32.\text{ Given that }\overrightarrow{F}(x,y,z)=\sin \left(y\right)\hat{i}+\cos \left(x\right)\hat{j}+5\hat{k},\text{ the integral }{∯}_S\overrightarrow{F}·d\overrightarrow{S}\text{ over the unit sphere }S\text{ centered at the origin evaluates to \_\_\_\_\_\_\_.}\text{ (Round off to one}

\text{decimal place)}

The correct answer is Option A: 0 (or simply 0).

To evaluate the surface integral of the vector field $\overrightarrow{F}(x,y,z)=\sin \left(y\right)\hat{i}+\cos \left(x\right)\hat{j}+5\hat{k}$ over the closed unit sphere $S$ centered at the origin, we can apply the Divergence Theorem.

The Divergence Theorem states that for a vector field $\overrightarrow{F}$ and a solid region $V$ enclosed by a closed surface $S$ with an outward orientation:
$${∯}_S\overrightarrow{F}·d\overrightarrow{S}={\iiint }_V(\nabla ·\overrightarrow{F})dV$$

First, let us calculate the divergence of the vector field $\overrightarrow{F}$, denoted as $\nabla ·\overrightarrow{F}$. The components of $\overrightarrow{F}=P\hat{i}+Q\hat{j}+R\hat{k}$ are:
$$P(x,y,z)=\sin \left(y\right)$$