Calculus III

Course Overview

Calculus III is a comprehensive course designed to cover multivariable and vector calculus. Because the sheer volume of material is so great, the course is designed to take an entire school year and is intended for students with a proven record of accomplishment and interest in mathematics. Students begin the course by covering vector operations such as dot and cross products, as well as sketching common quadric surfaces. These are considered the “mother graphs” of many of the surfaces we use later in the course. Students then move on to vector-valued functions, space curves and derivatives and integrals of vector-valued functions. The course then proceeds to what could be considered the differential calculus of multivariable functions. Students explore the ideas of limit, continuity, partial derivatives, tangent planes, and the chain rule. After learning these fundamentals, students use directional derivatives and gradients to optimize functions. Students learn Lagrange multipliers as a potentially easier and more elegant method for finding maxima and minima. Next, students move on to the integral calculus of multivariable functions. Students learn double and triple integrals, as well as methods such as integrating with spherical, cylindrical, and polar coordinates. Lastly, the surface area of solids is covered.

Students then move on to the concepts of vector calculus. They are introduced to vector fields, line integrals and the fundamental theorem of line integrals. More advanced topics including Green’s theorem, curl and divergence, parametric surfaces, surface integrals, Stokes’ theorem, and the divergence theorem conclude the course.

This course is designed to expose students to the more advanced topics of calculus and provide them with a strong computational and theoretical understanding of content that will be used subsequently in both advanced mathematics and physics courses.