Engineering Mathematics Vol I

Description

Cover Roadmap to the Syllabus Contents Preface About the Authors Chapter 1: Matrices 1.0 Introduction 1.1 Vector Worked examples 1.2 Eigen values and eigen vectors Worked examples Worked examples Exercise 1.1 1.3 Cayley-Hamilton theorem Worked examples Exercise 1.2 1.4 Similarity transformation and orthogonal transformation 1.4.1 Similar matrices 1.4.2 Diagonalisation of a square matrix 1.4.3 Computation of powers of a square matrix 1.4.4 Orthogonal matrix 1.4.5 Symmetric matrix 1.4.6 Diagonalisation by orthogonal transformation or orthogonal reduction Worked examples 1.5 Real quadratic form. Reduction to canonical form Worked examples Exercise 1.3 Part A: Questions and Answers Chapter 2: Sequence and Series 2.0 Introduction 2.1 Sequence 2.1.1 Infinite sequence 2.1.2 Finite sequence 2.1.3 Limit of a sequence 2.1.4 Convergent sequence 2.1.5 Oscillating sequence 2.1.6 Bounded sequence 2.1.7 Monotonic sequence Worked examples Exercise 2.1 2.2 Series 2.2.1 Convergent series 2.2.2 Divergent series 2.2.3 Oscillatory series 2.2.4 General properties of series 2.3 Series of positive terms 2.3.1 Necessary condition for convergence of a series 2.3.2 Test for convergence of positive term series 2.3.3 Comparison tests Worked examples Exercise 2.2 2.3.4 De' Alembert's ratio test Worked examples Exercise 2.3 2.3.5 Cauchy's root test Worked examples 2.3.6 Cauchy's integral test Worked examples Exercise 2.4 2.3.7 Raabe's test Worked examples Exercise 2.5 2.3.8 Logarithmic test Worked examples 2.4 Alternating series 2.4.1 Leibnitz's test Worked examples 2.5 Series of positive and negative terms 2.5.1 Absolute convergence and conditional convergence 2.5.2 Tests for absolute convergence Worked examples Exercise 2.6 2.6 Convergence of binomial series 2.7 Convergence of the exponential series 2.8 Convergence of the logarithmic series 2.9 Power series 2.9.1 Hadmard's formula 2.9.2 Properties of power series Worked examples Exercise 2.7 Part A: Questions and Answers Chapter 3: Applications of Differential Calculus 3.1 Curvature in cartesian coordinates 3.1.1 Introduction 3.1.2 Measure of curvature 3.1.3 Radius of curvature for cartesian equation of a given curve 3.1.4 Radius of curvature for parametric equations Worked examples 3.1.5 Centre of curvature and circle of curvature 3.1.6 Coordinates of centre of curvature Worked examples Exercise 3.1 3.2 Evolute 3.2.1 Procedure to find evolute Worked examples 3.3 Envelope 3.3.1 Method of finding envelope of single parameter family of curves Worked examples 3.3.2 Envelope of two parameter family of curves Worked examples 3.3.3 Evolute as the envelope of normals Worked examples Exercise 3.2 Part A: Questions and Answers Chapter 4: Differential Calculus of Several Variables 4.0 Introduction 4.1 Limit and continuity Worked examples Exercise 4.1 4.2 Partial derivatives 4.2.1 Geometrical meaning of ∂z/∂x, ∂z/∂y 4.2.2 Partial derivatives of higher order 4.2.3 Homogeneous functions and Euler's theorem Worked examples 4.2.4 Total derivatives Worked examples Exercise 4.2 4.3 Jacobians 4.3.1 Properties of Jacobians Worked examples 4.3.2 Jacobian of implicit functions Exercise 4.3 4.4 Taylor's expansion for function of two variables Worked examples Exercise 4.4 4.5 Maxima and minima for functions of two variables 4.5.1 Necessary conditions for maximum or minimum 4.5.2 Sufficient conditions for extreme values of f(x, y) 4.5.3 Working rule to find maxima and minima of f(x, y) Worked examples 4.5.4 Constrained Maxima and Minima 4.5.5 Method to decide maxima or minima Worked examples Exercise 4.5 Part A: Questions and Answers Chapter 5: Multiple Integrals 5.1 Double integration 5.1.1 Double integrals in cartesian coordinates 5.1.2 Evaluation of double integrals Worked examples Exercise 5.1 5.1.3 Change of order of integration Worked examples Exercise 5.2 5.1.4 Double integral in polar coordinates Worked examples 5.1.5 Change of variables in double integral Worked examples Exercise 5.3 5.1.6 Area as double integral Worked examples Exercise 5.4 Worked examples Exercise 5.4(A) 5.2 Triple integral in cartesian coordinates Worked examples 5.2.1 Volume as triple integral Worked examples Exercise 5.5 5.3 Area of a curved surface 5.3.1 Surface area of a curved surface 5.3.2 Derivation of the formula for surface area 5.3.3 Parametric representation of a surface Worked examples Exercise 5.6 Part A: Questions and Answers Formulae to Remember Index