Engineering Mathematics - III (Aditya)

Cover
About Pearson
Title
Copyright
Contents
Preface
1. Laplace Transforms
	1.0 Introduction
	1.1 Condition for Existence of Laplace Transform
	1.2 Laplace Transform of Some Elementary Functions
	1.3 Some Properties of Laplace Transform
		Worked Examples
		Worked Examples
		Exercise 1.1
		Answers to Exercise 1.11
	1.4 Differentiation and Integration of Transforms
		Worked Examples
		Exercise 1.2
		Answers to Exercise 1.2
	1.5 Laplace Transform of Derivatives and Integrals
		Worked Examples
		1.5.1 Evaluation of Improper Integrals Using Laplace Transform
			Worked Examples
	1.6 Laplace Transform of Periodic Functions and Other Special Type of Functions
		Worked Examples
		Worked Examples
		1.6.1 Laplace Transform of Unit Step Function
		1.6.2 Unit Impulse Function
		1.6.3 Dirac-Delta Function
		1.6.4 Laplace Transform of Delta Function
			Worked Examples
			Exercise 1.3
			Answers to Exercise 1.3
2. Inverse Laplace Transforms
	2.1 Inverse Laplace Transforms
		2.1.1 Type 1 – Direct and Shifting Methods
			Worked Examples
		2.1.2 Type 2 – Partial Fraction Method
			Worked Examples
		2.1.3 Type 3 – Multiplication by s and Division by s
			Worked Examples
		2.1.4 Type 4 – Inverse Laplace Transform of Logarithmic and Trigonometric Functions
			Worked Examples
			Exercise 2.1
			Answers to Exercise 2.1
		2.1.5 Type 5 – Method of Convolution
			Worked Examples
			Exercise 2.2
			Answers to Exercise 2.2
		2.7.6 Type 6: Inverse Laplace Transform as Contour Integral
			Worked Examples
			Exercise 2.3
			Answers to Exercise 2.3
	2.2 Application of Laplace Transform to the Solution of Ordinary Differential Equations
		2.2.1 First Order Linear Differential Equations with Constant Coefficients
			Worked Examples
		2.2.2 Ordinary Second and Higher Order Linear Differential Equations with Constant Coefficients
			Worked Examples
		2.2.3 Ordinary Second Order Differential Equations with Variable Coefficients
			Worked Examples
			Exercise 2.4
			Answers to Exercise 2.4
		2.2.4 Simultaneous Differential Equations
			Worked Examples
		2.2.5 Integral–Differential Equation
			Worked Examples
			Exercise 2.
5
			Answers to Exercise 2.5
3. Multiple Integrals and Beta Gamma Functions
	3.1 Double Integration
		3.1.1 Double Integrals in Cartesian Coordinates
		3.1.2 Evaluation of Double Integrals
			Worked Examples
			Exercise 3.1
			Answers to Exercise 3.1
		3.1.3 Change of Order of Integration
			Worked Examples
			Exercise 3.2
			Answers to Exercise 3.2
		3.1.4 Double Integral in Polar Coordinates
			Worked Examples
		3.1.5 Change of Variables in Double Integral
			Worked Examples
			Exercise 3.3
			Answers to Exercise 3.3
		3.1.6 Area as Double Integral
			Worked Examples
			Exercise 3.4
			Answers to Exercise 3.4
			Worked Examples
			Exercise 3.5
			Answers to Exercise 3.5
	3.2 Area of a Curved Surface
		3.2.1 Surface Area of a Curved Surface
		3.2.2 Derivation of the Formula for Surface Area
		3.2.3 Parametric Representation of a Surface
			Worked Examples
			Exercise 3.6
			Answers to Exercise 3.6
	3.3 Triple Integral in Cartesian Coordinates
		Worked Examples
		Exercise 3.7
		Answers to Exercise 3.7
		3.3.1 Volume as Triple Integral
			Worked Examples
			Exercise 3.8
			Answers to Exercise 3.8
	3.4 Beta and Gamma Functions
		3.4.1 Beta Function
		3.4.2 Symmetric Property of Beta Function
		3.4.3 Different Forms of Beta Function
	3.5 The Gamma Function
		3.5.1 Properties of Gamma Function
		3.5.2 Relation between Beta and Gamma Functions
			Worked Examples
			Exercise 3.9
			Answers to Exercise 3.9
4. Vector Differentiation
	4.0 Introduction
	4.1 Scalar and Vector Point Functions
		4.1.1 Geometrical Meaning of Derivative
	4.2 Differentiation Formulae
	4.3 Level Surfaces
	4.4 Gradient of a Scalar Point Function or Gradient of a Scalar Field
		4.4.1 Vector Differential Operator
		4.4.2 Geometrical Meaning of ___
		4.4.3 Directional Derivative
		4.4.4 Equation of Tangent Plane and Normal to the Surface
		4.4.5 Angle between Two Surfaces at a Common Point
		4.4.6 Properties of Gradients
			Worked Examples
			Exercise 4.1
			Answers to Exercise 4.1
	4.5 Divergence of a Vector Point Function or Divergence of a Vector Field
		4.5.1 Physical Interpretation of Divergence
	4.6 Curl of a Vector Point Function or Curl of a Vector Field
		4.6.1 Physical Meaning of Curl F
			Worked Examples
			Exercise 4.2
			Answers to Exercise 4.2
	4.7 Vector Identities
		Worked Examples
5. Vector Integration
	5.1 Integration of Vector Functions
		5.1.1 Line Integral
			Worked Examples
			Worked Examples
			Exercise 5.1
			Answers to Exercise 5.1
	5.2 Green’s Theorem in a Plane
		5.2.1 Vector Form of Green’s Theorem
			Worked Examples
	5.3 Surface Integrals
		5.3.1 Evaluation of Surface Integral
	5.4 Volume Integral
		Worked Examples
	5.5 Gauss Divergence Theorem
		5.5.1 Results Derived from Gauss Divergence Theorem
			Worked Examples
	5.6 Stoke’s Theorem
		Worked Examples
		Exercise 5.2
		Answers to Exercise 5.2
Appendix A Curve Tracing
	A.1 Curve Tracing
		A.1.1 Procedure for Tracing the Curve Given by the Cartesian Equation f (x, y) = 0
			Worked Examples
		A.1.2 Procedure for Tracing of Curve Given by Parametric Equations x = f (t), y = g(t)
			Worked Examples
		A.1.3 Procedure for Tracing of Curve Given by Equation in Polar Coordinates f (r, _
) = 0
			Worked Examples
			Exercise A.1
			Answers to Exercise A.1
Index