3

$\frac{\partial x}{\partial y} \cdot \frac{\partial y}{\partial z} \cdot \frac{\partial z}{\partial x} = -1$ according to the triple product rule
However, it would be 1, if derivatives behaved like fractions.

And in Nonstandard Calculus derivatives do behave like fractions. (it treats derivatives as fractions of actual infinitesimal quantities, rather than d/dx applied to a function)

So, how does Nonstandard Calculus get the correct answer here?
(which it must since it is equivalent to standard calculus)

ions me
  • 419
  • 2
  • 15
  • The wikipedia article https://en.wikipedia.org/wiki/Triple_product_rule has a derivation that may help. It doesn't use limits explicitly, so it might be compatible with non-standard calculus. – Peter Feb 10 '22 at 06:39
  • 3
    I believe that the two $\partial x$'s are different infinitesimals in a nonstandard treatment (and similar for the other variables) as they relate to different increments when different variables are held fixed. It is basically the same issue as the classical setting; the Leibniz notation just invites blurring these lines in both cases. This (or a related simplified version) is probably somewhere in Chapter 11 of the linked Keisler text. – leslie townes Feb 10 '22 at 06:50
  • @leslietownes Is there a simple way to keep track of that? I'm hoping that Nonstandard analysis provides a framework to easily find the answer to such things. Issues like this one are often quoted as the reason why d/dx is not a fraction. Yet so many proofs in physics are cavalier with d/dx "fractions". Occasionally that comes back to bite, as it does here. If the solution here is elegant, then I'd be convinced nonstandard analysis is worth learning – ions me Feb 10 '22 at 16:59

0 Answers0