In a previous post I discussed how we need ways to think about functions that are not curves on an x-y-plane. Well I have a seasonally-appropriate one for you: the Advent Calendar.
The advent calendar I have in mind is the kind where there is a little cardboard door for each day in December up to Christmas, and behind each door is a little chocolate. (Yes I know it might just have a picture, but seriously, the chocolate ones are better, right?)
Most of the time when we’re not drawing graphs, we talk about a function as a sort of machine, which takes some sort of input and produces some sort of output. This is a very dynamic view of function which I like very much. I imagine putting an object (usually a number) in a funnel at the top and the machine churns and whirs and gurgles until a new object (usually another number) shoots out of a chute at the bottom.
But a function doesn’t necessarily have this time element. The set-theory definition of function is simply a correspondence between one set and another, so that every object in a domain has associated with it one object in a codomain. (The domain is the set of things we usually call “inputs” and the codomain is the set of things from which we choose the “outputs”.) In this sense the “output” is there all the time whether we calculate it or not.
This is where the advent calendar comes in. The chocolate is there whether you open the door or not. Opening the door to see for yourself what shape the chocolate is corresponds to what we do when we calculate the value of the function. But fundamentally all the numbers in the domain have a value for the function before you calculate it, just like all the unopened doors have a chocolate.
I find this particular picture works well for vector calculus, where every point in a plane or in space has a function value, which may be a number (in the case of a “scalar field”) or a vector (in the case of a “vector field”). In the vector field case, the vector you find doesn’t really interact with the ordinary points, and indeed the vector “output” at one point doesn’t really interact with the ones at the other points. It’s almost as if at every point there is a little room where the vector lives all by itself. All we need is a little door to let us into this little room…