Syllabus

Welcome to Multivariable (Vector) Calculus!

My name is Gary Church and I will be your instructor for this important course. I've been teaching this class for many years and still find new and interesting things about it. I enjoy mathematics and spend time learning new mathematics or applications of math to other disciplines. In particular, topics in this course have significant applications to machine learning and artificial intelligence.

Other than math, I enjoy reading, playing chess and other board games, playing video games and VR games, biking on occasion and taking walks.

Being a fully online, asynchronous class, there are no scheduled class meetings. You will be learning the material by reading the textbook, studying the worked out examples in the book, watching videos and reading the notes of lectures I gave to a synchronous online math 253 class in the Spring 2021 semester. Of course, I will be available to answer any questions you have about the text readings, examples and homework problems through email, office hours on campus or scheduled over Zoom.

In this course we will learn to analyze functions with more than one input (multivariable functions). They will be described using the geometric language of points, curves, surfaces and volumes in multidimensional spaces; Not just one, two or three dimensional space, but n dimensional space, where "n" can be ANY positive integer.

These concepts will be encapsulated in algebraic constructs called vectors within the framework of vector spaces.

For example, consider the distribution of temperatures in the room in which you currently sit. The temperature will vary in different locations within the room and at different times. The location can be specified by three coordinates, x, y and z, measured from some convenient point (origin) within the room. The time requires another "coordinate" t, measured, say, with a clock. At a particular location and time within the room we measure a temperature T. We would say that the temperature T is a function of the four input variables x, y, z and t: T = T(x, y, z, t).

One way to understand this temperature function would be to study its graph. To graph a function we need axes for each input variable and an axis for the function value at a given collection of input variables. A function of one variable requires two axes (its graph is a curve that lies in a plane--a two dimensional space) and a function of two variables requires three axes (its graph is a surface that lies in three dimensional space). We (or computers) can draw pictures of the graphs of functions of one or two variables (curves in the plane or surfaces in space) but our example temperature function is a function of FOUR variables, so requires FIVE axes. Its graph lies in a five dimensional hyper-space and there is no way to draw a picture of it, either by hand or computer.

Although we can't study a picture of the graph of this function, we will develop a collection of mathematical tools (based on geometric vectors) and a language that will enable us to understand its graph, thereby giving us insight into the function itself.

Many of the concepts discussed in class will be illustrated using functions of only one or two variable inputs, since I can illustrate the ideas by actual drawings. The drawings are only a convenience to help understand the ideas; They are not needed to analyze the function.

I think you will find the course interesting but it is a challenging class and you will need to put in the time to really grasp the fundamental concepts.