Problem

請寫出計算及證明過程，勿只寫答案。

6. Let $\Sigma$ be the part of the paraboloid $z=x^2+y^2$ that lies within the cylinder $x^2+y^2=1$. Let $\mathbf{n}$ be its upward unit normal field. For the vector field

$$\mathbf{F}=\left(2x+e^y+z^2,\ \ln(z+2)-y,\ 2\right),$$

compute the flux integral $\iint_{\Sigma}\mathbf{F}\cdot\mathbf{n}\,dS$.