Engineering Mathematics Vol 1

Description

Cover Contents Preface to the Revised Edition Symbols and Basic Formulae Chapter 1: Sequences and Series 1.1 Sequences 1.2 Convergence of Sequences 1.3 The Upper and Lower Limits of a Sequence 1.4 Cauchy’s Principle of Convergence 1.5 Monotonic Sequence 1.6 Theorems on Limits 1.7 Subsequences 1.8 Series 1.9 Comparison Tests 1.10 D’alemberi’s Ratio Test 1.11 Cauchy’s Root Test 1.12 Raabe’s Test 1.13 Logarithmic Test 1.14 De Morgan–Berirand Test 1.15 Gauss’s Test 1.16 Cauchy’s Integral Test 1.17 Cauchy’s Condensation Test 1.18 Kummer’s Test 1.19 Alternating Series 1.20 Absolute Convergence of a Series 1.21 Convergence of the Series of the Type 1.22 Derangement of Series 1.23 Nature of Non-Absolutely Convergent Series 1.24 Effect of Derangement of Non-Absolutely Convergent Series 1.25 Uniform Convergence 1.26 Uniform Convergence of a Series of Functions 1.27 Properties of Uniformly Convergent Series 1.28 Power Series Exercises Chapter 2: Successive Differentiation, Mean Value Theorems and Expansion of Functions 2.1 Successive Differentiation 2.2 Leibnitz’s Theorem and its Applications 2.3 General Theorems 2.4 Taylor’s Infinite Series and Power Series Expansion 2.5 Maclaurin’s Infinite Series 2.6 Expansion of Functions 2.7 Indeterminate Forms Exercises Chapter 3: Curvature 3.1 Radius of Curvature of Intrinsic Curves 3.2 Radius of Curvature for Cartesian Curves 3.3 Radius of Curvature for Parametric Curves 3.4 Radius of Curvature for Pedal Curves 3.5 Radius of Curvature for Polar Curves 3.6 Radius of Curvature at the Origin 3.7 Center of Curvature 3.8 Evolutes and Involutes 3.9 Equation of the Circle of Curvature 3.10 Chords of Curvature Parallel to the Coordinate Axes 3.11 Chord of Curvature in Polar Coordinates 3.12 Miscellaneous Examples Exercises Chapter 4: Asymptotes and Curve Tracing 4.1 Determination of Asymptotes When the Equation of the Curve in Cartesian form is Givens 4.2 The Asymptotes of the General Rational Algebraic Curve 4.3 Asymptotes Parallel to Coordinate Axes 4.4 Working Rule for Finding Asymptotes of Rational Algebraic Curve 4.5 Intersection of a Curve and its Asymptotes 4.6 Asymptotes by Expansion 4.7 Asymptotes of the Polar Curves 4.8 Circular Asymptotes 4.9 Concavity, Convexity and Singular Points 4.10 Curve Tracing (Cartesian Equations) 4.11 Curve Tracing (Polar Equations) 4.12 Curve Tracing (Parametric Equations) Exercises Chapter 5: Functions of Several Variables 5.1 Continuity of a Function of two Variables 5.2 Differentiability of a Function of two Variables 5.3 The Differential Coefficients 5.4 Distinction Between Derivatives and Differential Coefficients 5.5 Higher-Order Partial Derivatives 5.6 Envelopes and Evolutes 5.7 Homogeneous Functions and Euler’s Theorem 5.8 Differentiation of Composite Functions 5.9 Transformation from Cartesian to Polar Coordinates and Vice Versa 5.10 Taylor’s Theorem for Functions of Several Variables 5.11 Approximation of Errors 5.12 General Formula for Errors 5.13 Tangent Plane and Normal to a Surface 5.14 Jacobians 5.15 Properties of Jacobian 5.16 Necessary and Sufficient Conditions for Jacobian to Vanish 5.17 Differentiation Under the Integral Sign 5.18 Miscellaneous Examples 5.19 Extreme Values 5.20 Lagrange’s