I am trying to undersatnd a proof and I am stuck on the derivative part in this proof. I can't understand why is there still a Fx and Fy in the polar coordinate.
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$\begingroup$ Because you have to use the chain rule. $\endgroup$– user474330Oct 14, 2017 at 23:12
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1$\begingroup$ I'm wondering why the author writes $\partial f/\partial \Omega$ instead of $\partial f/\partial \theta$, and calls polar coordinates in the plane “spherical coordinates” (and writes $cos$ instead of $\cos$)... And I'm also wondering what exactly your question is. Are you asking why there are first derivatives $f_x$ and $f_y$ in the expression for the second derivative? $\endgroup$– Hans LundmarkOct 15, 2017 at 6:53
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The author is describing how partial derivatives work when they say "fixing $\theta$ and letting $r$ vary." The rest is indeed the chain rule $$ \frac{\partial f}{\partial r}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial r}\\ =f_x\cos\theta+f_y\sin \theta $$
answered Oct 14, 2017 at 23:19