31. Use Stokes' theorem to evaluate curl F. ds, where F(x, y, z) = −y² î + x + z²k and S is the part of plane x+y+z=1 in the Dositive octant and oriented counterclockwise x ≥ 0, y ≥ 0, z ≥ 0.
31. Use Stokes' theorem to evaluate curl F. ds, where F(x, y, z) = −y² î + x + z²k and S is the part of plane x+y+z=1 in the Dositive octant and oriented counterclockwise x ≥ 0, y ≥ 0, z ≥ 0.
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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Question
31
![**Problem 31:**
Use Stokes' theorem to evaluate the surface integral:
\[
\iint_S \text{curl} \ \vec{F} \cdot d\vec{S}
\]
where
\[
\vec{F}(x, y, z) = -y^2 \, \hat{i} + x \, \hat{j} + z^2 \, \hat{k}
\]
and \( S \) is the part of the plane \( x + y + z = 1 \) in the positive octant, which is oriented counterclockwise with \( x \geq 0 \), \( y \geq 0 \), \( z \geq 0 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F07dc3289-c262-41f1-9fc5-a8d18500de1b%2F95628cc0-d626-43e2-85e9-e1561e71290f%2Fy19tujc_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 31:**
Use Stokes' theorem to evaluate the surface integral:
\[
\iint_S \text{curl} \ \vec{F} \cdot d\vec{S}
\]
where
\[
\vec{F}(x, y, z) = -y^2 \, \hat{i} + x \, \hat{j} + z^2 \, \hat{k}
\]
and \( S \) is the part of the plane \( x + y + z = 1 \) in the positive octant, which is oriented counterclockwise with \( x \geq 0 \), \( y \geq 0 \), \( z \geq 0 \).
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