Method of Undetermined Multipliers Exercises Chapter 6: Tangents and Normals 6.1 Introduction 6.2 Equation of the Tangent at a Point of a Curve 6.3 Equation of the Normal at a Point of a Curve 6.4 Lengths of Tangent, Normal, Sub-Tangent and Subnormal at any Point of a Curve Exercises Chapter 7: Beta and Gamma Functions 7.1 Beta Function 7.2 Properties of Beta Function 7.3 Gamma Function 7.4 Properties of Gamma Function 7.5 Relation Between Beta and Gamma Functions 7.6 Dirichlet’s and Liouville’s Theorems 7.7 Miscellaneous Examples Exercises Chapter 8: Reduction Formulas 8.1 Reduction Formulas for sinn x dx and cosn x dx 8.2 Reduction Formula for sinm x cosn x dx 8.3 8.3 Reduction Formulas for tann x dx and secn x dx 8.5 8.4 Reduction Formulas for xn sinmx dx and xn cosmx dx 8.5 Reduction Formulas for x n eaxdx and xm (log x)n dx 8.6 Reduction Formula for Imn = cosm x sin nx dx. 8.7 Reduction Formula For Exercises Chapter 9: Quadrature and Rectification 9.1 Quadrature 9.1.1 Area of a Curve Given by the Cartesian Equation 9.1.2 Area of a Curve Given by Polar Equation 9.2 Rectification 9.2.1 Length of a Curve Exercises Chapter 10: Centre of Gravity and Momentof Inertia 10.1 Centre of Gravity 10.2 Moment of Inertia 10.3 Mean Values of a Function Exercises Chapter 11: Volumes and Surfaces of Solids of Revolution 11.1 Volume of the Solid of Revolution (Cartesian Equations) 11.2 Volume of the Solid of Revolution (Parametric Equations) 11.3 Volume of the Solid of Revolution (Polar Curves) 11.4 Surface of the Solid of Revolution (Cartesian Equations) 11.5 Surface of the Solid of Revolution (Parametric Equations) 11.6 Surface of the Solid of Revolution (Polar Curves) Exercises Chapter 12: Multiple Integrals 12.1 Double Integrals 12.2 Properties of a Double Integral 12.3 Evaluation of Double Integrals (Cartesian Coordinates) 12.4 Evaluation of Double Integrals (Polar Coordinates) 12.5 Change of Variables in a Double Integral 12.6 Change of Order of Integration 12.7 Area Enclosed by Plane Curves (Cartesian and Polar Coordinates) 12.8 Volume and Surface Area as Double Integrals 12.9 Triple Integrals and their Evaluation 12.10 Change to Spherical Polar Coordinates from Cartesian Coordinates in a Triple Integral 12.11 Volume as a Triple Integral 12.12 Miscellaneous Examples Exercises Chapter 13: Vector Calculus 13.1 Differentiation of a Vector 13.2 Partial Derivatives of a Vector Function 13.3 Gradient of a Scalar Field 13.4 Geometrical Interpretation of a Gradient 13.5 Properties of a Gradient 13.6 Directional Derivatives 13.6.1 Directional Derivatives along Coordinate Axes 13.7 Divergence of a Vector-Point Function 13.8 Physical Interpretation of Divergence 13.9 Curl of a Vector-Point Function 13.10 Physical Interpretation of Curl 13.11 The Laplacian Operator 13.12 Properties of Divergence and Curl 13.13 Integration of Vector Functions 13.14 Line Integral 13.15 Work Done by a Force 13.16 Surface Integral 13.17 Volume Integral 13.18 Gauss’s Divergence Theorem 13.19 Green’s Theorem in a Plane 13.20 Stoke’s Theorem 13.21 Miscellaneous Examples Exercises Chapter 14: Three-Dimensional Geometry 14.1 Coordinate Planes 14.2 Distance Between Two Points 14.3 Direction Ratios and Direction Cosines of a Line 14.4 Section Formulae—Internal