A note on functions

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 f:\mathbb{R} \to \mathbb{R}).  In a more general world, we might like to study functions f: \mathbb{R}^m \to \mathbb{R}^n (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 f:\mathbb{R}^2 \to \mathbb{R}, interpreted as a surface, later looking at functions T:\mathbb{R}^3 \to \mathbb{R}, 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 f:\mathbb{R}^n \to \mathbb{R} 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 \alpha: \mathbb{R} \to \mathbb{R}^2 could be interpreted as a line in the plane, and would write \alpha(t) = (x(t),y(t)), so that at any moment in time, the curve is at some point in the plane.  Similarly, a function f: \mathbb{R}^2 \to \mathbb{R}^3 could be construed as a parametrization of a surface, and we would write f(u,v) = (x(u,v),y(u,v),z(u,v)).  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 f: \mathbb{R}^m \to \mathbb{R}^n, which, notationally, we can write

\mathbf{f}(x_1,\ldots,x_m) = (f^1(x_1,\ldots,x_m),\ldots,f^n(x_1,\ldots,x_m)).

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!

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