Engineering Mathematics

About Pearson
Copyright
Dedication
Brief Contents
Contents
Preface
About the Authors
1. Matrices
	1.0 Introduction
	1.1 Basic Concepts
		1.1.1 Basic Operations on Matrices
		1.1.2 Properties of Addition, Scalar Multiplication and Multiplication
	1.2 Complex Matrices
		Worked Examples
		Exercise 1.1
		Answers to Exercise 1.1
	1.3 Rank of a Matrix
		Worked Examples
		Exercise 1.2
		Answers to Exercise 1.2
	1.4 Solution of System of Linear Equations
		1.4.1 Non-homogeneous System of Equations
		1.4.2 Homogeneous System of Equations
		1.4.3 Type 1: Solution of Non-homogeneous System of Equations
			Worked Examples
		1.4.4 Type 2: Solution of Non-homogeneous Linear Equations Involving Arbitrary Constants
			Worked Examples
		1.4.5 Type 3: Solution of the System of Homogeneous Equations
			Worked Examples
		1.4.6 Type 4: Solution of Homogeneous System of Equation Containing Arbitrary Constants
			Worked Examples
			Exercise 1.3
			Answers to Exercise 1.3
	1.5 Matrix Inverse by Gauss–Jordan method
		Worked Examples
		Exercise 1.4
		Answers to Exercise 1.4
	1.6 Eigen Values and Eigen Vectors
		1.6.0 Introduction
		1.6.1 Vector
			Worked Examples
		1.6.2 Eigen Values and Eigen Vectors
		1.6.3 Properties of Eigen Vectors
			Worked Examples
		1.6.4 Properties of Eigen Values
			Worked Examples
			Exercise 1.5
			Answers to Exercise 1.5
		1.6.5 Cayley-Hamilton Theorem
			Worked Examples
			Exercise 1.6
			Answers to Exercise 1.6
	1.7 Similarity Transformation and Orthogonal Transformation
		1.7.1 Similar Matrices
		1.7.2 Diagonalisation of a Square Matrix
		1.7.3 Computation of the Powers of a Square Matrix
		1.7.4 Orthogonal Matrix
		1.7.5 Properties of Orthogonal Matrix
		1.7.6 Symmetric Matrix
		1.7.7 Properties of Symmetric Matrices
		1.7.8 Diagonalisation by Orthogonal Transformation or Orthogonal Reduction
			Worked Examples
	1.8 Real Quadratic Form. Reduction to Canonical Form
		Worked Examples
		Exercise 1.7
		Answers to Exercise 1.7
	Short Answer Questions
	Objective Type Questions
	Answers
2. Sequences 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
			Answers to 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
			Answers to Exercise 2.2
		2.3.4 De’ Alembert’s Ratio Test
			Worked Examples
			Exercise 2.3
			Answers to Exercise 2.3
		2.3.5 Cauchy’s Root Test
			Worked Examples
		2.3.6 Cauchy’s Integral Test
			Worked Examples
			Exercise 2.4
			Answers to Exercise 2.4
		2.3.7 Raabe’s Test
			Worked Examples
			Exercise 2.5
			Answers to 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
			Answers to 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
			Answers to Exercise 2.7
	Short Answer Questions
	Objective Type Questions
	Answers
3. Differential Calculus
	3.0 Introduction
	3.1 Successive Differentiation
		Worked Examples
		Exercise 3.1
		3.1.1 The nth Derivative of Standard Functions
			Worked Examples
			Exercise 3.2
			Answers to Exercise 3.2
			Worked Examples
			Exercise 3.3
	3.2 Applications of Derivative
		3.2.1 Geometrical Interpretation of Derivative
		3.2.2 Equation of the Tangent and the Normal to the Curve y = f(x)
			Worked Examples
			Exercise 3.4
			Answers to Exercise 3.4
		3.2.3 Length of the Tangent, the Sub-Tangent, the Normal and the Sub-normal
			Worked Examples
			Exercise 3.5
			Answers to Exercise 3.5
		3.2.4 Angle between the Two Curves
			Worked Examples
			Exercise 3.6
			Answers to Exercise 3.6
	3.3 Mean-value Theorems of Derivatives
