- #1
draco193
- 7
- 0
Homework Statement
Use Divergence theorem to determine an alternate formula for [tex]\int\int u \nabla^2 u dx dy dz[/tex] Then use this to prove laplaces equation [tex]\nabla^2 u = 0[/tex] is unique. u is given on the boundary.
Homework Equations
[tex]u \nabla^2 u = \nabla * (u \nabla u) -(\nabla u)^2[/tex]
The Attempt at a Solution
Using Divergence theorem, I get that the new equation should be [tex]\oint (u \nabla u) *n -\oint (\nabla u)^2[/tex] where n is the normal vector.
I wanted to make sure that I had applied this correctly before moving onto the next part.
My plan for the next part would then be to say that u=v-w, where v and w are arbitrary vectors, and show that v =w to show uniqueness of u.