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
- 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
- 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
- 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
- Transcendental Functions– Five Lectures
- Inverse functions
- Logarithmic and Exponential Functions
- Inverse Trigonometric Function
- Hyperbolic Functions and Inverse Hyperbolic Functions
- More Techniques of Integration
- Analytical Geometry– Six Lectures
- Three Dimensional Geometry
- Vectors in Spaces
- Vector Calculus
- Directional Derivatives
- Divergence, Curl of a Vector Field
- Multivariable Functions
- Partial Derivatives
- 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