Multivariable Calculus

A function is a set of instructions involving two sets (called the domain and codomain). A function assigns to each element of the domain \(D\) exactly one element in the codomain \(R\text{.}\) We'll often refer to the codomain \(R\) as the target space. We'll write

when we want to remind ourselves of the domain and target space. In this class, we will study what happens when the domain and target space are subsets of \({\mathbb{R}}^n\) (Euclidean \(n\)-space). In particular, we will study functions of the form

when \(m\) and \(n\) are 3 or less. The value of \(n\) is the dimension of the input vector (or number of inputs). The number \(m\) is the dimension of the output vector (or number of outputs). Our goal is to understand uses for each type of function, and be able to construct graphs to represent the function.

We will focus most of our time this semester on two- and three-dimensional problems. However, many problems in the real world require a higher number of dimensions. When you hear the word “dimension”, it does not always represent a physical dimension, such as length, width, or height. If a quantity depends on 30 different measurements, then the problem involves 30 dimensions. As a quick illustration, the formula for the distance between two points depends on 6 numbers, so distance is really a 6-dimensional problem. As another example, if a piece of equipment has a color, temperature, age, and cost, we can think of that piece of equipment being represented by a point in four-dimensional space (where the coordinate axes represent color, temperature, age, and cost).

The next several sections will explore different sorts of functions with a variety of \(n\) and \(m\) combinations. We will start with these same questions as we work to understand the functions. Later on in the semester, you can use these questions to help understand what sort of function you are studying.