Engineering Mathematics I (Aditya)

Cover
About Pearson
Title
Copyright
Contents
Preface
1. Differential Equations of First Order and First Degree
	1.0 Introduction
	1.1 Formation of Differential Equations
		Worked Examples
		Exercise 1.1
		Answers to Exercise 1.1
	1.2 First Order and First Degree Differential Equations
		1.2.1 Type I Variable Separable Equations
			Worked Example
			Exercise 1.2
			Answers to Exercise 1.2
		1.2.2 Type II Homogeneous Equation
			Worked Examples
			Exercise 1.3
			Answers to Exercise 1.3
		1.2.3 Type III Non-Homogenous Differential Equations of the First Degree
			Worked Examples
			Exercise 1.4
			Answers to Exercise 1.4
		1.2.4 Type IV Linear Differential Equation
			Worked Examples
			Exercise 1.5
			Answers to Exercise 1.5
		1.2.5 Type V Bernoulli’s Equation
			Worked Examples
			Exercise 1.6
			Answers to Exercise 1.6
		1.2.6 Type VII First Order Exact Differential Equations
			Worked Examples
			Exercise 1.7
			Answers to Exercise 1.7
	1.3 Integrating Factors
		Worked Examples
		1.3.1 Rules for Finding the Integrating Factor for Non-Exact Differential Equation Mdx + Ndy = 0
			Worked Examples
			Exercise 1.8
			Answers to Exercise 1.8
	1.4 Application of Ordinary Differential Equations
		1.4.1 Introduction
	1.5 Applications of Ordinary Differential Equations of First Order
		1.5.1 Law of Growth and Decay
		1.5.2 Newton’s Law of Cooling of Bodies
			Worked Examples
			Exercise 1.9
			Answers to Exercise 1.9
		1.5.3 Chemical Reaction and Solutions
			Worked Examples
			Exercise 1.10
			Answers to Exercise 1.10
		1.5.4 Simple Electric Circuit
			Worked Examples
			Exercise 1.11
			Answers to Exercise 1.11
		1.5.5 Geometrical Applications
		1.5.5(a) Orthogonal Trajectories in Casterian Coordinates
			Worked Examples
		1.5.5(b) Orthogonal Trajectories in Polar Coordinates
			Worked Examples
			Exercise 1.12
			Answers to Exercise 1.12
2. Linear Differential Equations of Higher Order
	2.0 Introduction
	2.1 Linear Differential Equation with Constant Coefficients
		2.1.1 Complementary Function
		2.1.2 Particular Integral
			Worked Examples
			Worked Examples
			Worked Examples
			Worked Examples
			Exercise 2.1
			Answers to Exercise 2.1
	2.2 Method of Variation of Parameters
		2.2.1 Working Rule
			Worked Examples
			Exercise 2.2
			Answers to Exercise 2.2
	2.3 Method of Undetermined Coefficients
		Worked Examples
		Exercise 2.3
		Answers to Exercise 2.3
		2.3.1 Electric Circuits
			Worked Examples
			Exercise 2.4
			Answers to Exercise 2.4
		2.3.2 Simple Harmonic Motion (S.H.M)
			Worked Examples
			Exercise 2.5
			Answers to Exercise 2.5
3. Linear System of Equations
	3.0 Introduction
	3.1 Basic Concepts
		3.1.1 Basic Operations on Matrices
		3.1.2 Properties of Addition, Scalar Multiplication and Multiplication
	3.2 Complex Matrices
		Worked Examples
		Exercise 3.1
		Answers to Exercise 3.1
	3.3 Rank of a Matrix
		Worked Examples
		Worked Examples
		Exercise 3.2
		Answers to Exercise 3.2
	3.4 Solution of System of Linear Equations
		3.4.1 Non-homogeneous System of Equations
		3.4.2 Homogeneous System of Equations
		3.4.3 Type 1: Solution of Non-homogeneous System of Equations
			Worked Examples
		3.4.4 Type 2: Solution of Non-homogeneous Linear Equations Involving Arbitrary Constants
			Worked Examples
		3.4.5 Type 3: Solution of the System of Homogeneous Equations
			Worked Examples
		3.4.6 Type 4: Solution of Homogeneous System of Equation Containing Arbitrary Constants
