During the Fall 2022 term, Calculus III is coordinated by Prof. Charmaine Sia (sia@cims.nyu.edu).
Welcome to Calculus III!
Even before the development of calculus, Galileo appreciated that “the book of the universe is written in the language of mathematics.” The four hundred years since have proven this true again and again, and now is no different. All realms of modern knowledge rely on quantitative descriptions of quantities that change with time, with space, and with changes in other quantities. Calculus is the framework for describing and understanding these important problems.
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. Why study multi-variable 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:
- 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.