A quick word on functions that I have noticed often confuses students.

A function is (roughly) a rule for associating objects in one set with objects in another. The calculus sequence focuses almost exclusively on *real-valued functions on the real line*. That is to say, functions that eat a real number, and give back a real number (one writes ). In a more general world, we might like to study functions (where *m* and *n* might not be 1). Indeed, some common courses will study functions where either *m* or *n* is not equal to 1.

In *multivariable* calculus, one often moves to functions that eat a vector and give back a real number. A typical starting place might be functions , interpreted as a surface, later looking at functions , which might be thought of as, say, temperature at a point in space. In reality, once the student understands the ideas behind partial differentiation and integration over a region, the only to not proceed immediately to functions is concreteness. Indeed, I would argue that students who take linear algebra before calculus would benefit from this approach. Note that the *range* of the function remains one-dimensional, which allows us to continue talking intuitively about those calculus stalwarts, rates of change and area under a graph.

In a differential geometry class (and maybe the last few weeks of a calculus course), we also deal with functions whose *range* has a higher dimension. The function might then be called a *vector-valued* function, or *parametrized* line/surface/hypersurface/region etc. That is to say, a function could be interpreted as a line in the plane, and would write , so that at any moment in time, the curve is at some point in the plane. Similarly, a function could be construed as a parametrization of a surface, and we would write . Discussing proper notions of “derivative” and “integral” for these functions is a topic for another time, but it is worth it to realize that this might not be entirely straightforward.

A very general setting for theorems in calculus is for functions , which, notationally, we can write

I’m sure I will discuss calculus in this context in the future. I guess the point for now is, as usual, is to mind your *m*‘s and *n*‘s.

Also, happy New Year!