		3.3.1 Rolle’s Theorem
			Worked Examples
		3.3.2 Lagrange’s Mean Value Theorem
			Worked Examples
		3.3.3 Cauchy’s Mean Value Theorem
			Worked Examples
			Exercise 3.7
			Answers to Exercise 3.7
	3.4 Monotonic Functions
		3.4.1 Increasing and Decreasing Functions
		3.4.2 Piece−wise Monotonic Function
		3.4.3 Test for Increasing or Decreasing Functions
			Worked Examples
			Exercise 3.8
			Answers to Exercise 3.8
	3.5 Generalised Mean Value Theorem
		3.5.1 Taylor’s Theorem with Lagrange’s form of Remainder
		3.5.2 Taylor’s Series
		3.5.3 Maclaurin’s Theorem with Lagrange’s form of Remainder
		3.5.4 Maclaurin’s Series
			Worked Examples
			Exercise 3.9
			Answers to Exercise 3.9
		3.5.5 Expansion by Using Maclaurin’s Series of Some Standard Functions
			Worked Examples
		3.5.6 Expansion of Certain Functions Using Differential Equations
			Worked Examples
			Exercise 3.10
			Answers to Exercise 3.10
	3.6 Indeterminate Forms
		3.6.1 General L’Hopital’s Rule for 0/0 form
			Worked Examples
			Exercise 3.11
			Answers to Exercise 3.11
	3.7 Maxima and Minima of a Function of One Variable
		3.7.1 Geometrical Meaning
		3.7.2 Tests for Maxima and Minima
			Summary
			Worked Examples
			Exercise 3.12
			Answers to Exercise 3.12
	3.8 Asymptotes
		Worked Examples
		3.8.1 A General Method
		3.8.2 Asymptotes Parallel to the Coordinates Axes
			Worked Examples
		3.8.3 Another Method for Finding the Asymptotes
			Worked Examples
		3.8.4 Asymptotes by Inspection
			Worked Examples
		3.8.5 Intersection of a Curve and Its Asymptotes
			Worked Examples
			Exercise 3.13
			Answers to Exercise 3.13
	3.9 Concavity
		Worked Examples
		Exercise 3.14
		Answers to Exercise 3.14
	3.10 Curve Tracing
		3.10.1 Procedure for Tracing the Curve Given by the Cartesian Equation f(x, y) = 0.
			Worked Examples
		3.10.2 Procedure for Tracing of Curve Given by Parametric Equations x = f(t), y = g(t)
			Worked Examples
		3.10.3 Procedure for Tracing of Curve Given by Equation in Polar Coordinates f(r, θ) = 0
			Worked Examples
			Exercise 3.15
			Answers to Exercise 3.15
	Short Answer Questions
	Objective Type Questions
	Answers
4. Applications of Differential Calculus
	4.1 Curvature in Cartesian Coordinates
		4.1.0 Introduction
		4.1.1 Measure of Curvature
		4.1.2 Radius of Curvature for Cartesian Equation of a Given Curve
		4.1.3 Radius of Curvature for Parametric Equations
			Worked Examples
		4.1.4 Centre of Curvature and Circle of Curvature
		4.1.5 Coordinates of the Centre of Curvature
			Worked Examples
			Exercise 4.1
			Answers to Exercise 4.1
		4.1.6 Radius of Curvature in Polar Coordinates
			Worked Examples
		4.1.7 Radius of Curvature at the Origin
			Worked Examples
		4.1.8 Pedal Equation or p – r Equation of a Curve
			Worked Examples
		4.1.9 Radius of Curvature Using the p – r Equation of a Curve
			Worked Examples
			Exercise 4.2
			Answers to Exercise 4.2
	4.2 Evolute
		4.2.1 Properties of Evolute
		4.2.2 Procedure to Find the Evolute
			Worked Examples
			Exercise 4.3
			Answers to Exercise 4.3
	4.3 Envelope
		4.3.1 Method of Finding Envelope of Single Parameter Family of Curves
			Worked Examples
		4.3.2 Envelope of Two Parameter Family of Curves
			Worked Examples
		4.3.3 Evolute as the Envelope of Normals
			Worked Examples
			Exercise 4.4
			Answers to Exercise 4.4
	Short Answer Questions
	Objective Type Questions
	Answers
5. Differential Calculus of Several Variables
	5.0 Introduction
