Math 55 - Calculus III
Course Description
The goal of this course is to extend these ideas to three dimensions and to other coordinate systems. Topics for this course include vector
functions and their derivatives and integrals; multivariate functions and their limits and continuity; partial derivatives;
directional derivatives; extrema of multivariate functions; multiple integrals.
After completing this course, you will broaden and deepen your knowledge and perspective of calculus and its
applications. The topics you will learn will enable you to solve real-world problems where more than two quantities
change over time. This course aims to further sharpen your analytical and computational skills as you solve these
problems with some of the advanced methods in calculus.
Course Learning Outcomes
After completion of the course, the student should be able to:
- Discuss the underlying concepts in the calculus of multivariable functions and vector functions;
- Explain the main results in multivariable calculus and their implication and applications;
- Perform various differentiation and integration techniques on multivariable functions and vector
functions;
- Construct mathematical models for various application areas; and
- Engage in problem solving using multivariable calculus concepts, models, and tools learned.
Course Outline
UNIT 1. Vector Functions
- Vector Functions and Space Curves
- Derivatives and Integrals of Vector Functions
- Arc Length and Curvature
- Motion in Space: Velocity and Acceleration
UNIT 2. Partial Derivatives
- Functions of Several Variables
- Limits and Continuity
- Partial Derivatives
- Tangent Planes and Linear Approximations
- The Chain Rule
- Directional Derivatives and the Gradient Vector
- Extrema of Functions of Several Variables
- Lagrange Multipliers
UNIT 3. Multiple Integrals
- Double Integrals Over Rectangles
- Iterated Integrals
- Double Integrals Over General Regions
- Double Integrals in Polar Coordinates
- Applications of Double Integrals
- Triple Integrals
- Triple Integrals in Cylindrical Coordinates
- Triple Integrals in Spherical Coordinates
- Change in Variables in Multiple Integrals
UNIT 4. Vector Calculus
- Vector Fields
- Curl and Divergence
- Line Integrals
- The Fundamental Theorem for Line Integrals
- Parametric Surfaces and Their Areas
- Surface Integral
- Green’s Theorem
- Stoke’s Theorem
- Divergence Theorem