I have found the following integral to be zero, but i don't think its correct. $$ C = \oint_K d\mathbf r\cdot \mathbf A $$ Where $\mathbf A = \frac 1 2 \mathbf n \times \mathbf r$ and $\mathbf n \cdot \mathbf n=1$.
Taking $K$ as a circle with radius $R$ and $\mathbf n$ is the normal to the plane where the circle lives.
Any kind of help is appreciated . The problem is that when i found $A$ it came out to be zero. i found normal as $\nabla f$ where $f=x^2+y^2-1$ .
Progress : I applied stokes theorem, and also $\nabla \times \mathbf A =2\mathbf n$ , then i get $C=\pi R^2$ . am i right ?? how do i do it using line integral ?