Calculus 3 is a third-semester calculus course for students who have a good knowledge of differential and integral calculus for functions of a single variable. In this course, we will figure out how to generalize these concepts for functions of two, three, or potentially many variables. Why study multivariable calculus? Simply put, in the real world any given quantity depends on many other ones. By the end of the course, you will understand how to mathematically describe such systems.

Some key topics, roughly in order of their appearance in the course, include:

• Geometry of three-dimensional space and vectors
• Vector functions or space curves, and their calculus
• Functions of several variables, partial derivatives, and gradients
• Multiple integration, including in different coordinate systems
• Vector fields, their derivatives (divergence and curl) and their integrals (line and surface integrals)
• Fundamental theorems of vector calculus (Green’s, Gauss’, and Stokes’)

The material we take up in this course has applications in physics, chemistry, biology, environmental science, astronomy, economics, statistics, and just about everything else. We want you to leave the course not only with computational ability, but with the ability to use these notions in their natural scientific contexts, and with an appreciation of their mathematical beauty and power.

By the end of the course, students will be able to:

• Investigate higher-dimensional geometry using the concept of a vector
• Understand the concept of a function when extended to multiple inputs and outputs
• Compute and use limits in higher dimensions
• Compute and use derivatives in higher dimensions (partial, directional, total, gradient, divergence, curl, etc)
• Compute and use integrals in higher dimensions (area, volume, path, surface, flux, etc)
• Communicate mathematically, including understanding, making, and critiquing mathematical arguments