Gradient and Divergent Identities

[bugatti79]

Homework Statement

I need to show that $\displaystyle\int_\Omega (\nabla G)w dxdy=-\int_\Omega (\nabla w) G dxdy+\int_\Gamma \hat{n} w G ds$ given

$\displaystyle \int_\Omega \nabla F dxdy=\oint_\Gamma \hat{n} F ds$ where $\Omega$ and $\Gamma$ are the domain and boundary respectively. F,G and w are scalar functions...any ideas?

I attempted to expand the LHS but I didnt feel it was leading me anywhere...

Homework Equations

The Attempt at a Solution

$\displaystyle \int_\Omega (\hat{e_x}\frac{\partial G}{\partial x}+\hat{e_y}\frac{\partial G}{\partial y})w dx dy$...

NOTE: I have posted this query on MHF 3 days ago and nobody has answered. Here is the link just in case somebody has replied. thanks http://mathhelpforum.com/calculus/200911-gradient-divergent-identities.html

 

[HallsofIvy]

Integration by parts, taking u= w, dv= \nabla G dx dy.

 

[bugatti79]

Homework Statement

I need to show that $\displaystyle\int_\Omega (\nabla G)w dxdy=-\int_\Omega (\nabla w) G dxdy+\int_\Gamma \hat{n} w G ds$ given

$\displaystyle \int_\Omega \nabla F dxdy=\oint_\Gamma \hat{n} F ds$ where $\Omega$ and $\Gamma$ are the domain and boundary respectively. F,G and w are scalar functions...any ideas?

Homework Equations

The Attempt at a Solution

Integration by parts, taking u= w, dv= \nabla G dx dy.

If we let $u=w$ then $du=dw=\nabla w$?

$\displaystyle dv=\nabla G dxdy$ then

$\displaystyle v=\int_\Omega \nabla G dxdy=\int_\Gamma (\hat{n_x} \hat{e_x}+ \hat{n_y} \hat{e_y})Gds$

Thus

$\displaystyle ∫_Ω(∇G)wdxdy= \int_\Gamma (\hat{n_x} \hat{e_x}+ \hat{n_y} \hat{e_y})G w ds- \int \int_\Gamma (\hat{n_x} \hat{e_x}+ \hat{n_y} \hat{e_y})G \nabla w ds$

Clearly I have gone wrong somewhere...?