# Directional derivative and Laplace operator

Let $R$ be a region in the plane satisfying Green's theorem, with boundary $C$. Let $S$ be an open set containing $R$, and let $\phi$ be a scalar field $\phi: S \rightarrow \mathbb{R}$ with continuous first and second partial derivatives.

If $x$ is a point of $C$, let $\mathbf{n}$ be a unit normal, i.e. a unit vector normal to the tangent line at $C$. Denote by $D_{\mathbf{n}} \phi(x)$ the directional derivative of $\phi$ in the direction of $\mathbf{n}$. Recall that $D_{\mathbf{n}} \phi(x)= \nabla \phi (x) \cdot \mathbf{n}$.

a) I showed that $\displaystyle\int_CD_{\mathbf{n}} \phi ds=\int\!\!\! \int (\nabla^2 \phi)dxdy$
b) I want to show now that for a scalar field $\psi: S \rightarrow \mathbb{R}$ with properties same as $\phi$, we have:

$\displaystyle\int_C \psi D_{\mathbf{n}} \phi ds=\int\!\!\! \int (\psi \nabla^2 \phi + \nabla \psi \cdot \nabla \phi)dxdy$.

Thank you.

• Can somebody help me with this ? – amir Apr 28 '13 at 0:18

The same method you used for a) will work. Write this as the flux of $\mathbf F = \psi\nabla\phi$ and take the divergence.