This specific post is in response to a query about a derivation of what’s commonly known as Green’s Theorem that I had my MVC class work through, but the general ideas are applicable to a very wide range of ideas/theorems/developments in math and science–and even other areas.
Laura had a question about the “Changing Dimensions” lab that it might be helpful to send out to the rest of you. Somewhat paraphrase, she said, “I get how it works, but why did anyone ever start to think with something like the partial derivative of L with respect to y?”
My response, somewhat abbreviated, was that even great mathematicians almost never start with something like that. They usually have some idea of what the outcome will be, and someone (maybe Mr. Green) thought that, “Hmmm. I bet what happens on the boundary has to be the same as the sum of all the infinitesimal changes inside…”
And he might have then thought, “what happens on the boundary” is often a line integral around the circumference of a region.” And, “infinitesimal changes inside often means the derivative of something.” And, “the sum of infinitesimal changes sounds like the integral of a derivative to me.”
And then, the question is “which line intgral?” and “which derivative?”
Well, for divergence, you want “amount of stuff moving across a boundary,” and an amount is a scalar, for that’s why we wanted a scalar derivative for divergence. On the other hand, if we’re talking about movement, rather than amount, we want a vector derivative. Hmmm. And if the stuff can’t move across a boundary but has to stay inside, we probably want the derivative of one component of the vector field with respect to the other. Because when we wanted to know how much stuff was moving across a boundary, we did the derivative of the x-component with respect to x –change in a component with respect to its own direction and then the same thing for all 3 directions. But if we want movement around within a boundary, it might make sense to look at the change in a component with respect to a different variable. That insight caused someone, again perhaps Mr. Green, to think about the various ways you can take derivatives of components with respect to the other axial variables, which might eventually lead him (or you) to mess around with and other such “cross-component derivatives.”
And so, what looks like either brilliant insight or lucky pulling it out of his butt was probably actually neither. It probably started with an idea of what might work and then formalizing a statement of that, and then working backwards to see how in the hell such a result might have come about.
In spite of how math is usually taught, the above is a much better schematic of what usually actually happened before it got “prettied up” for presentation to an audience—or for publication.