
kinetic energy PDE
How can I show that the kinetic energy for an incompressible, irrotational fluid with no sources or sinks is equal to
(delta/2)*(double integral of u*(du/dn))dS where delta is the density, u is the velocity potential, and S is a closed surface bounding the volume V. Assume that the density is constant. du/dn is a partial derivative
I know that kinetic energy = (1/2) triple integral((v^2*delta)dt).
I'm sorry for the typing; can anyone help me? I'm a little lost.

Since the fluid is incompressible without singularities, we have $\displaystyle p=const$, and since it is irrational, we have $\displaystyle v=\nabla u$ for the velocity $\displaystyle v$. Now, apply Gauss's theorem on the triple integral representing the kinetic energy.