Suppose $F =(2x−4y)i +(x+3y)j$. Use Stokes' Theorem to make the following circulation calculations:
(a) Find the circulation of $F$ around the circle $C$ of radius $10$ centered at the origin in the $xy$-plane, oriented clockwise as viewed from the positive $z$-axis.
Circulation = $\int_CF\cdot dr$
Here is my work: Please tell me the correct answer and what I am doing wrong:
$$ \begin{align*} \int_C F\cdot dr&= \int\int_S \text{curl} F · dS & \text{by Stokes' Theorem} \\ &= \int\int\langle 0, 0, 5\rangle \cdot \langle 0, 0, 1\rangle~dA & \text{since the circle lies on }z = 0 \\ &= 5 * (\text{Area of }C) \\ &= 5 * 100π \\ &= 500π. \end{align*} $$