In a thermodynamics question, I was recently perplexed slightly by some partial derivative questions, both on notation and on physical meaning.(adsbygoogle = window.adsbygoogle || []).push({});

I believe my questions are best posed as examples. Suppose we have an equation, [itex](\frac{\partial x(t)}{\partial t}) = \frac{1}{y}[/itex], where y is a function, not necessarily of t.

When we solve for y, does this become [itex]y = (\frac{\partial x(t)}{\partial t})^{-1}[/itex]?

In general, does [itex](\frac{\partial x(t)}{\partial t})^{-1} \neq \frac{\partial t}{\partial x(t)}[/itex]? I was under the impression that they are, in general, not equal.

Suppose we had a function that is dependent on both position in space (x,y,z) and time (t). I shall denote it f(x,y,z,t). If we take [itex]g = \frac{\partial f}{\partial t}[/itex], the others are said to be held constant. Can g be seen as the change in f with respect to time at a specific point, or is it simply talking about the average change in f over an infinitesimal change in time? I thought that the former of the two (change at a point) was true, but now I'm not sure.

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# Partial Derivatives Questions

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