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!