multivar help

[oohaysomeone]

Determine whether the function u = tan^-1(y/x) is a solution of
Laplace?s equation, u[sub:2ww9gh0r]xx[/sub:2ww9gh0r] + u[sub:2ww9gh0r]yy[/sub:2ww9gh0r] = 0

so, my question is how do you find the second partial derivitives required for the answer?

 

[tkhunny]

First find the first derivatives?

What have you tried?

\(\displaystyle \frac{d}{dt}\left(atan(t)\right)\;=\;\frac{1}{t^{2}+1}\)

If you don't recall that derivative, it is quite a bit more difficult.

 

[BigGlenntheHeavy]

\(\displaystyle u \ = \ arctan(\frac{y}{x})\)

\(\displaystyle u_x \ = \ \frac{-y}{x^{2}+y^{2}} \ \ \ u_y \ = \ \frac{x}{x^{2}+y^{2}}\)

\(\displaystyle u_{xx} \ = \ \frac{2xy}{(x^{2}+y^{2})^{2}} \ \ \ u_{yy} \ = \ \frac{-2xy}{(x^{2}+y^{2})^{2}}\)

\(\displaystyle Hence, \ u_{xx}+u_{yy} \ = \ 0\)

 

[oohaysomeone]

thank you. i missed the negative at the beginning which threw me off.