Division of a Line by a Point on the Line 14.4.1 External Division of a Line by a Point on the Extended Line 14.5 Straight Line in Three Dimensions 14.6 Angle Between Two Lines 14.7 Shortest Distance Between Two Skew Lines 14.8 Equation of a Plane 14.9 Equation of a Plane Passing through a Given Point and Perpendicular to a Given Direction 14.10 Equation of a Plane Passing through Three Points 14.11 Equation of a Plane Passing through a Point and Parallel to Two Given Vectors 14.12 Equation of a Plane Passing through Two Point and Parallel to a Line 14.13 Angle Between Two Planes 14.14 Angle Between a Line and a Plane 14.15 Perpendicular Distance of a Point From a Plane 14.16 Planes Bisecting the Angles Between Two Planes 14.17 Intersection of Planes 14.18 Planes Passing through the Intersection of Two Given Planes 14.19 Sphere 14.20 Equation of a Sphere Whose Diameter is the Line Joining Two Given Points 14.21 Equation of a Sphere Passing through Four Points 14.22 Equation of the Tangent Plane to a Spherem 14.23 Condition of Tangency 14.24 Angle of Intersection of Two Spheres 14.25 Condition of Orthogonality of Two Spheres 14.26 Cylinder 14.27 Equation of a Cylinder with Given Axis and Guiding Curves 14.28 Right Circular Cylinder 14.29 Cone 14.30 Equation of a Cone with its Vertex at the Origin 14.31 Equation of a Cone with Given Vertex and Guiding Curve 14.32 Right Circular Cone 14.33 Right Circular Cone with Vertex (α, β, γ), Semi-Vertical Angle and the (l, m, n) Direction Cosines of the Axis. 14.34 Conicoids 14.35 Shape of an Ellipsoid 14.36 Shape of the Hyperboloid of One Sheet 14.37 Shape of the Hyperboloid of Two Sheets 14.38 Shape of the Elliptic Cone 14.39 Intersection of a Conicoid and a Line 14.40 Tangent Plane at a Point of Central Conicoid 14.41 Condition of Tangency 14.42 Equation of Normal to the Central Conicoid at any Point (α, β, γ) on it 14.43 Miscellaneous Examples Exercises Chapter 15: Logic 15.1 Propositions 15.2 Basic Logical Operations 15.2.1 Translating from English to Symbols 15.2.2 Truth Table for Exclusive OR 15.3 Logical Equivalence Involving Tautologies and Contradictions 15.4 Conditional Propositions Exercises Chapter 16: Elements of Fuzzy Logic 16.1 Fuzzy Set 16.2 Standard Operations on a Fuzzy Set 16.3 Many Valued Logic 16.4 Fuzzy Logic 16.5 Fuzzy Propositions Exercises Chapter 17: Graphs 17.1 Definitions and Basic Concepts 17.2 Special Graphs 17.3 Subgraphs 17.4 Isomorphisms of Graphs 17.5 Walks, Paths and Circuits 17.6 Eulerian Paths and Circuits 17.6.1 Methods for Finding Euler Circuit 17.7 Hamiltonian Circuits 17.7.1 Travelling Salesperson Problem 17.8 Matrix Representation of Graphs 17.9 Planar Graphs 17.10 Colouring of Graph 17.11 Directed Graphs 17.12 Trees 17.13 Isomorphism of Trees 17.14 Representation of Algebraic Expressions by Binary Trees 17.15 Spanning Tree of a Graph 17.16 Shortest Path Problem 17.16.1 Dijkstra’s Shortest Path Algorithm 17.16.2 Shortest Path if All Edges Have Length 1 17.17 Minimal Spanning Tree 17.17.1 Prim Algorithm 17.17.2 Kruskal’s Algorithm 17.18 Cut Sets 17.18.1 Relation Between Spanning Trees, Circuits and Cut Sets 17.19 Tree Searching 17.19.1 Procedure to Evaluate an Expression Given in Polish Form 17.20 Transport Networks Exercises Index