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given that $z=f(x,y)$ and $x =r\cos(\theta)$ and y = $r\sin(\theta)$

$d^2(z)/d(r)^2$ and $d^2(z)/d(r)$

are both second derivatives of the function $z$? I am getting a little confused with all the notations.

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up vote 2 down vote accepted

The notation for the first means $$ \frac{\partial^2 z}{\partial r^2} = \frac{\partial}{\partial r} \left( \frac{\partial z}{\partial r} \right),$$ so you take the partial derivative with respect to $r$ of the partial derivative of $z$ with respect to $r$, hence the "second derivative" of $z$ with respect to $r$.

The notation for the second quantity does not make sense, at least not according to anything I'm familiar with.

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