## COURSE OBJECTIVES

This course provides an introduction to the theory, solution, and application of ordinary differential equations. Topics discussed in the course include methods of solving first-order differential equations, existence and uniqueness theorems, second-order linear equations, power series solutions, higher-order linear equations, and their applications.

### COURSE LEARNING OUTCOMES (CLO)

CLO: 1. Explain the ideas of rate of change and derivatives using the concept of limits and continuity. (C2)
CLO: 2. Apply the derivatives for solving different problems arising in engineering sciences (C3).
CLO: 3. Select the techniques of integration for solving problems in integral calculus (C4).
CLO: 4. Apply the concept of vector calculus and analytical geometry in multiple dimensions (C3).

## COURSE CONTENTS

1. Limits and Continuity– Four Lectures
• Introduction to Limits
• Rates of Change and Limits
• One-Sided Limits, Infinite Limits
• Continuity, Continuity at a Point, Continuity on an interval
2. Differentiation– Six Lectures
• Definition and Examples
• Relation Between Differentiability and Continuity
• Derivative as slope, as rate of change (graphical representation).
• The Chain Rule
• Applications of Ordinary Derivatives
3. Integration– Five Lectures
• Indefinite Integrals
• Different Techniques for Integration
• Definite Integrals
• Riemann Sum, Fundamental Theorem of Calculus
• Area Under the Graph of a Nonnegative Function
• Improper Integrals
4. Transcendental Functions– Five Lectures
• Inverse functions
• Logarithmic and Exponential Functions
• Inverse Trigonometric Function
• Hyperbolic Functions and Inverse Hyperbolic Functions
• More Techniques of Integration
5. Analytical Geometry– Six Lectures
• Three Dimensional Geometry
• Vectors in Spaces
• Vector Calculus
• Directional Derivatives
• Divergence, Curl of a Vector Field
• Multivariable Functions
• Partial Derivatives
6. Analytical Geometry– Six Lectures
• Conic Sections
• Parameterizations of Plane Curves
• Vectors in Plane, Vectors in space
• Dot Products, Cross Products
• Lines and Planes in Space
• Spherical, Polar and Cylindrical Coordinates.
• Vector-Valued Functions and Space Curves
• Arc-Length and Tangent Vector
• Curvature, Torsion and TNB Frame
• Fubini’s Theorem for Calculating Double Integrals
• Areas Moments and Centers of Mass
• Triple Integrals, Volume of a Region in Space