How is it 'algebraically' justified that, for $f(t)=(x(t),y(t))$ $$\frac{\partial f}{\partial t} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial t}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial t}$$ when otherwise it's ok to abuse notation as in cases with variable substitution?
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I don't really know what you mean by "when otherwise ..." but the short answer to the question is "the chain rule in several variables" (which is proved in every textbook on the subject, and you can surely find many proofs on the web as well). – Hans Lundmark Jun 15 '15 at 21:44
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@HansLundmark Basically when we have a function $y(x)$ and we set $u=g(x)$ so that $dx = \frac{du}{g'(x)}$. Then we are abusing notation in the case of variable substitution since $d/dx$ is an operator not a factor. I know that above is a special case of the chain rule in multivariable calc., but really what concerns me is when is it justified to abuse notation and when is it not. – Lozansky Jun 15 '15 at 22:07
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When we do substitution in a (single-variable) integral, we are using the single-variable chain rule. Similar questions have been asked over and over again on this site. Try searching for "differential", "chain rule", etc. (and sort the results by the number of votes, to find the most relevant questions). Some suggestions: http://math.stackexchange.com/questions/21199/is-frac-textrmdy-textrmdx-not-a-ratio, http://math.stackexchange.com/questions/1252405/is-it-mathematically-valid-to-separate-variables-in-a-differential-equation – Hans Lundmark Jun 16 '15 at 07:13