So I was looking at this reply to a question that was asked, and I kind of skipped a bunch of multivariable calculus in classes due to a thing and essentially learned most of what I could on my own, but I don't understand why we can just do the thing that I've mentioned in the question statement. Is it just that we're treating the derivatives as fractions or is there a proper reasoning for this? Could someone please explain?
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1See e.g. https://math.stackexchange.com/questions/1169942/why-can-partial-derivatives-be-exchanged – Minus One-Twelfth Oct 25 '21 at 19:16
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Ahhh yesss okay, looking at both of these right now and these seem to be making sense. Thanks guys! – Applesauce44 Oct 25 '21 at 19:24
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From what I understand, the general idea is that if a function is separable for its variables i.e. $$f(x,y)=\sum_i g_i(x)h_i(y)$$ then the order of derivatives is interchangeable since: $$\partial_x\partial_y(g_ih_i)=\partial_xg_i\partial_yh_i=\partial_y\partial_x(g_ih_i)$$ however I am still trying to find a reference for this
Henry Lee
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1Why is a reference needed for this? All the steps are very straightforward. – Vercassivelaunos Oct 25 '21 at 19:24
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Ahh yess I see. So I wanted to know the reasoning behind this, but this is helpful too, thank you! – Applesauce44 Oct 25 '21 at 19:28
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This is actually not always the case. There are various ways of formulating the constraint for this to be the case.
There's an article about it on Wikipedia here. https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives
buddhabrot
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1The link to https://math.stackexchange.com/questions/1169942/why-can-partial-derivatives-be-exchanged actually refers to an answer that also has a blog post link for the same theorem, yet a little less complete than the above.. – buddhabrot Oct 25 '21 at 19:19
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