let-f-x-y-ln-x-2-y-2-calculate-2-f-x-2-x-y-2-f-y-2-

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

l e t f (x, y) = l n (\sqrt{x^{2} + y^{2}})

c a l c u l a t e \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

w e h a v e f (x, y) = \frac{1}{2} l n (x^{2} + y^{2}) \Rightarrow

\frac{\partial f}{\partial x} (x, y) = \frac{x}{x^{2} + y^{2}} \Rightarrow \frac{\partial}{\partial x} (\frac{\partial f}{\partial x} (x, y)) = \frac{x^{2} + y^{2} - 2 x^{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}} f i s s y m e t r i c s o

\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