			Worked Examples
			Exercise 3.3
			Answers to Exercise 3.3
	3.5 Matrix Inverse by Gauss–Jordan Method
		Worked Examples
		Exercise 3.4
		Answers to Exercise 3.4
	3.6 Direct Methods
		3.6.1 Matrix Inversion Method
			Worked Examples
		3.6.2 Gauss Elimination Method
			Worked Examples
		3.6.3 Jordan Modification to Gauss Method
			Worked Examples
		3.6.4 Triangularization (Triangular Factorization) Method
			Worked Examples
		3.6.5 Triangularization of Symmetric Matrix
			Worked Examples
		3.6.6 Crout’s Method
			Worked Examples
	3.7 Iterative Methods for Linear Systems
		3.7.1 Jacobi Iteration Method
		3.7.2 Gauss–Seidel Method
			Worked Examples
		3.7.3 Convergence of Iteration Method
			Exercise 3.86
			Answers to Exercise 3.88
4. Eigen Values – Eigen Vectors and Quadratic Forms
	4.1 Eigen Values and Eigen Vectors
		4.1.0 Introduction
		4.1.1 Vector
			Worked Examples
		4.1.2 Eigen Values and Eigen Vectors
		4.1.3 Properties of Eigen Vectors
			Worked Examples
		4.1.4 Properties of Eigen Values
			Worked Examples
			Exercise 4.1
			Answers to Exercise 4.1
		4.1.5 Cayley-Hamilton Theorem
			Worked Examples
			Exercise 4.2
			Answers to Exercise 4.2
	4.2 Similarity Transformation and Orthogonal Transformation
		4.2.1 Similar Matrices
		4.2.2 Diagonalisation of a Square Matrix
		4.2.3 Computation of the Powers of a Square Matrix
		4.2.4 Orthogonal Matrix
		4.2.5 Properties of Orthogonal Matrix
		4.2.6 Symmetric Matrix
		4.2.7 Properties of Symmetric Matrices
		4.2.8 Diagonalisation by Orthogonal Transformation or Orthogonal Reduction
			Worked Examples
	4.3 Real Quadratic Form. Reduction to Canonical Form
		Worked Examples
		Exercise 4.3
		Answers to Exercise 4.3
5. Partial Differentiation and Partial Differential Equations
	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 dz/dx, dz/dy
		5.2.2 Partial Derivatives of Higher Order
		5.2.3 Homogeneous Functions and Euler’s Theorem
			Worked Examples
			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 Introduction of Partial Differential Equations
	5.7 Order and Degree of Partial Differential Equations
	5.8 Linear and Non-linear Partial Differential Equations
	5.9 Formation of Partial Differential Equations
		Worked Examples
		Exercise 5.6
		Answers to Exercise 5.6
	5.10 Solutions of Partial Differential Equations
		5.10.1 Procedure to Find General Integral and Singular Integral for a First Order Partial Differential Equation
			Worked Examples
			Exercise 5.7
			Answers to Exercise 5.7
		5.10.2 First Order Non-linear Partial Differential Equation of Standard Types
			Worked Examples
			Worked Examples
			Exercise 5.8
			Answers to Exercise 5.8
			Worked Examples
			Worked Examples
			Worked Examples
			Worked Examples
		5.10.3 Equations Reducible to Standard Forms
			Worked Examples
			Exercise 5.9
			Answers to Exercise 5.9
	5.11 Lagrange’s Linear Equation
		Worked Examples
		Exercise 5.10
		Answers to Exercise 5.10
Appendix A Higher Order Partial Differential Equations
	A.1 Homogeneous Linear Partial Differential Equations of the Second and Higher Order with Constant Coefficients
		A.1.1 Working Procedure to Find Complementary Function
		A.1.2 Working Procedure to Find Particular Integral
			Worked Examples
			Exercise A.1
			Answers to Exercise A.1
	A.2 Non-homogeneous Linear Partial Differential Equations of the Second and Higher Order with Constant Coefficients
		Worked Examples
		Exercise A.2
		Answers to Exercise A.2
Index