Problem

Part II Fill in the blanks

14. Let $E$ be a tetrahedron（四面體）in $\mathbb{R}^3$ bounded by the planes $x+y+z=3$, $x=2z$, $y=0$ and $z=0$. Let also $\mathbf{F}:=(x-y)\mathbf{i}+(y^2+z^2)\mathbf{j}+e^{x^3}\mathbf{k}$.

(a) $\operatorname{curl}\mathbf{F}=\underline{\quad(18)\quad}$.

(b) If $S_1$ is the boundary surface of $E$ (including all faces) endowed with the outward orientation, one has $\iint_{S_1}\operatorname{curl}\mathbf{F}\cdot d\mathbf{S}=\underline{\quad(19)\quad}$.

(c) If $S_2$ is the surface obtained from $S_1$ by removing the face in the $zy$-plane while keeping the orientation from $S_1$ on all other faces, one then has $\iint_{S_2}\operatorname{curl}\mathbf{F}\cdot d\mathbf{S}=\underline{\quad(20)\quad}$.