	5.1 Limit and Continuity
		Worked Examples
		Exercise 5.1
		Answers to Exercise 5.1
	5.2 Partial Derivatives
		5.2.1 Geometrical Meaning of ∂z/∂x, ∂z/∂y
		5.2.2 Partial Derivatives of Higher Order
		5.2.3 Homogeneous Functions and Euler’s Theorem
			Worked Examples
		5.2.4 Total Derivatives
			Worked Examples
			Exercise 5.2
			Answers to Exercise 5.2
	5.3 Jacobians
		5.3.1 Properties of Jacobians
			Worked Examples
		5.3.2 Jacobian of Implicit Functions
			Worked Examples
			Exercise 5.3
			Answers to Exercise 5.3
	5.4 Taylor’s Series Expansion for Function of Two Variables
		Worked Examples
		Exercise 5.4
		Answers to Exercise 5.4
	5.5 Maxima and Minima for Functions of Two Variables
		5.5.1 Necessary Conditions for Maximum or Minimum
		5.5.2 Sufficient Conditions for Extreme Values of f (x, y ).
		5.5.3 Working Rule to find Maxima and Minima of f (x, y )
			Worked Examples
		5.5.4 Constrained Maxima and Minima
		5.5.5 Lagrange’s Method of (undetermined) Multiplier
		5.5.6 Method to Decide Maxima or Minima
			Worked Examples
			Exercise 5.5
			Answers to Exercise 5.5
	5.6 Errors and Approximations
		Worked Examples
		Exercise 5.6
		Answers to Exercise 5.6
	Short Answer Questions
	Objective Type Questions
	Answers
6. Integral Calculus
	6.0 Introduction
	6.1 Indefinite Integral
		6.1.1 Properties of Indefinite Integral
		6.1.2 Integration by Parts
		6.1.3 Bernoulli’s Formula
		6.1.4 Special Integrals
			Worked Examples
			Exercise 6.1
			Answers to Exercise 6.1
	6.2 Definite Integral (Newton–Leibnitz Formula)
		6.2.1 Properties of Definite Integral
			Worked Examples
			Exercise 6.2
			Answers to Exercise 6.2
	6.3 Definite Integral b∫a f(x)dx as Limit of a Sum
		6.3.1 Working Rule
			Worked Examples
			Exercise 6.3
			Answers to Exercise 6.3
	6.4 Reduction Formulae
		6.4.1 The Reduction Formula for (a) ∫sinn xdx and (b) ∫cosn xdx
		6.4.2 The Reduction Formula for (a) ∫tann xdx and (b) ∫cotn xdx
		6.4.3 The Reduction Formula for (a) ∫secn xdx and (b) ∫cosecnxdx
			Worked Examples
		6.4.4 The Reduction Formula for ∫sinmxcosnxdx, Where m,n are Non-negative Integers
			Worked Examples
		6.4.5 The Reduction Formula For (a) ∫xm(log x)ndx, (b) ∫ xn sin mx dx, (c) ∫ xn cos mx dx
		6.4.6 The Reduction Formula for (a) ∫eax sinmxdx and (b) ∫eax cosnxdx
		6.4.7 The Reduction Formula for (a) ∫ cosmxsinnxdx and (b) ∫cosmxcosnxdx
			Exercise 6.4
			Answers to Exercise 6.4
	6.5 Application of Integral Calculus
		6.5.1 Area of Plane Curves
			6.5.1 (a) Area of Plane Curves in Cartesian Coordinates
				Worked Examples
				Exercise 6.5
				Answers to Exercise 6.5
			6.5.1 (b) Area in Polar Coordinates
				Worked Examples
				Exercise 6.6
				Answers to Exercise 6.6
		6.5.2 Length of the Arc of a Curve
			6.5.2 (a) Length of the Arc in Cartesian Coordinates
				Worked Examples
				Exercise 6.7
				Answers to Exercise 6.7
			6.5.2 (b) Length of the Arc in Polar Coordinates
				Worked Examples
				Exercise 6.8
				Answers to Exercise 6.8
		6.5.3 Volume of Solid of Revolution
			6.5.3(a) Volume in Cartesian Coordinates
				Worked Examples
				Exercise 6.9
				Answers to Exercise 6.9
			6.5.3 (b) Volume in Polar Coordinates
				Worked Examples
				Exercise 6.10
				Answers to Exercise 6.10
		6.5.4 Surface Area of Revolution
			6.5.4(a) Surface Area of Revolution in Cartesian Coordinates
				Worked Examples
				Exercise 6.11
				Answers to Exercise 6.11
			6.5.4 (b) Surface Area in Polar Coordinates
				Worked Examples
