How to integrate by parts when del operator is involved?

[iScience]

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]

Looks like:
http://en.wikipedia.org/wiki/Integration_by_parts#Higher_dimensions

 

[iScience]

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]

^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]

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

I'm sorry, but you did ask:

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

... and the answer to that question is to follow the formula for integration by parts. Now you say you know the formula?

Please show us your best attempt using your knowledge of the formula, so that we may better understand the question.