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Question Number 126635 by BHOOPENDRA last updated on 22/Dec/20

Answered by Ar Brandon last updated on 22/Dec/20

$${\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2={\mathrm{r}}^2{\cos }^2\theta {\cos }^2\varphi +{\mathrm{r}}^2{\cos }^2\theta {\sin }^2\varphi +{\mathrm{r}}^2\sin \theta$$$$={\mathrm{r}}^2[{\cos }^2\theta \underset{1}{\underbrace{({\cos }^2\varphi +{\sin }^2\varphi )}}+{\sin }^2\theta ]$$$$={\mathrm{r}}^2[{\cos }^2\theta +{\sin }^2\theta ]={\mathrm{r}}^2$$$$w(x,y,z)=f(x^2+y^2+z^2)=f\left(r^2\right)$$$$\frac{\partial w}{\partial \theta }=\frac{\partial f\left(r^2\right)}{\partial \theta }=0=\frac{\partial w}{\partial \varphi }$$

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