Homework Help: Integrating until symmetric bilinear form

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1. Nov 25, 2014

TheFerruccio

1. The problem statement, all variables and given/known data
I am looking for some quick methods to integrate while leaving each step in its vector form without drilling down into component-wise integration, and I am wondering whether it is possible here.

Suppose I have a square domain over which I am integrating two functions w and v. Using integration by parts, what steps can I use to ensure that I end up with a symmetric blinear form + boundary terms?

2. Relevant equations

$\exists$ a region $\Omega$ enclosed by $\Gamma$ and on $\Omega$ $$\int_\Omega{Tv\nabla^2 w d\Omega}=\int_\Omega{v p(x,y) d\Omega}$$

3. The attempt at a solution
Using integration by parts, whereby I feel as if I am completely guessing here:
$$T\left[v\nabla w\right]_\Gamma-T\int_\Omega{\nabla v \cdot \nabla w dV} = \int_\Omega{v p(x,y) d\Omega}$$

Is this the correct symmetric bilinear form that I am looking for? How would that simplified boundary term expression be evaluated explicitly?

2. Nov 30, 2014