Topics in Multivariable and Vector Calculus
Compared to the familiar single-variable territory of Calculus I and II, multivariable calculus is a foreign land. Imagine, if you will, that instead of a function taking a single input and producing a single output, we either use one input and get multiple outputs (vector functions) or use several inputs and get one output (multivariable functions). And yes, there are even functions that have several inputs and multiple outputs! In this new realm, we will investigate lines and planes, curves and surfaces, and multidimensional generalizations of these objects, with a focus on those functions that can be visualized in three dimensions. For both vector and multivariable functions, we will address the basic questions of calculus: How do we measure rates of change? How do we find areas and volumes? How can we interpret derivatives and integrals both geometrically and for practical purposes? Fascinatingly, each of these questions has more than one answer. We will examine gradients and directional derivatives, maxima and minima and saddle points, double and triple integrals, integrals taken along curves, and more—as time permits. This seminar is essential for students intending to pursue engineering, physics, mathematics, graduate study in economics, or rocket science and is recommended for students pursuing chemistry or computer science. Prerequisites: Calculus I and Calculus II.