 258 8.7 Reduction Formula For......Page 259 Exercises......Page 260 9.1.1 Area of a Curve Given by the Cartesian Equation......Page 262 9.1.2 Area of a Curve Given by Polar Equation......Page 267 9.2.1 Length of a Curve......Page 270 Exercises......Page 275 10.1 Centre of Gravity......Page 278 10.2 Moment of Inertia......Page 281 10.3 Mean Values of a Function......Page 282 Exercises......Page 283 11.1 Volume of the Solid of Revolution (Cartesian Equations)......Page 284 11.2 Volume of the Solid of Revolution (Parametric Equations)......Page 289 11.3 Volume of the Solid of Revolution (Polar Curves)......Page 291 11.4 Surface of the Solid of Revolution (Cartesian Equations)......Page 292 11.5 Surface of the Solid of Revolution (Parametric Equations)......Page 294 11.6 Surface of the Solid of Revolution (Polar Curves)......Page 296 Exercises......Page 297 12.1 Double Integrals......Page 300 12.3 Evaluation of Double Integrals (Cartesian Coordinates)......Page 301 12.4 Evaluation of Double Integrals (Polar Coordinates)......Page 305 12.5 Change of Variables in a Double Integral......Page 307 12.6 Change of Order of Integration......Page 311 12.7 Area Enclosed by Plane Curves (Cartesian and Polar Coordinates)......Page 315 12.8 Volume and Surface Area as Double Integrals......Page 318 12.9 Triple Integrals and their Evaluation......Page 325 12.10 Change to Spherical Polar Coordinates from Cartesian Coordinates in a Triple Integral......Page 329 12.11 Volume as a Triple Integral......Page 332 12.12 Miscellaneous Examples......Page 336 Exercises......Page 338 Chapter 13: Vector Calculus......Page 344 13.1 Differentiation of a Vector......Page 354 13.2 Partial Derivatives of a Vector Function......Page 361 13.3 Gradient of a Scalar Field......Page 362 13.5 Properties of a Gradient......Page 363 13.6.1 Directional Derivatives along Coordinate Axes......Page 364 13.8 Physical Interpretation of Divergence......Page 369 13.11 The Laplacian Operator......Page 371 13.12 Properties of Divergence and Curl......Page 375 13.14 Line Integral......Page 380 13.15 Work Done by a Force......Page 384 13.16 Surface Integral......Page 386 13.17 Volume Integral......Page 390 13.18 Gauss’s Divergence Theorem......Page 392 13.19 Green’s Theorem in a Plane......Page 398 13.20 Stoke’s Theorem......Page 402 13.21 Miscellaneous Examples......Page 407 Exercises......Page 415 14.3 Direction Ratios and Direction Cosines of a Line......Page 424 14.4 Section Formulae—Internal Division of a Line by a Point on the Line......Page 425 14.4.1 External Division of a Line by a Point on the Extended Line......Page 426 14.5 Straight Line in Three Dimensions......Page 429 14.6 Angle Between Two Lines......Page 432 14.7 Shortest Distance Between Two Skew Lines......Page 434 14.8 Equation of a Plane......Page 441 14.10 Equation of a Plane Passing through Three Points......Page 442 14.12 Equation of a Plane Passing through Two Point and Parallel to a Line......Page 443 14.13 Angle Between Two Planes......Page 447 14.15 Perpendicular Distance of a Point From a Plane......Page 449 14.16 Planes Bisecting the Angles Between Two Planes......Page 450 14.18 Planes Passing through the Intersection of Two Given Planes......Page 452 14.19 Sphere......Page 454 14.20 Equation of a Sphere Whose Diameter is the Line Joining Two Given Points......Page 457 14.21 Equation of a Sphere Passing through Four Points......Page 456 14.24 Angle of Intersection of Two Spheres......Page 459 14.25 Condition of Orthogonality of Two Spheres......Page 460 14.27 Equation of a Cylinder with Given Axis and Guiding Curves......Page 463 14.28 Right Circular Cylinder......Page 464 14.31 Equation of a Cone with Given Vertex and Guiding Curve......Page 466 14.32 Right Circular Cone......Page 468 14.33 Right Circular Cone with Vertex (α, β, γ), Semi-Vertical Angle and the (l, m, n) Direction Cosines of the Axis.......Page 469 14.34 Conicoids......Page 470 14.36 Shape of the Hyperboloid of One Sheet......Page 471 14.38 Shape of the Elliptic Cone......Page 472 14.40 Tangent Plane at a Point of Central Conicoid......Page 473 14.42 Equation of Normal to the Central Conicoid at any Point (α, β, γ) on it......Page 474 14.43 Miscellaneous Examples......Page 477 Exercises......Page 479 15.1 Propositions......Page 490 15.2 Basic Logical Operations......Page 491 15.2.1 Translating from English to Symbols......Page 492 15.2.2 Truth Table for Exclusive OR......Page 493 15.3 Logical Equivalence Involving Tautologies and Contradictions......Page 495 15.4 Conditional Propositions......Page 496 Exercises......Page 507 16.1 Fuzzy Set......Page 510 16.2 Standard Operations on a Fuzzy Set......Page 512 16.3 Many Valued Logic......Page 514 Exercises......Page 515 17.1 Definitions and Basic Concepts......Page 518 17.2 Special Graphs......Page 520 17.3 Subgraphs......Page 523 17.4 Isomorphisms of Graphs......Page 525 17.5 Walks, Paths and Circuits......Page 527 17.6 Eulerian Paths and Circuits......Page 531 17.6.1 Methods for Finding Euler Circuit......Page 536 17.7 Hamiltonian Circuits......Page 538 17.7.1 Travelling Salesperson Problem......Page 543 17.8 Matrix Representation of Graphs......Page 544 17.9 Planar Graphs......Page 546 17.10 Colouring of Graph......Page 553 17.11 Directed Graphs......Page 556 17.12 Trees......Page 560 17.13 Isomorphism of Trees......Page 565 17.14 Representation of Algebraic Expressions by Binary Trees......Page 568 17.15 Spanning Tree of a Graph......Page 571 17.16.1 Dijkstra’s Shortest Path Algorithm......Page 573 17.16.2 Shortest Path if All Edges Have Length 1......Page 576 17.17.1 Prim Algorithm......Page 577 17.17.2 Kruskal’s Algorithm......Page 580 17.18 Cut Sets......Page 582 17.18.1 Relation Between Spanning Trees, Circuits and Cut Sets......Page 584 17.19 Tree Searching......Page 586 17.19.1 Procedure to Evaluate an Expression Given in Polish Form......Page 587 17.20 Transport Networks......Page 588 Exercises......Page 595 Index......Page 602