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Question Number 36178 by prof Abdo imad last updated on 30/May/18

$$letf(x,y)=ln\left(\sqrt{x^2+y^2}\right)$$$$calculate\frac{{\partial }^{2}f}{\partial x^2}(x,y)+\frac{{\partial }^{2}f}{\partial y^2}$$

Commented by abdo imad last updated on 31/May/18

$$wehavef(x,y)=\frac{1}{2}ln(x^2+y^2)\Rightarrow$$$$\frac{\partial f}{\partial x}(x,y)=\frac{x}{x^2+y^2}\Rightarrow \frac{\partial }{\partial x}\left(\frac{\partial f}{\partial x}(x,y)\right)=\frac{x^2+y^2-2x^2}{{(x^2+y^2)}^2}$$$$\Rightarrow \frac{{\partial }^{2}f}{\partial x^2}(x,y)=\frac{-x^2+y^2}{{(x^2+y^2)}^2}fissymetricso$$$$\frac{{\partial }^{2}f}{\partial x^2}(x,y)=\frac{-y^2+x^2}{(x^2+y^2)}\Rightarrow \frac{{\partial }^{2}f}{\partial x^2}(x,y)+\frac{{\partial }^{2}f}{\partial y^2}(x,y)=0$$

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