Use the extended divergence theorem to compute the total flux of the vector field F(x, y,...

Use the extended divergence theorem to compute the total flux of the vector field

F(x, y, z) = −3x2 + 3xz − y, 2y3 − 6y, 9x2 + 4z2 − 3x outward from the region F that lies inside the sphere x2 + y2 + z2 = 25 and outside the solid cylinder x2 + y2 = 4 with top at z = 1 and bottom at z = −1.

Expert Solution

genius_generous answered 2 years ago

Verify the Divergence Theorem for the vector field F(x, y, z) = < y, x ,...

Verify the Divergence Theorem for the vector field F(x, y, z) = < y, x , z^2 > on the region E bounded by the planes y + z = 2, z = 0 and the cylinder x^2 + y^2 = 1. By Surface Integral: By Triple Integral:

Use the divergence theorem to calculate the flux of the vector field F⃗ (x,y,z)=−5xyi⃗ +2yzj⃗ +4xzk⃗ through...

Use the divergence theorem to calculate the flux of the vector field F⃗ (x,y,z)=−5xyi⃗ +2yzj⃗ +4xzk⃗ through the sphere S of radius 2 centered at the origin and oriented outward. ∬SF⃗ ⋅dA⃗ =

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Verify that the Divergence Theorem is true for the vector field F on the region E....

Verify that the Divergence Theorem is true for the vector field F on the region E. Give the flux. F(x, y, z) = xyi + yzj + zxk, E is the solid cylinder x2 + y2 ≤ 144, 0 ≤ z ≤ 4.

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Compute the line integral of the vector field F(x, y, z) = ⟨−y, x, z⟩ along the curve which is given by the intersection of the cylinder x 2 + y 2 = 4 and the plane x + y + z = 2 starting from the point (2, 0, 0) and ending at the point (0, 2, 0) with the counterclockwise orientation.

Use the Divergence Theorem to evaluate S F · dS, where F(x, y, z) = z2xi...

Use the Divergence Theorem to evaluate S F · dS, where F(x, y, z) = z2xi + y3 3 + cos z j + (x2z + y2)k and S is the top half of the sphere x2 + y2 + z2 = 4. (Hint: Note that S is not a closed surface. First compute integrals over S1 and S2, where S1 is the disk x2 + y2 ≤ 4, oriented downward, and S2 = S1 ∪ S.)

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Use the extended divergence theorem to compute the total flux of the vector field F(x, y,...

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