				Exercise 6.12
				Answers to Exercise 6.12
	Short Answer Questions
	Objective Type Questions
	Answers
7. Improper Integrals
	7.1 Improper Integrals
		7.1.1 Kinds of Improper Integrals and Their Convergence
			Worked Examples
			Exercise 7.1
			Answers to Exercise 7.1
		7.1.2 Tests of Convergence of Improper Integrals
			Worked Examples
			Exercise 7.2
			Answers to Exercise 7.2
	7.2 Evaluation of Integral by Leibnitz’s Rule
		7.2.1 Leibnitz’s Rule—Differentiation Under Integral Sign for Variable Limits
			Worked Examples
			Exercise 7.3
			Answers to Exercise 7.3
	7.3 Beta and Gamma functions
		7.3.1 Beta Function
		7.3.2 Symmetric property of beta function
		7.3.3 Different forms of beta function
	7.4 The Gamma Function
		7.4.1 Properties of Gamma Function
		7.4.2 Relation between Beta and Gamma Functions
			Worked Examples
			Exercise 7.4
			Answers to Exercise 7.4
	7.5 The Error Function
		7.5.1 Properties of Error Functions
		7.5.2 Series expansion for error function
		7.5.3 Complementary error function
			Worked Examples
			Exercise 7.5
			Answers to Exercise 7.5
	Short Answer Questions
	Objective Type Questions
	Answers
8. Multiple Integrals
	8.1 Double Integration
		8.1.1 Double Integrals in Cartesian Coordinates
		8.1.2 Evaluation of Double Integrals
			Worked Examples
			Exercise 8.1
			Answers to Exercise 8.1
		8.1.3 Change of Order of Integration
			Worked Examples
			Exercise 8.2
			Answers to Exercise 8.2
		8.1.4 Double Integral in Polar Coordinates
			Worked Examples
		8.1.5 Change of Variables in Double Integral
			Worked Examples
			Exercise 8.3
			Answers to Exercise 8.3
		8.1.6 Area as Double Integral
			Worked Examples
			Exercise 8.4
			Answers to Exercise 8.4
			Worked Examples
			Exercise 8.5
			Answers to Exercise 8.5
	8.2 Area of a Curved Surface
		8.2.1 Surface Area of a Curved Surface
		8.2.2 Derivation of the Formula for Surface Area
		8.2.3 Parametric Representation of a Surface
			Worked Examples
			Exercise 8.6
			Answers to Exercise 8.6
	8.3 Triple Integral in Cartesian Coordinates
		Worked Examples
		Exercise 8.7
		Answers to Exercise 8.7
		8.3.1 Volume as Triple Integral
			Worked Examples
			Exercise 8.8
			Answers to Exercise 8.8
	Short Answer Questions
	Objective Type Questions
	Answers
9. Vector Calculus
	9.0 Introduction
	9.1 Scalar and Vector Point Functions
		9.1.1 Geometrical Meaning of Derivative
	9.2 Differentiation Formulae
	9.3 Level Surfaces
	9.4 Gradient of a Scalar Point Function or Gradient of a Scalar Field
		9.4.1 Vector Differential Operator
		9.4.2 Geometrical Meaning of "Ф
		9.4.3 Directional Derivative
		9.4.4 Equation of Tangent Plane and Normal to the Surface
		9.4.5 Angle between Two Surfaces at a Common Point
		9.4.6 Properties of gradients
			Worked Examples
			Exercise 9.1
			Answers to Exercise 9.1
	9.5 Divergence of a Vector Point Function or Divergence of a Vector Field
		9.5.1 Physical Interpretation of Divergence
	9.6 Curl of a Vector Point Function or Curl of a Vector Field
		9.6.1 Physical Meaning of Curl F
			Worked Examples
			Exercise 9.2
			Answers to Exercise 9.2
	9.7 Vector Identities
		Worked Examples
	9.8 Integration of Vector Functions
		9.8.1 Line Integral
			Worked Examples
			Exercise 9.3
			Answers to Exercise 9.3
	9.9 Green’s Theorem in a Plane
		9.9.1 Vector Form of Green’s Theorem
			Worked Examples
	9.10 Surface Integrals
		9.10.1 Evaluation of Surface Integral
	9.11 Volume Integral
		Worked Examples
