Let be given the unit disc $S$ and the functions $f$ and $ \varphi$ on $S$:
$f(x,y) = (3x+e^y, e^x – y)$
$\varphi(x,y) = \sqrt{x^2+y^2} * e^{x^2+y^2} \ \ \ \ (*) $
Now there is to compute: $ \int\limits_{ S} f * \nabla \varphi \ \ d S $
First one starts out verifying that in fact the divergence theorem can be used (also Fubini and the Transformation theorem at the appropriate positions), since the functions are all defined on a compact set, they have a continuous derivative and the set has a smooth boundary.
Integrating $(*)$ by parts gives:
$ \int\limits_{S} \varphi * div f \ \ dS $ = $ \int\limits_{\partial S} f * \varphi \ \ d \partial S $ – $ \int\limits_{S} f * \nabla \varphi \ \ dS $
$ \Leftrightarrow \int\limits_{S} f * \nabla \varphi \ \ dS =- \int\limits_{S} \varphi * div f \ \ dS + \int\limits_{\partial S} f * \varphi \ \ d \partial S $
However I don't see a good way to compute $ \int\limits_{\partial S} f * \varphi \ \ d \partial S $ , etc. Those terms look way too ugly to be useful.
Now I am wondering. Did I overlook something? Do I approach this problem from the wrong side straight into a dead end? If yes, what is a good approach?
Therefore I would be very happy about any sort of constructive hint, comment or answer.
Best Answer
The requested integral could be written as $$ \int_{\partial S} \mathbf{n}\cdot\mathbf{f}\,\varphi \, dl $$ where $$ \mathbf{n} = \left(\frac{x}{\sqrt{x^2+y^2}},\frac{y}{\sqrt{x^2+y^2}}\right) $$ is the unit normal outgoing vector.
The integral on the boundary, using the parametric equation of the boundary $ (x,y) = (\cos t, \sin t)$ becomes $$ \int_0^{2\pi} (\cos t,\sin t)\cdot(3\cos t+e^{\sin t},e^{\cos t}-\sin t)e^{1}dt = \\ = e\int_0^{2\pi} (3\cos^2 t+\cos t\,e^{\sin t}+\sin t\,e^{\cos t}-\sin^2 t)dt $$ from now on you can easily procede.