Engineering Mathematics-I : For RTU (Subject Code: 102)

Cover
Contents
Preface
Roadmap to the Syllabus
Symbols and Basic Formulae
Chapter 1: Curvature
	1.1 Radius of Curvature of Intrinsic Curves
	1.2 Radius of Curvature for Cartesian Curves
	1.3 Radius of Curvature for Parametric Curves
	1.4 Radius of Curvature for Pedal Curves
	1.5 Radius of Curvature for Polar Curves
		1.5.1 Second Method
	1.6 Radius of Curvature at the Origin
		1. Newton’s Method
		2. Method of Expansion
	1.7 Center of Curvature
	1.8 Evolutes and Involutes
	1.9 Equation of the Circle of Curvature
	1.10 Chords of Curvature Parallel to the Coordinate Axes
	1.11 Chord of Curvature in Polar Coordinates
	1.12 Miscellaneous Examples
	Exercises
Chapter 2: Asymptotes and Curve Tracing
	2.1 Determination of Asymptotes when the Equation of the Curve in Cartesian Form is Given
	2.2 The Asymptotes of the General Rational Algebraic Curve
	2.3 Asymptotes Parallel to Coordinate Axes
		(I) Asymptotes Parallel to Y-Axis of a Rational Algebraic Curve
		(II) Asymptotes Parallel to the X-Axis of a Rational Algebraic Curve
	2.4 Working Rule for Finding Asymptotes of Rational Algebraic Curve
	2.5 Intersection of a Curve and its Asymptotes
	2.6 Asymptotes by Expansion
	2.7 Asymptotes of the Polar Curves
	2.8 Circular Asymptotes
	2.9 Concavity, Convexity and Singular Points
	2.10 Curve Tracing (Cartesian Equations)
	2.11 Curve Tracing (Polar Equations)
	2.12 Curve Tracing (Parametric Equations)
	Exercises
Chapter 3: Functions of Several Variables
	3.1 Continuity of a Function of Two Variables
	3.2 Differentiability of a Function of Two Variables
	3.3 The Differential Coefficients
	3.4 Distinction between Derivatives and Differential Coefficients
	3.5 Higher-Order Partial Derivatives
	3.6 Envelopes and Evolutes
	3.7 Homogeneous Functions and Euler’s Theorem
	3.8 Differentiation of Composite Functions
	3.9 Transformation from Cartesian to Polar Coordinates and Vice Versa
	3.10 Taylor’s Theorem for Functions of Several Variables
	3.11 Extreme Values
		Necessary and sufficient conditions for extremevalues
	3.12 Lagrange’s Method of Undetermined Multipliers
	3.13 Jacobians
	3.14 Properties of Jacobians
	3.15 Necessary and Sufficient Conditions for a Jacobian to Vanish
	3.16 Differentiation Under the Integral Sign
	3.17 Approximation of Errors
	3.18 General Formula for Errors
	3.19 Miscellaneous Examples
	Exercises
Chapter 4: Quadrature and Rectification
	4.1 Quadrature
		4.1.1 Area of a Curve Given by the Cartesian Equation
		4.1.2 Area of a Curve Given by Polar Equation
	4.2 Rectification
		4.2.1 Length of a Curve
	Exercises
Chapter 5: Volumes and Surfaces of Solids of Revolution
	5.1 Volume of the Solid of Revolution (Cartesian Equations)
	5.2 Volume of the Solid of Revolution (Parametric Equations)
	5.3 Volume of the Solid of Revolution (Polar Curves)
	5.4 Surface of the Solid of Revolution (Cartesian Equations)
	5.5 Surface of the Solid of Revolution (Parametric Equations)
	5.6 Surface of the Solid of Revolution (Polar Curves)
	Exercises
Chapter 6: Beta and Gamma Functions
	6.1 Beta Function
	6.2 Properties of Beta Function
	6.3 Gamma Function
	6.4 Properties of Gamma Function
	6.5 Relation between Beta and Gamma Functions
	6.6 Dirichlet’s and Liouville’s Theorems
	6.7 Miscellaneous Examples
	Exercises
Chapter 7: Multiple Integrals
	7.1 Double Integrals
	7.2 Properties of a Double Integral
	7.3 Evaluation of Double Integrals (Cartesian Coordinates)
	7.4 Evaluation of Double Integrals (Polar Coordinates)
	7.5 Change of Variables in a Double Integral
	7.6 Change of Order of Integration
	7.7 Area Enclosed by Plane Curves (Cartesian and Polar Coordinates)
	7.8 Volume and Surface Area as Double Integrals
	7.9 Triple Integrals and their Evaluation
	7.10 Change to Spherical Polar Coordinates from Cartesian Coordinates in a Triple Integral
	7.11 Volume as a Triple Integral
	7.12 Miscellaneous Examples
	Exercises
Chapter 8: Ordinary Differential Equations
	8.1 Definitions and Examples
	8.2 Formulation of Differential Equation
	8.3 Solution of Differential Equation
	8.4 Differential Equations of First Order
	8.5 Separable Equations
	8.6 Homogeneous Equations
	8.7 Equations Reducible to Homogeneous Form
	8.8 Linear Differential Equations
	8.9 Equations Reducible to Linear Differential Equations
	8.10 Exact Differential Equation
	8.11 The Solution of Exact Differential Equation
	8.12 Equations Reducible to Exact Equation
	8.13 Applications of First Order and First Degree Equations
		(A) Problems Related to Electric Circuits
		(B) Problems Related to Newton’s Law of Cooling
		(C) Problems Relating to Heat Flow
		(D) Rate Problems
		(E) Falling Body Problems
		(F) Orthogonal Trajectories
	8.14 Linear Differential Equations
	8.15 Solution of Homogeneous Linear Differential Equation with Constant Coefficients
		Case I. Distinct Real Roots
		Case II. Repeated Real Roots
		Case III. Conjugate Complex Roots
	8.16 Complete Solution of Linear Differential Equation with Constant Coefficients
		8.16.1 Standard Cases of Particular Integrals
	8.17 Method of Variation of Parameters to Find Particular Integral
	8.18 Differential Equations with Variable Coefficients
		(A) Method of Solution by Changing Independent Variable
		(B) Method of Solution by Changing the Dependent Variable
		(C) Method of Undetermined Coefficients
		(D) Method of Reduction of Order
		(E) Cauchy–Euler Homogeneous Linear Equation
		(F) Legendre’s Linear Equation
	8.19 Miscellaneous Examples
	Exercises
Solved Question Papers
Index