	9.12 Gauss Divergence Theorem
		9.12.1 Results Derived from Gauss Divergence Theorem
			Worked Examples
	9.13 Stoke’s Theorem
		Worked Examples
		Exercise 9.4
		Answers to Exercise 9.4
	Short Answer Questions
	Objective Type Questions
	Answers
10. Ordinary First Order Differential Equations
	10.0 Introduction
	10.1 Formation of Differential Equations
		Worked Examples
		Exercise 10.1
		Answers to Exercise 10.1
	10.2 First Order and First Degree Differential Equations
		10.2.1 Type I Variable Separable Equations
			Worked Example
			Exercise 10.2
			Answers to Exercise 10.2
		10.2.2 Type II Homogeneous Equation
			Worked Examples
			Exercise 10.3
			Answers to Exercise 10.3
		10. 2.3 Type III Non-Homogenous Differential Equations of the First Degree
			Worked Examples
			Exercise 10.4
			Answers to Exercise 10.4
		10.2.4 Type IV Linear Differential Equation
			Worked Examples
			Exercise 10.5
			Answers to Exercise 10.5
		10.2.5 Type V Bernoulli’s Equation
			Worked Examples
			Exercise 10.6
			Answers to Exercise 10.6
		10.2.6 Type VI Riccati Equation
			Worked Examples
			Exercise 10.7
			Answers to Exercise 10.7
		10.2.7 Type VII First Order Exact Differential Equations
			Worked Examples
			Exercise 10.8
			Answers to Exercise 10.8
	10.3 Integrating Factors
		Worked Examples
		10.3.1 Rules for Finding the Integrating Factor for Non-Exact Differential Equation Mdx + Ndy = 0
			Worked Examples
			Exercise 10.9
			Answers to Exercise 10.9
	10.4 Ordinary Differential Equations of the First Order but of Degree Higher than One
		10.4.1 Type 1 Equations Solvable for p
			Worked Examples
			Exercise 10.10
			Answers to Exercise 10.10
		10.4.2 Type 2 Equations Solvable for y
			Worked Examples
		10.4.3 Type 3 Equations Solvable for x
			Worked Examples
			Exercise 10.11
			Answers to Exercise 10.11
		10.4.4 Type 4 Clairaut’s Equation
			Worked Examples
			Exercise 10.12
			Answers to Exercise 10.12
	Short Answer Questions
	Objective Type Questions
	Answers
11. Ordinary Second and Higher Order Differential Equations
	11.0 Introduction
	11.1 Linear Differential Equation with Constant Coefficients
		11.1.1 Complementary Function
		11.1.2 Particular Integral
			Worked Examples
			Exercise 11.1
			Answers to Exercise 11.1
	11.2 Linear Differential Equations with Variable Coefficients
		11.2.1 Cauchy’s Homogeneous Linear Differential Equations
			Worked Examples
		11.2.2 Legendre’s Linear Differential Equation
			Worked Examples
			Exercise 11.2
			Answers to Exercise 11.2
	11.3 Simultaneous Linear Differential Equations with Constant Coefficients
		Worked Examples
		Exercise 11.3
		Answers to Exercise 11.3
	11.4 Method of Variation of Parameters
		11.4.1 Working Rule
			Worked Examples
			Exercise 11.4
			Answers to Exercise 11.4
	11.5 Method of Undetermined Coefficients
		Worked Examples
		Exercise 11.5
		Answers to Exercise 11.5
	Short Answers Questions
	Objective Type Questions
	Answers
12. Applications of Ordinary Differential Equations
	12.0 Introduction
	12.1 Applications of Ordinary Differential Equations of First Order
		12.1.1 Law of Growth and Decay
		12.1.2 Newton’s Law of Cooling of Bodies
			Worked Examples
			Exercise 12.1
			Answers To Exercise 12.1
		12.1.3 Chemical Reaction and Solutions
			Worked Examples
			Exercise 12.2
			Answers to Exercise 12.2
		12.1.4 Simple Electric Circuit
			Worked Examples
			Exercise 12.3
			Answers to Exercise 12.3
		12.1.5 Geometrical Applications
