If-V-log-x-2-y-2-then-V-xx-V-yy-

Question Number 295 by defg last updated on 25/Jan/15

If V = \log (x^{2} + y^{2}) then V_{x x} + V_{y y} = ?

Answered by 123456 last updated on 19/Dec/14

V_{x} = \frac{2 x}{x^{2} + y^{2}}

V_{x x} = \frac{2 (x^{2} + y^{2}) - 2 x \cdot 2 x}{{(x^{2} + y^{2})}^{2}} = \frac{- 2 x^{2} + 2 y^{2}}{x^{4} + 2 x^{2} y^{2} + y^{4}}

V_{y} = \frac{2 y}{x^{2} + y^{2}}

V_{y y} = \frac{2 (x^{2} + y^{2}) - 2 y \cdot 2 y}{{(x^{2} + y^{2})}^{2}} = \frac{2 x^{2} - 2 y^{2}}{x^{4} + 2 x^{2} y^{2} + y^{4}}

V_{x x} + V_{y y} = \frac{- 2 x^{2} + 2 y^{2}}{x^{4} + 2 x^{2} y^{2} + y^{4}} + \frac{2 x^{2} - 2 y^{2}}{x^{4} + 2 x^{2} y^{2} + y^{4}} = 0

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