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 the tool needed.
The integral asks for the flux of \(\vec{F}\) through a closed surface, the unit sphere. The divergence theorem converts a closed surface integral into a volume integral of the divergence:
\[ \oiint_S \vec{F}\cdot d\vec{s}=\iiint_V (\nabla\cdot\vec{F})\,dV \]

Step 2: Compute the divergence of \(\vec{F}\).
\[ \vec{F}(x,y,z)=\sin(y)\,\hat{x}+\cos(x)\,\hat{y}+5\,\hat{z} \]
\[ \nabla\cdot\vec{F}=\frac{\partial}{\partial x}\big(\sin(y)\big)+\frac{\partial}{\partial y}\big(\cos(x)\big)+\frac{\partial}{\partial z}(5) \]

Step 3: Evaluate each partial derivative.
\(\sin(y)\) does not depend on \(x\), so \(\dfrac{\partial}{\partial x}\big(\sin(y)\big)=0\). \(\cos(x)\) does not depend on \(y\), so \(\dfrac{\partial}{\partial y}\big(\cos(x)\big)=0\). The constant \(5\) does not depend on \(z\), so \(\dfrac{\partial}{\partial z}(5)=0\).

Step 4: Add up the divergence.
\[ \nabla\cdot\vec{F}=0+0+0=0 \]
The divergence is zero everywhere in space, not just on the sphere.

Step 5: Apply the divergence theorem.
Since \(\nabla\cdot\vec{F}=0\) at every point inside the unit ball,
\[ \iiint_V (\nabla\cdot\vec{F})\,dV=\iiint_V 0\,dV=0 \]

Step 6: Final conclusion.
\[ \boxed{0.0} \]