Verify partial differentiation.

I just want to verify



For Polar coordinates, ##r^2=x^2+y^2## and ##x=r\cos \theta##, ##y=r\sin\theta##



##x(r,\theta)## and## y(r,\theta)## are not independent to each other like in rectangular.



In rectangular coordinates, ##\frac{\partial y}{\partial x}=\frac{dy}{dx}=0##



But in Polar coordinates,

[tex]\frac{\partial r}{\partial x}=\cos\theta,\;\frac{\partial \theta}{\partial x}=-\frac{\sin\theta}{r}[/tex]

[tex]\frac{\partial y(r,\theta)}{\partial x(r,\theta)}=\frac{\partial y(r,\theta)}{\partial r}\frac{\partial r}{\partial x(r,\theta)}+\frac{\partial y(r,\theta)}{\partial \theta}\frac{\partial \theta}{\partial x(r,\theta)}=



(\cos\theta) \frac{\partial y(r,\theta)}{\partial r}-\left(\frac{\sin\theta}{r}\right)\frac{\partial y(r,\theta)}{\partial \theta}[/tex]



Thanks

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