Let $F: \mathbb{R}^3 \to \mathbb{R}^3, F(x, y, z)=(y+yze^{xz}, e^{xz}+3y^2, -\sin(xy))$ be a vector field.
The question is: find $\iiint_V \operatorname{curl}(F)\cdotp\mathbb{a} \space dV$, where $V=\{(x, y, z) \in \mathbb{R}^3 \mid x^2+y^2+z^2<1, z\geq0 \}$, and $$\mathbb{a}(x, y, z)= \left( \frac{x}{\sqrt{1-x^2-y^2}}, \frac{y}{\sqrt{1-x^2-y^2}}, 1 \right), x^2+y^2<1.$$
So far, I've tried computing $\mathbb{curl}(F)$ and computing the dot product, but the resulting integral looks very unapproachable. For what it's worth, $\mathbb{curl}(F)=(-x\cos{xy}-xe^{xz}, yz^{xz}-y\cos{xy}, -1)$, barring any arithmetic mistakes on my part.
I've also tried finding a 2-form whose differential might be the resulting expression, but with no luck. I've also noticed that $\mathbb{a}$ is actually the projection of the hemisphere onto the plane $z=1$, if that's of any use.
I'm unsure of how to proceed from here.