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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.

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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.

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The question itself is straightforward. I think it hints at something more. – isomorphismes Oct 4 '13 at 21:45
@isomorphismes: No more than any other elementary manipulation hints at something more. – Brian M. Scott Oct 4 '13 at 21:47
O.K. That's your opinion. I'd like to hear an answer from someone who agrees there is something more interesting underlying this standard exercise. – isomorphismes Oct 4 '13 at 21:50
@isomorphismes: There simply isn’t anything more interesting underlying the substitution itself. – Brian M. Scott Oct 4 '13 at 21:52
Maybe if I put the question this way it will sound more interesting. If you agree I'll try to work it into the question body. """Given a function $f$ with any domain and codomain as well as functions $g,h,i,j,k, \ldots$, give a general account of when free parameters of $f$ can be reduced/eliminated using relations given by $g,h,i,j,k,\ldots$.""" – isomorphismes Oct 4 '13 at 21:55

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