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Find $\large\frac{\partial w}{\partial r}$ and $\large\frac{\partial w}{\partial \theta}$ if $\;w=\log(x^{2}+y^{2})$where $\;x=r\cos\theta,y=r\sin\theta$

This question has multiple parts. Therefore each part has been answered as a separate question on Clay6.com
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Toolbox:
  • Chain rule for partial derivatives:
  • If $w=f(u,v),u=g(x,y),v=u(x,y)$ then,
  • $\large\frac{\partial w}{\partial x}=\frac{\partial w}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial w}{\partial v}\frac{\partial v}{\partial x},\frac{\partial w}{\partial y}=\frac{\partial w}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial w}{\partial v}\frac{\partial v}{\partial y}$
$w=\log (x^2+y^2),\; x=r \cos \theta,\;y=r \sin \theta$
Step 1:
$\large\frac{\partial w}{\partial r}=\frac{\partial w}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial r}$
$\qquad=\large\frac{2x}{x^2+y^2} $$\cos \theta +\large\frac{2y}{x^2+y^2} $$\sin \theta$
$\qquad=\large\frac{2 r \cos \theta.\cos \theta}{r^2 \cos ^2 \theta +r^2 \sin ^2 \theta}+\frac{2 r \sin \theta.\sin \theta}{r^2 \cos ^2 \theta +r^2 \sin ^2 \theta}$
$\qquad=\large\frac{2 r (\cos ^2\theta+ \sin ^2 \theta)}{r^2(\cos ^2 \theta+\sin ^2 \theta)}=\frac{2}{r}$
Step 2:
$\large\frac{\partial w}{\partial \theta}=\frac{\partial w}{\partial x}\frac{\partial x}{\partial \theta}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial \theta}$
$\qquad=\large\frac{2x}{x^2+y^2} $$(-r \sin \theta) +\large\frac{2y}{x^2+y^2} $$r \cos \theta$
$\qquad=\large\frac{-2 r \cos \theta.r \sin \theta}{r^2 \cos ^2 \theta +r^2 \sin ^2 \theta}+\frac{2 r \sin \theta.r \cos \theta}{r^2 \cos ^2 \theta +r^2 \sin ^2 \theta}$$=0$
answered Aug 13, 2013 by meena.p
 

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