			12.1.5 (a) Orthogonal Trajectories in Casterian Coordinates
				Worked Examples
			12.1.5 (b) Orthogonal Trajectories in Polar Coordinates
				Worked Examples
				Exercise 12.4
				Answers to Exercise 12.4
	12.2 Applications of Second Order Differential Equations
		12.2.1 Bending of Beams
			Worked Examples
		12.2.2 Electric Circuits
			Worked Examples
			Exercise 12.5
			Answers to Exercise 12.5
		12.2.3 Simple Harmonic Motion (S.H.M)
			Worked Examples
			Exercise 12.6
			Answers to Exercise 12.6
	Objective Type Questions
	Answers
13. Series Solution of Ordinary Differential Equations and Special Functions
	13.0 Introduction
	13.1 Power Series Method
		13.1.1 Analytic Function
		13.1.2 Regular Point
		13.1.3 Singular Point
		13.1.4 Regular and Irregular Singular Points
			Worked Examples
			Exercise 13.1
			Answers to Exercise 13.1
	13.2 Frobenius Method
		Worked Examples
		Exercise 13.2
		Answers to Exercise 13.2
	13.3 Special Functions
	13.4 Bessel Functions
		13.4.1 Series Solution of Bessel’s Equation
		13.4.2 Bessel’s Functions of the First Kind
			Worked Examples
		13.4.3 Some Special Series
		13.4.4 Recurrence Formula for Jn (x)
		13.4.5 Generating Function for Jn (x) of Integral Order
			Worked Examples
		13.4.6 Integral Formula for Bessel’s Function Jn (x)
			Worked Examples
		13.4.7 Orthogonality of Bessel’s Functions
		13.4.8 Fourier–Bessel Expansion of a Function f(x)
			Worked Examples
		13.4.9 Equations Reducible to Bessel’s Equation
			Worked Examples
			Exercise 13.3
			Answers to Exercise 13.3
	13.5 Legendre Functions
		13.5.1 Series Solution of Legendre’s Differential Equation
		13.5.2 Legendre Polynomials
		13.5.3 Rodrigue’s Formula
			Worked Examples
		13.5.4 Generating Function for Legendre Polynomials
			Worked Examples
		13.5.5 Orthogonality of Legendre Polynomials in [-1, 1]
			Worked Examples
		13.5.6 Fourier–Legendre Expansion of f(x) in a Series of Legendre Polynomials
			Worked Examples
			Exercise 13.4
			Answers to Exercise 13.4
14. Partial Differential Equations
	14.0 Introduction
	14.1 Order and Degree of Partial Differential Equations
	14.2 Linear and Non-linear Partial Differential Equations
	14.3 Formation of Partial Differential Equations
		Worked Examples
		Exercise 14.1
		Answers to Exercise 14.1
	14.4 Solutions of Partial Differential Equations
		14.4.1 Procedure to Find General Integral and Singular Integral for a First Order Partial Differential Equation
			Worked Examples
			Exercise 14.2
			Answers to Exercise 14.2
		14.4.2 First Order Non-linear Partial Differential Equation of Standard Types
			Worked Examples
			Exercise 14.3
			Answers to Exercise 14.3
			Worked Examples
		14.4.3 Equations Reducible to Standard Forms
			Worked Examples
			Exercise 14.4
			Answers to Exercise 14.4
	14.5 Lagrange’s Linear Equation
		Worked Examples
		Exercise 14.5
		Answers to Exercise 14.5
	14.6 Homogeneous Linear Partial Differential Equations of the Second and Higher Order with Constant Coefficients
		14.6.1 Working Procedure to Find Complementary Function
		14.6.2 Working Procedure to Find Particular Integral
			Worked Examples
			Exercise 14.6
			Answers to Exercise 14.6
	14.7 Non-homogeneous Linear Partial Differential Equations of the Second and Higher Order with Constant Coefficients
		Worked Examples
		Exercise 14.7
		Answers to Exercise 14.7
	Short Answer Questions
	Objective Type Questions
	Answers
15. Analytic Functions
	15.0 Preliminaries
	15.1 Function of a Complex Variable
