Engineering Mathematics II

Cover
Dedications
Roadmap to the Syllabus
Contents
Preface
About the Authors
Chapter 1: Vector Calculus
	1.0 Introduction
	1.1 Scalar and vector point functions
	1.2 Differentiation formulae
	1.3 Level surfaces
	1.4 Gradient of a scalar point function or gradient of a scalar field
		1.4.1 Vector differential operator
		1.4.2 Geometrical meaning of ∇Ф
		1.4.3 Directional derivative
		1.4.4 Equation of tangent plane and normal to the surface
		1.4.5 Angle between two surfaces at a common point
		1.4.6 Properties of gradients
		Worked examples
		Exercise 1.1
	1.5 Divergence of a vector point function or divergence of a vector field
		1.5.1 Physical interpretation of divergence
	1.6 Curl of a vector point function or curl of a vector field
		Worked examples
		Exercise 1.2
	1.7 Vector identities
		Worked examples
	1.8 Integration of vector functions
		1.8.1 Line integral
		Worked examples
		Worked examples
		Exercise 1.3
	1.9 Green’s theorem in a plane
		Worked examples
	1.10 Surface integrals
		1.10.1 Evaluation of surface integral
	1.11 Volume integral
		Worked examples
	1.12 Gauss divergence theorem
		Worked examples
	1.13 Stoke’s theorem
		Worked examples
		Exercise 1.4
		Part A Questions and Answers
Chapter 2: Ordinary Differential Equations
	2.0 Introduction
	2.1 Linear differential equation with constant coefficients
		2.1.1 Complementary function
		2.1.2 Particular integral
		Type 1: Q(x) = eαx
		Worked examples
		Type 2: Q(x) = sin αx or cos αx
		Worked examples
		Type 3: Q(x) = xm, where m is a positive integer.
		Worked examples
		Type 4: Q(x) = eαx g(x)
		Type 5: Q(x) = xmcos αx or xmsin αx
		Worked examples
		Exercise 2.1
	2.2 Linear differential equations with variable coefficients
		2.2.1 Cauchy’s homogeneous linear differential equations
		Worked examples
		2.2.2 Legendre’s linear differential equation
		Worked examples
		Exercise 2.2
	2.3 Simultaneous linear differential equations with constant coefficients
		Worked examples
		Type I
		Type II
		Type III
		Exercise 2.3
	2.4 Method of variation of parameters
		2.4.1 Working rule
		Worked examples
		Exercise 2.4
		Part A Questions and Answers
Chapter 3: Laplace Transforms
	3.0 Introduction
	3.1 Condition for existence of Laplace transform
	3.2 Laplace transform of some elementary functions
	3.3 Some properties of Laplace transform
		Worked examples
		Worked examples
		Exercise 3.1
	3.4 Differentiation and integration of transforms
		Worked examples
		Exercise 3.2
	3.5 Laplace transform of derivatives and integrals
		Worked examples
		3.5.1 Evaluation of improper integrals using Laplace transform
		Worked examples
	3.6 Laplace transform of periodic functions and other special type of functions
		Worked examples
		Worked examples
		3.6.1 Laplace transform of unit step function
		3.6.2 Unit impulse function
		3.6.3 Dirac-delta function
		3.6.4 Laplace transform of delta function
		Worked examples
		Exercise 3.3
	3.7 Inverse Laplace transforms
		Worked examples
		3.7.1 Type 1 – Direct and shifting methods
		3.7.2 Type 2 – Partial fraction method
		3.7.3 Type 3 – 1. multiplication by s, 2. Division by s
		Worked examples
		3.7.4 Type 4 – Inverse Laplace transform of logarithmic and trigonometric functions
		Worked examples
		Exercise 3.4
		3.7.5 Type 5 – Method of convolution
		Worked examples
		Exercise 3.5
	3.8 Application of Laplace transform to the solution of ordinary linear second order differential equations
		Worked examples
		Exercise 3.6
		Part A Questions and Answers
Chapter 4: Analytic Functions
	4.0 Preliminaries
	4.1 Function of a complex variable
		4.1.1 Geometrical representation of complex function or mapping
		4.1.2 Extended complex number system
		4.1.3 Neighbourhood of a point and region
	4.2 Limit of a function
		4.2.1 Continuity of a function
		4.2.3 Derivative of f(z)
		4.2.4 Differentiation formulae
	4.3 Analytic function
		4.3.1 Necessary and sufficient condition for f(z) to be analytic
		4.3.2 C-R equations in polar form
		Worked examples
		Exercise 4.1
	4.4 Harmonic functions and properties of analytic function
		4.4.1 Construction of an analytic function whose real or imaginary part is given. Milne-Thomson Method
		Worked examples
		Exercise 4.2
	4.5 Conformal mapping
		4.5.1 Angle of rotation
		4.5.2 Mapping by elementary functions
		I. Translation w=z+b
		Worked examples
		II. The transformation w = αz
		Worked examples
		III. The Transformation w=1/z
		Worked examples
		IV. The Transformation w=z2
		Worked examples
		(B) The Transformation w=z2 in polar form
		Worked examples
		V. The Transformation w=ez
		Worked examples
		Exercise 4.3
		4.5.3 Bilinear transformation
		Worked examples
		Exercise 4.4
		Part A Questions and Answers
Chapter 5: Complex Integration
	5.0 Introduction
	5.1 Contour Integral
		5.1.1 Properties of contour integrals
		Worked examples
		5.1.2 Simply connected and multiply connected domains
	5.2 Cauchy’s integral theorem or Cauchy’s fundamental theorem
		5.2.1 Cauchy-Goursat integral theorem
	5.3 Cauchy’s integral formula
		5.3.1 Cauchy’s integral formula for derivatives
		Worked examples
		Exercise 5.1
	5.4 Taylor’s series and Laurent’s series
		5.4.1 Taylor’s series
		5.4.2 Laurent’s series
		Worked examples
		Exercise 5.2
	5.5 Classification of singularities
	5.6 Residue
		5.6.1 Methods of finding residue
	5.7 Cauchy’s residue theorem
		5.7.1 Working rule for detecting the type of singularity
		Worked examples
		Exercise 5.3
	5.8 Application of residue theorem to evaluate real integrals
		5.8.1 Type 1.
		Worked examples
		5.8.2 Type 2.
		Worked examples
		5.8.3 Type 3.
		Worked examples
		Exercise 5.4
		Part A Questions and Answers
Formulae to Remember
Index