The two-dimensional divergence theorem is given by $$\iint_S{(\vec{\nabla} \cdot \vec{F}) \mathrm{d}A} = \oint_C{(\vec{F} \cdot \hat{n}) \mathrm{d}l}. $$
For the line element $\mathrm{d}l$ of a circle lying in the x-y plane the normal vector is clearly not uniquely defined, but assuming it to lie in the x-y plane gives $$\hat{n}=\frac{1}{a}(x\hat{x}+y\hat{y}),$$ where $a$ is the radius of the circle. By choosing $$\vec{F}=\frac{1}{2}(x\hat{x} + y\hat{y}),$$ I can calculate $$\iint_S{\mathrm{d}A}=\frac{1}{2 a}\oint_C{(x^2+y^2)\mathrm{d}l}=\frac{a}{2} \oint_C{\mathrm{d}l}=\pi a^2.$$ If instead however, $$\vec{F}=\frac{1}{3}(x\hat{x}+y\hat{y}+z\hat{z}),$$ it follows that $$\iint_S{\mathrm{d}A}=\frac{1}{3 a}\oint_C{(x^2+y^2)\mathrm{d}l}=\frac{a}{3} \oint_C{\mathrm{d}l}=\frac{2}{3}\pi a^2,$$ which is a clear inconsistency. Additionally, the above formulation implies that the surface area of any arbitrary shape bounded by the circle of radius $a$ (think of the upper half of a sphere for example) in the x-y plane has identical surface area, which cannot be possible. Where does my mistake lie?
In particular, I thought I might easily calculate the surface area of a spherical cap using this two-dimensional divergence theorem.