Let S be the surface shown in the following figure in the first octant in the xy-plane, and let 7 be a continuously differentiable vector field on R³ such that V x F = 2K on S. If the circulation of F is 12 around C₂, 157 around C³, and 14 around C4, determine the circulation of F around C₁. (Assume that the radii of C₂, C3, and C are equal to 1.) y 2 C₁ 48
Let S be the surface shown in the following figure in the first octant in the xy-plane, and let 7 be a continuously differentiable vector field on R³ such that V x F = 2K on S. If the circulation of F is 12 around C₂, 157 around C³, and 14 around C4, determine the circulation of F around C₁. (Assume that the radii of C₂, C3, and C are equal to 1.) y 2 C₁ 48
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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Transcribed Image Text:Let \( S \) be the surface shown in the following figure in the first octant in the \( xy \)-plane, and let \( \mathbf{F} \) be a continuously differentiable vector field on \( \mathbb{R}^3 \) such that \( \nabla \times \mathbf{F} = 2k \) on \( S \). If the circulation of \( \mathbf{F} \) is \( 12\pi \) around \( C_2 \), \( 15\pi \) around \( C_3 \), and \( 14\pi \) around \( C_4 \), determine the circulation of \( \mathbf{F} \) around \( C_1 \). (Assume that the radii of \( C_2 \), \( C_3 \), and \( C_4 \) are equal to 1.)
### Diagram Explanation:
- The diagram depicts a quarter-circle sector in the first octant of the \( xy \)-plane.
- The outer boundary of the sector forms an arc from \( (6,0) \) to \( (0,6) \), centered at the origin.
- Inside the sector, there are three smaller circles, labeled \( C_2 \), \( C_3 \), and \( C_4 \).
- \( C_2 \) is located in the upper left region, near the \( y \)-axis.
- \( C_3 \) is positioned in the lower left region near the origin.
- \( C_4 \) is on the right, near the midpoint of the \( x \)-axis.
- Arrows indicate the direction of circulation around each circle.
- The inner part of the sector is labeled \( S \).
### Assumptions:
- The radii of the circles \( C_2 \), \( C_3 \), and \( C_4 \) are each 1 unit.
- The circulation values for \( C_2 \), \( C_3 \), and \( C_4 \) are provided, and the problem requires determining the circulation for the larger arc \( C_1 \).
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