# Harmonic Functions (Partial Derivatives)

• Oct 26th 2010, 10:16 PM
ZeroVector
Harmonic Functions (Partial Derivatives)
The Laplace operator \$\displaystyle \Delta\$ is defined by \$\displaystyle \Delta f = f_{xx} + f_{yy}\$. A function \$\displaystyle u(x,y)\$ satisfying the Laplace equation \$\displaystyle \Delta u = 0\$ is called harmonic.
Show that \$\displaystyle u(x,y) = x\$ is harmonic.

First thing, though: What does it mean for a function to be harmonic? The explanation they give is confusing to me. Can you show me how to do this so I can understand how to do my other problems related to this?
• Oct 27th 2010, 02:02 AM
drumist
Suppose \$\displaystyle f(x,y) = 2x^2 - 2y^2\$. (This is just an example.)

Then \$\displaystyle f_{xx} = 4\$ and \$\displaystyle f_{yy} = -4\$, which means \$\displaystyle \Delta f = f_{xx} + f_{yy} = 4 - 4 = 0\$. Therefore, \$\displaystyle f(x,y)\$ is harmonic. If this value had been nonzero, then \$\displaystyle f(x,y)\$ would not be harmonic.

Does this make sense?
• Oct 27th 2010, 04:45 AM
HallsofIvy
You are told that \$\displaystyle \nabla f= f_{xx}+ f_{yy}\$ and you were told that u being "harmonic" means that u satisfies \$\displaystyle \nabla u= 0\$. Putting those together, u(x,y) is "harmonic" if and only if \$\displaystyle u_{xx}+ u_{yy}= 0\$. Is that true for u(x,y)= x?