I am trying to solve following problem
Let $u(x,y) = (x+e^y \sin(y), -y + e^x \cos(x))$ and $S$ be the unit circle with center at $(0,0)$. compute \begin{align} \int_S u(x,y) \cdot n(x,y) ds \end{align} where $n(x,y)$ is the unit outer normal vector to $S$ and $ds$ is the line element on $S$.
First the normal vector of circle, $F=x^2+y^2-1$ is given by $n(x,y) = \frac{\nabla F}{|\nabla F|} = \frac{(x,y)}{\sqrt{x^2+y^2}} = (x, y)$
Then \begin{align} \int_S (u,x,y) \cdot n(x,y) ds &= \int_S (x^2 + x e^y \sin(y)) dx + (-y^2 + e^x y \cos(x)) dy \end{align} My first trial is plug $x=\cos(t)$, $y=\sin(t)$ with $t\in [0,2\pi]$, this seems complicated since I have $e^{\sin(t)} \sin(\cos(t))$. ...
My second trial is use Green's theorem, then \begin{align} = \int_{D} \left[ ye^x ( \cos(x) - \sin(x)) - x e^y ( \sin(y) + \cos(y) ) \right] dxdy \end{align} and i tried to solve this by using polar coordinates, then I got more complicated expression.
Am i mislead the problem? How one can solve this problem and is there any strategy for this kinds of computation?