Use the Divergence Theorem to evaluate ∬SF⋅dS∬SF⋅dS , where F(x,y,z)=⟨2x(y−z),x2−y2,z⟩F(x,y,z)=⟨2x(y−z),x2−y2,z⟩ and SS is the portion of the sphere x2+y2+z2=9x2+y2+z2=9 satisfying z≥0z≥0. Assume an outward orientation.∬SF⋅dS=∬SF⋅dS= Can you check the solution for the second one I solve and get 16pi but is not correct? ### Divergence Theorem Verification for a Given Vector Field**Problem Statement:**Confirm that the Divergence Theorem holds for the vector field $\mathbf{F}(x, y, z) = \langle -y, x, 2z \rangle$ and the solid $B$ that is bounded by the graphs of $z = 4$ and $z = x^2 + y^2$, oriented outwards.**Task:**Calculate the common value of the two integral expressions.\[ \iint\limits_{\partial B} \mathbf{F} \cdot d\mathbf{S} = \iiint\limits_B (\nabla \cdot \mathbf{F}) \, dV \]**Solution:**Enter the common value:\[ \boxed{-16\pi}\] ### Divergence Theorem Problem**Problem:** Use the Divergence Theorem to evaluate \[ \iint_S \mathbf{F} \cdot d\mathbf{S} \]where \[ \mathbf{F}(x, y, z) = (2x(y - z), x^2 - y^2, z) \]and $ S $ is the portion of the sphere \[ x^2 + y^2 + z^2 = 9 \]satisfying $ z \geq 0 $. Assume an outward orientation.\[ \iint_S \mathbf{F} \cdot d\mathbf{S} = \] *[Input Box]*

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Answered: Use the Divergence Theorem to evaluate…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Use the Divergence Theorem to evaluate ∬SF⋅dS∬SF⋅dS , where F(x,y,z)=⟨2x(y−z),x2−y2,z⟩F(x,y,z)=⟨2x(y−z),x2−y2,z⟩ and SS is the portion of the sphere x2+y2+z2=9x2+y2+z2=9 satisfying z≥0z≥0. Assume an outward orientation.

∬SF⋅dS=∬SF⋅dS=

Can you check the solution for the second one I solve and get 16pi but is not correct?

### Divergence Theorem Verification for a Given Vector Field

**Problem Statement:**
Confirm that the Divergence Theorem holds for the vector field $\mathbf{F}(x, y, z) = \langle -y, x, 2z \rangle$ and the solid $B$ that is bounded by the graphs of $z = 4$ and $z = x^2 + y^2$, oriented outwards.

**Task:**
Calculate the common value of the two integral expressions.

\[
\iint\limits_{\partial B} \mathbf{F} \cdot d\mathbf{S} = \iiint\limits_B (\nabla \cdot \mathbf{F}) \, dV
\]

**Solution:**
Enter the common value:

\[
\boxed{-16\pi}
\]

### Divergence Theorem Problem

**Problem:**
Use the Divergence Theorem to evaluate
\[ \iint_S \mathbf{F} \cdot d\mathbf{S} \]
where
\[ \mathbf{F}(x, y, z) = (2x(y - z), x^2 - y^2, z) \]
and $ S $ is the portion of the sphere
\[ x^2 + y^2 + z^2 = 9 \]
satisfying $ z \geq 0 $. Assume an outward orientation.

\[ \iint_S \mathbf{F} \cdot d\mathbf{S} = \]
*[Input Box]*

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