		15.1.1 Geometrical Representation of Complex Function or Mapping
		15.1.2 Extended Complex Number System
		15.1.3 Neighbourhood of a Point and Region
	15.2 Limit of a Function
		15.2.1 Continuity of a Function
		15.2.2 Derivative of f(z)
		15.2.3 Differentiation Formulae
	15.3 Analytic Function
		15.3.1 Necessary and Sufficient Condition for f(z) to be Analytic
		15.3.2 C-R Equations in Polar Form
			Worked Examples
			Exercise 15.1
			Answers to Exercise 15.1
	15.4 Harmonic Functions and Properties of Analytic Function
		15.4.1 Construction of an Analytic Function Whose Real or Imaginary Part is Given Milne-Thomson Method
			Worked Examples
			Exercise 15.2
			Answers to Exercise 15.2
	15.5 Conformal Mapping
		15.5.1 Angle of Rotation
		15.5.2 Mapping by Elementary Functions
			Worked Examples
			Exercise 15.3
			Answers to Exercise 15.3
		15.5.3 Bilinear Transformation
			Worked Examples
			Exercise 15.4
			Answers to Exercise 15.4
	Short Answer Questions
	Objective Type Questions
	Answers
16. Complex Integration
	16.0 Introduction
	16.1 Contour Integral
		16.1.1 Properties of Contour Integrals
			Worked Examples
		16.1.2 Simply Connected and Multiply Connected Domains
	16.2 Cauchy’s Integral Theorem or Cauchy’s Fundamental Theorem
		16.2.1 Cauchy-Goursat Integral Theorem
	16.3 Cauchy’s Integral Formula
		16.3.1 Cauchy’s Integral Formula for Derivatives
			Worked Examples
			Exercise 16.1
			Answers to Exercise 16.1
	16.4 Taylor’s Series and Laurent’s Series
		16.4.1 Taylor’s Series
		16.4.2 Laurent’s Series
			Worked Examples
			Exercise 16.2
			Answers to Exercise 16.2
	16.5 Classification of Singularities
	16.6 Residue
		16.6.1 Methods of Finding Residue
	16.7 Cauchy’s Residue Theorem
		Worked Examples
		Exercise 16.3
		Answers to Exercise 16.3
	16.8 Application of Residue Theorem to Evaluate Real Integrals
		16.8.1 Type 1
			Worked Examples
		16.8.2 Type 2. Improper Integrals of Rational Functions
			Worked Examples
		16.8.3 Type 3
			Worked Examples
			Exercise 16.4
			Answers to Exercise 16.4
	Short Answer Questions
	Objective Type Questions
	Answers
17. Fourier Series
	17.0 Introduction
	17.1 Fourier series
		17.1.1 Dirichlet’s Conditions
		17.1.2 Convergence of Fourier Series
			Worked Examples
	17.2 Even and Odd Functions
		17.2.1 Sine and Cosine Series
			Worked Examples
			Exercise 17.1
			Answers to Exercise 17.1
	17.3 Half-Range Series
		17.3.1 Half-range Sine Series
		17.3.2 Half-range Cosine Series
			Worked Examples
			Exercise 17.2
			Answers to Exercise 17.2
	17.4 Change of Interval
		Worked Examples
	17.5 Parseval’s Identity
		Worked Examples
		Exercise 17.3
		Answers to Exercise 17.3
	17.6 Complex Form of Fourier Series
		Worked Examples
		Exercise 17.4
		Answers to Exercise 17.4
	17.7 Harmonic Analysis
		17.7.1 Trapezoidal Rule
			Worked Examples
			Exercise 17.5
			Answers to Exercise 17.5
	Short Answer Questions
	Objective Type Questions
	Answers
18. Fourier Transforms
	18.0 Introduction
	18.1 Fourier Integral Theorem
		18.1.1 Fourier Cosine and Sine Integrals
			Worked Examples
		18.1.2 Complex Form of Fourier Integral
	18.2 Fourier Transform Pair
		18.2.1 Properties of Fourier Transforms
			Worked Examples
			Exercise 18.1
			Answers to Exercise 18.1
	18.3 Fourier Sine and Cosine Transforms
		18.3.1 Properties of Fourier Sine and Cosine Transforms
			Worked Examples
			Exercise 18.2
			Answers to Exercise 18.2
	18.4 Convolution Theorem
