MATH-UA 123 Multivariable and Vector Calculus

Term: Fall 2022
Instructor: Dr. Daniel Stein
Level: Undergraduate

Topics

Functions of several variables. Vectors in the plane and space. Partial derivatives with applications, especially Lagrange multipliers. Double and triple integrals. Spherical and cylindrical coordinates. Surface and line integrals. Divergence, gradient, and curl. Theorem of Gauss and Stokes.

Description

Calculus III 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.

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

• the 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 different coordinate systems,
• vector fields, their derivatives (divergence and curl) and their integrals (line and surface integrals), and
• the 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,
• learn about and compute limits in higher dimensions,
• learn about and compute derivatives in higher dimensions (partial, directional, total, gradient, divergence, curl, etc),
• learn about and compute integrals in higher dimensions (area, volume, path, surface, flux, etc), and
• communicate mathematically, including understanding, making, and critiquing mathematical arguments.