"Let $F(x,y) = x^2-y^2$. Evaluate:
$\oint \frac{\partial F}{\partial n}ds$ around the circle $x^2+y^2 = 1$. Here, $\frac{\partial F}{\partial n}$ is the directional derivative of $F$ along the outer normal and $dS$ = $|dS|$."
I can't use the fact that this is a conservative vector field. I know the answer should be $0$. I have to do the question using first principles. How would I go about doing this though? I know I have to parametrize the circle but how do I incorporate it into the integral? I am a little confused about that.
Can someone guide me in the right direction? Thanks!
For a point on the unit circle, the unit normal vector would be $\hat{n} =(x,y)$.
$\frac{\partial F}{\partial n}= \nabla F \cdot \hat{n}=2x^2-2y^2$
Your integral would then become $\int_{c}(2x^2-2y^2)ds$.
Can you get it from here?