Given a continuously differentiable vector field $v:\mathbb{R}^3\to\mathbb{R}^3$ with the property $\nabla\times v=(\arctan(2z),2e^{2z},0)^{\top}$. The Surface $\mathcal{F_a}$ with $a\in\mathbb{R}$ is define with $\mathcal{F}_a=\{(x,y,z)^\top|x^2+y^2\le1,z=a\cdot(1-x^2-y^2)\cdot[\ln{(1+\sqrt[3]{1+x^2})}]+e^{x+\sin{(2y)}}\}$
Prove or disprove that $\int_{\mathcal{F}_a}\nabla\times v\cdot do=0,\forall a\in\mathbb{R}$