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Question Number 164121 by mkam last updated on 14/Jan/22

$$ifw=f(x,y)andx=rcos\theta ,y=rsin\theta$$$$$$$$thenprovethat{w}_{rr}+w_{\theta \theta }=0?$$

Commented by mkam last updated on 14/Jan/22

$$????$$

Commented by mkam last updated on 15/Jan/22

$$???$$

Commented by mr W last updated on 15/Jan/22

$$youcantprove!$$$${it}^{\prime }sfalse!$$

Answered by mr W last updated on 15/Jan/22

$$w_r=f_x\cos \theta +f_y\sin \theta$$$$w_{rr}=f_{xx}{\cos }^2\theta +f_{xy}\cos \theta \sin \theta +f_{yx}\sin \theta \cos \theta +f_{yy}{\sin }^2\theta$$$$w_{\theta }=-f_{x}r\sin \theta +f_{y}r\cos \theta$$$$w_{\theta \theta }=-f_{x}r\cos \theta -r\sin \theta (-f_{xx}r\sin \theta +f_{xy}r\cos \theta )-f_{y}r\sin \theta +r\cos \theta (-f_{xy}r\sin \theta +f_{yy}r\cos \theta )$$$$w_{\theta \theta }=-f_{x}r\cos \theta +f_{xx}{r}^2{\sin }^2\theta -f_{xy}{r}^2\sin \theta \cos \theta -f_{y}r\sin \theta -f_{xy}{r}^2\sin \theta \cos \theta +f_{yy}{r}^2{\cos }^2\theta$$$$w_{\theta \theta }=-f_{x}r\cos \theta -f_{y}r\sin \theta +f_{xx}{r}^2{\sin }^2\theta +f_{yy}{r}^2{\cos }^2\theta -2f_{xy}{r}^2\sin \theta \cos \theta$$$$w_{rr}+w_{\theta \theta }\ne 0$$

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