Consider the following two surfaces, shown below: • S₁ be the surface given by the part of the sphere of radius √2 centered at (0,0,1) (i.e., x² + y² + (z − 1)² = 2) above the xy-plane. • S₂ is the upside-down cone with vertex at (0, 0, 1), height 1, and radius 1, above the xy-plane (i.e., x² + y² = (1 - 2)²). 3 3 Let F(x, y, z) = (sin(xy) +e², x+ye, z³ - cos(y)) be a vector field. Let G = curl(F). Explain in your own words, using complete sentences, and without computing any integrals, why JI G. ds = [G 1 G.ds. You may assume that both S₁ and S₂ are oriented with outward-pointing normal vectors.
Consider the following two surfaces, shown below: • S₁ be the surface given by the part of the sphere of radius √2 centered at (0,0,1) (i.e., x² + y² + (z − 1)² = 2) above the xy-plane. • S₂ is the upside-down cone with vertex at (0, 0, 1), height 1, and radius 1, above the xy-plane (i.e., x² + y² = (1 - 2)²). 3 3 Let F(x, y, z) = (sin(xy) +e², x+ye, z³ - cos(y)) be a vector field. Let G = curl(F). Explain in your own words, using complete sentences, and without computing any integrals, why JI G. ds = [G 1 G.ds. You may assume that both S₁ and S₂ are oriented with outward-pointing normal vectors.
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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Please explain and solve this multi variable calculus problem. Thank you!
Transcribed Image Text:Consider the following two surfaces, shown below:
• S₁ be the surface given by the part of the sphere of radius √2 centered at (0,0,1)
(i.e., x² + y² + (z − 1)² = 2) above the xy-plane.
• S₂ is the upside-down cone with vertex at (0, 0, 1), height 1, and radius 1, above
the xy-plane (i.e., x² + y² = (1 - 2)²).
3
2
3
JI G. ds = [G
1
Let F(x, y, z) = (sin(xy) +e², x+ye, z³ - cos(y)) be a vector field. Let G = curl(F).
Explain in your own words, using complete sentences, and without computing any
integrals, why
G.dS.
2
You may assume that both S₁ and S₂ are oriented with outward-pointing normal
vectors.
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