# How to integrate by parts when del operator is involved?

i'm trying to integrate this:

$$W=\frac{ε}{2}\int{\vec{∇}\cdot\vec{E})Vdτ}$$

where ε is a constant, E= -∇V, τ is a volume element

how do i end up with the following via integration by parts?

$$W=\frac{ε}{2}[-\int{\vec{E}\cdot(\vec{∇}V)dτ}+\oint{V\vec{E}\cdot d\vec{a}}$$]

where the vector a is an area element

thanks

Simon Bridge
Homework Helper
I know the formula for integration by parts, i don't know what to do with the del operator.

ie, I don't know what to make my 'u' and 'dv'.

lurflurf
Homework Helper
^If you know the formula why are you asking?
There are some variations the one you want is

$$\int_{\mathcal{V}} \! k \, (\nabla \cdot \mathbf{A}) \, \mathrm{d}\mathcal{V}=\oint_{\partial \mathcal{V}} \! k \, \mathbf{A} \, \mathrm{d}\mathcal{S}-\int_{\mathcal{V}} \! \mathbf{A} \cdot (\nabla k) \, \mathrm{d}\mathcal{V} \\ \text{which is like} \\ \int u \, \mathrm{d}v=u \, v-\int v \, \mathrm{d}u \\ \text{with} \\u=k \\v=\mathbf{A}$$

Simon Bridge