Compute Curl Integral of Vector Field and Ellipse Consider a vector field : ${\bf F}=P(x){\bf i}+ Q(y){\bf j}$, for some functions : $P(x), Q(y)$ which have continuous partial derivatives everywhere.
Let: $C$ stand for the ellipse : $\{ 4x^2+9y^2=1\}$. Then : $\int_C {\bf F}\cdot d{\bf r}$
 equals:
I can't determine an answer with this data but the options are: 
 $P(\frac{1}{2}) - Q(\frac{1}{2}), 0, P(\frac{1}{2}),P(\frac{1}{2}) + Q(\frac{1}{2})$ 
Which is the right answer?

Notice that
$\nabla \times \mathbf F = \dfrac{\partial Q(y)}{\partial x} - \dfrac{\partial P(x)}{\partial y} = 0; \tag 1$
next recall that, via Stokes theorem,
$\displaystyle \int_\xi \nabla \times \mathbf F dA = \int_{\partial \xi} \mathbf F \cdot d \mathbf r = \int_C \mathbf F \cdot d \mathbf r, \tag 2$
since
$\partial \xi = C; \tag 3$
here $\xi$ is the region bounded by the ellipse $C$, and $dA$ is the area element on $\Bbb R^2$.  Together (1) and (2) yield
$\displaystyle \int_C \mathbf F \cdot d \mathbf r = 0, \tag 4$
which is thus the correct option.
