In elementary college-level calculus courses, I've given students a problem which reduces to this:
Given $f(p,q)$ and a relation $p=g(q)$ use substitution to derive $\mathfrak{f}(p)$ then proceed with optimisation as usual.
There is definitely something more general going on here, but what is it? This problem glosses over
- when the substitutions can be done
- how to handle multiple substitutions
and it feels like a "peek" into issues that are probably quite large if dealt with in their entirety.
My guesses are "invariant theory" or something about invertibility or ¿lambda calculus? but I'm sure someone here knows better.
Edit: Reading this question again many months later, I can see why it wasn't very popular. I didn't do a very good job of describing the "more" I intuitively see "behind" the problem. Apologies for not asking a clear question.
It’s just a straightforward substitution: since $p$ is a function of $q$, $f$ is also just a function of $q$; if we denote this function by $\mathfrak{f}$, it’s given by $\mathfrak{f}(q)=f\big(g(q),q\big)$. Of course you can change the name of the independent variable and write it as $\mathfrak{f}(p)=f\big(g(p),p\big)$, or as $\mathfrak{f}(x)=f\big(g(x),x\big)$, etc. Nothing is being glossed over.