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If $z = f(x, y)$ and we know that $f$ has second derivative and that $y = r\cos\theta$, $x = r\sin\theta$ then calculate: $$\frac{∂^2z}{∂r^2}$$

I myself have basically written that: $$∂\frac{\frac{∂z}{∂x}cos(θ)+\frac{∂z}{∂y}sin(θ)}{∂r}$$ Don't know what do next.

Nosrati
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ATheCoder
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  • Is your statement correct? Maybe $x=r\cos (\theta)$. – Abelois Apr 11 '17 at 08:40
  • It seems the increment theorem may be of use here. Alternatively, see: https://en.wikipedia.org/wiki/Polar_coordinate_system#Differential_calculus. – Bilbottom Apr 11 '17 at 09:19
  • @Abelois Yes it is correct, I double checked – ATheCoder Apr 11 '17 at 09:59
  • Also see http://math.stackexchange.com/questions/432955/how-do-we-take-second-order-of-total-differential and http://math.stackexchange.com/questions/1391991/second-order-total-differential-definition-unclear for total second derivatives. – Bilbottom Apr 11 '17 at 10:18

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