Question

Given that \(\vec{F}(x,y,z)=\sin(y)\,\hat{x}+\cos(x)\,\hat{y}+5\,\hat{z}\), the integral \(\oiint_S \vec{F}(x,y,z)\cdot d\vec{s}\) over the unit sphere \(S\) centered at the origin evaluates to
(Round off to one decimal place)

Hint:

Apply the divergence theorem and check each partial derivative of F directly; the cross-mixed dependence of sin(y) and cos(x) makes the divergence vanish.

Answer 1

Correct Answer: 0

Step 1: Recognize this as a closed-surface flux problem.
The sphere $S$ is a closed surface, so instead of computing the surface integral piece by piece, use Gauss's divergence theorem to turn it into a volume integral:
\[ \oiint_S \vec F\cdot d\vec s=\iiint_V(\nabla\cdot \vec F)\,dV \]

Step 2: Take the divergence term by term.
$\vec F=\sin(y)\hat x+\cos(x)\hat y+5\hat z$. The $x$-component depends only on $y$, so differentiating it with respect to $x$ gives $0$. The $y$-component depends only on $x$, so differentiating it with respect to $y$ gives $0$. The $z$-component is a plain constant, so differentiating it with respect to $z$ gives $0$.

Step 3: Sum the three zero terms.
\[ \nabla\cdot\vec F=0+0+0=0 \]

Step 4: Integrate zero over the unit ball.
A volume integral of a function that is zero everywhere is zero, regardless of the shape or size of the region:
\[ \iiint_V 0\,dV=0 \]

Step 5: State the result.
The field is divergence-free everywhere, so the net outward flux through any closed surface, including this unit sphere, comes out to zero.
\[ \boxed{0.0} \]