Harmonic function satisfies Laplace equation and have continuous 1st and 2nd partial derivatives. Laplace equation is [itex]\nabla^2 u=0[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

Using Green's 1st identity:

[tex]\int_{\Omega} v \nabla^2 u \;+\; \nabla u \;\cdot \; \nabla v \; dx\;dy \;=\; \int_{\Gamma} v\frac{\partial u}{\partial n} \; ds [/tex]

[tex] v=1 \;\Rightarrow\; \int_{\Omega} \nabla^2 u \; dx\;dy \;=\; \int_{\Gamma} \frac{\partial u}{\partial n} \; ds = 0 \;\hbox { if } \;u \;\hbox{ is a harmonic function .} [/tex]

Why is it equal zero if u is harmonic function? Why is:

[tex]\int_{\Omega} \nabla^2 u \; dx\;dy =0 \hbox { if } \nabla^2 u =0 [/tex]

Or more basic question:

What is [itex]\int_{\Gamma} 0 dxdy[/itex]? Is it not zero?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Green's identity help.

**Physics Forums | Science Articles, Homework Help, Discussion**