Let S₁ be the portion of the unit sphere with z≥ 0, oriented with normal vector pointing up and out of the sphere. Let S₂ be the portion of the unit sphere with z≤ 0, oriented with normal vector pointing up and into the sphere. Finally, let F = 2y i + 2zj + 2x k. (a) Find a vector field G such that the curl V×G is equal to F. (There are infinitely many possibilities, but I've set the problem up so that some are particularly easy to find.) (b) Use Stokes' theorem to explain why the flux integral of F across S₁ is equal to the flux integral of F across S2. (c) Use the divergence theorem to explain why the flux integral of F across S₁ is equal to the flux integral of F across S2.

Let S₁ be the portion of the unit sphere with z≥ 0, oriented with normal vector pointing up and out of the sphere. Let S₂ be the portion of the unit sphere with z≤ 0, oriented with normal vector pointing up and into the sphere. Finally, let F = 2y i + 2zj + 2x k. (a) Find a vector field G such that the curl V×G is equal to F. (There are infinitely many possibilities, but I've set the problem up so that some are particularly easy to find.) (b) Use Stokes' theorem to explain why the flux integral of F across S₁ is equal to the flux integral of F across S2. (c) Use the divergence theorem to explain why the flux integral of F across S₁ is equal to the flux integral of F across S2.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.2: Inner Product Spaces

Problem 101E: Consider the vectors u=(6,2,4) and v=(1,2,0) from Example 10. Without using Theorem 5.9, show that...

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