MATH 201 Multivariable Calculus
Overview
MATH 201 Multivariable Calculus is a required course for students in several majors in the Division of Natural and Applied Sciences, and an elective for students in other majors.
Course Description
In MATH 101 Introductory Calculus or in MATH 105 Calculus you have learned about single variable Calculus — the study of functions that depend on a single variable and whose output is a single number. This is the basis for large parts of modern Mathematics and its applications. However, it does not take long to realize that, despite all its intricacies, single variable calculus is not sufficient for describing many aspects of our world: planets and particles move along trajectories in three-dimensional space (or four-dimensional if you take the Theory of General Relativity into account); climate models describe the state of the atmosphere through quantities such as temperature, atmospheric pressure, and wind velocity — all of them functions of position (longitude, latitude, height) and time; wind velocity itself is not a single quantity but a vector — we need to know both how strong the wind is and what is its direction; neural networks, used in machine learning, are trained by minimizing cost functions that depend on millions of variables; in electromagnetism the electric and magnetic fields are described by vectors that depend on the position in three-dimensional space and time; volumes, surfaces, and curves in computer graphics and animation are represented using concepts from multivariable calculus; and, from a purely mathematical point of view, Multivariable Calculus is a subject that opens the door to almost all of modern mathematics — a beautiful topic with close connections to Geometry.
A very useful mathematical concept for working with many variables is vectors. In this course we consider vectors and the basic operations between them. Then we define vector functions that describe a curve in space and the motion of a particle along such a curve. After that, we consider functions that depend on more than one variables (for simplicity, most of the discussion will be for two or three variables but the concepts can be generalized to any number of variables) and how to differentiate them, leading to the notions of partial derivatives, directional derivatives, and the gradient, and we use these to find minima and maxima of such functions. Then we take a look at how to compute integrals for functions that depend on two or three variables (double and triple integrals). We close the course with three fundamental results on integration (Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem) that generalize to more than one dimensions the Fundamental Theorem of Calculus.
Syllabus
Literature
Calculus Volume 3. OpenStax (2016).