Use Stokes' theorem to evaluate // V x 7. ds where S is the hemisphere x² + y² + z² = 16, x ≥ 0, oriented frontward. Here F = (yz, x sin(z), xyz²).
Use Stokes' theorem to evaluate // V x 7. ds where S is the hemisphere x² + y² + z² = 16, x ≥ 0, oriented frontward. Here F = (yz, x sin(z), xyz²).
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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![**Problem Statement: Application of Stokes' Theorem**
Use Stokes' theorem to evaluate the surface integral:
\[
\iint_{S} (\nabla \times \vec{F}) \cdot \hat{n} \, dS
\]
where \( S \) is the hemisphere defined by the equation \( x^2 + y^2 + z^2 = 16 \), with the condition \( x \geq 0 \), and it is oriented frontward. The vector field \( \vec{F} \) is given by:
\[
\vec{F} = \langle yz, \, x \sin(z), \, xyz^2 \rangle
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc34e9aed-6aed-49a2-aae3-6d378ffef9e3%2F6065cd2a-f38e-46a1-ab9d-8098ccbfd564%2Fzb3qwh_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Application of Stokes' Theorem**
Use Stokes' theorem to evaluate the surface integral:
\[
\iint_{S} (\nabla \times \vec{F}) \cdot \hat{n} \, dS
\]
where \( S \) is the hemisphere defined by the equation \( x^2 + y^2 + z^2 = 16 \), with the condition \( x \geq 0 \), and it is oriented frontward. The vector field \( \vec{F} \) is given by:
\[
\vec{F} = \langle yz, \, x \sin(z), \, xyz^2 \rangle
\]
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