I was given the following problems as practice, and I've sold all but one. However, I am not sure that my answers are correct.
Calculate $(\partial r/\partial x)_y$ , $(\partial r/\partial y)_x$ , $(\partial θ/\partial x)_y$ ,$(\partial y/\partial x)_r$, $(\partial r/\partial \theta)_x$
$x=r\cos(\theta)$, $y=r\sin(\theta)$
I have that
$(\partial r/\partial x)_y=\cos(\theta)$
$(\partial r/\partial y)_x=\sin(\theta)$
$(\partial y/\partial x)_r=-\cot(\theta)$
$(\partial \theta/\partial x)y=-\sin(\theta)/r$
Would $(\partial r/\partial \theta)_x$ then be equal to $r\tan\theta$?