		18.4.1 Definition: Convolution of Two Functions
		18.4.2 Theorem 18.1: Convolution Theorem or Faltung Theorem
		18.4.3 Theorem 18.2 : Parseval’s Identity for Fourier Transforms or Energy Theorem
			Worked Examples
			Exercise 18.3
			Answers to Exercise 18.3
	Short Answer Questions
	Objective Type Questions
	Answers
19. Laplace Transforms
	19.0 Introduction
	19.1 Condition for Existence of Laplace Transform
	19.2 Laplace Transform of Some Elementary Functions
	19.3 Some Properties of Laplace Transform
		Worked Examples
		Exercise 19.1
		Answers to Exercise 19.1
	19.4 Differentiation and Integration of Transforms
		Worked Examples
		Exercise 19.2
		Answers to Exercise 19.2
	19.5 Laplace Transform of Derivatives and Integrals
		Worked Examples
		19.5.1 Evaluation of Improper Integrals Using Laplace Transform
			Worked Examples
	19.6 Laplace Transform of Periodic Functions and Other Special Type of Functions
		Worked Examples
		19.6.1 Laplace Transform of Unit Step Function
		19.6.2 Unit Impulse Function
		19.6.3 Dirac-delta Function
		19.6.4 Laplace Transform of Delta Function
			Worked Examples
			Exercise 19.3
			Answers to Exercise 19.3
	19.7 Inverse Laplace Transforms
		19.7.1 Type 1 – Direct and Shifting Methods
			Worked Examples
		19.7.2 Type 2 – Partial Fraction Method
			Worked Examples
		19.7.3 Type 3 – 1. Multiplication by s and 2. Division by s
			Worked Examples
		19.7.4 Type 4 – Inverse Laplace Transform of Logarithmic and Trigonometric Functions
			Worked Examples
			Exercise 19.4
			Answers to Exercise 19.4
		19.7.5 Type 5 – Method of Convolution
			Worked Examples
			Exercise 19.5
			Answers to Exercise 19.5
		19.7.6 Type 6: Inverse Laplace Transform as Contour Integral
			Worked Examples
			Exercise 19.6
			Answers to Exercise 19.6
	19.8 Application of Laplace Transform to the Solution of Ordinary Differential Equations
		19.8.1 First Order Linear Differential Equations with Constant Coefficients
			Worked Examples
		19.8.2 Ordinary Second and Higher Order Linear Differential Equations with Constant Coefficients
			Worked Examples
		19.8.3 Ordinary Second Order Differential Equations with Variable Coefficients
			Worked Examples
			Exercise 19.7
			Answers to Exercise 19.7
		19.8.4 Simultaneous Differential Equations
			Worked Examples
		19.8.5 Integral–Differential Equation
			Worked Examples
			Exercise 19.8
			Answers to Exercise 19.8
	Short Answer Questions
	Objective Type Questions
	Answers
20. Applications of Partial Differential Equations
	20.0 Introduction
	20.1 One Dimensional Wave Equation – Equation of Vibrating String
		20.1.1 Derivation of Wave Equation
		20.1.2 Solution of One-Dimensional Wave Equation by the Method of Separation of Variables (or the Fourier Method)
			Worked Examples
			Exercise 20.1
			Answers to Exercise 20.1
		20.1.3 Classification of Partial Differential Equation of Second Order
			Worked Examples
			Exercise 20.2
			Answers to Exercise 20.2
	20.2 One-Dimensional Equation of Heat Conduction (In a Rod)
		20.2.1 Derivation of Heat Equation
		20.2.2 Solution of Heat Equation by Variable Separable Method
			Worked Examples
			Exercise 20.3
			Answers to Exercise 20.3
			Worked Examples
			Exercise 20.4
			Answers to Exercise 20.4
	20.3 Two Dimensional Heat Equation in Steady State
		20.3.1 Solution of Two Dimensional Heat Equation
			Worked Examples
			Exercise 20.5
			Answers to Exercise 20.5
	Short Answer Questions
	Objective Type Questions
	Answers
Index