Consider the vector field F(x, y, z) = (5xyz, 5xyz, 5xyz), the cube D = = {(x, y, z) | 0 ≤ x ≤ 4,0<≤ y ≤ 4,0 ≤ y ≤ 4} = [0,4] × [0,4] × [0,4] and the opposite pair of vertices = (0,0,0) and P = (4, 4, 4) of D. The boundary of the cube D consists of six square faces: three faces sharing a corner at O: one face on the plane z = 0 (the xy coordinate plane); - another on the plane y = 0 (the xz coordinate plane); - the third on the plane x = = 0 (the yz coordinate plane). three faces sharing a corner at P: - - - one on the plane z = 4; another on the plane y = 4; the third on the plane x = 4. To visualize all of this, pick up any cube such as a six sided game die, and imagine that O and P are opposite corners of that cube. (a) Calculate the flux of ♬ through the boundary of D by applying the divergence theorem. (A triple integral calculation is needed). (b) Using a separate calculation, verify that the flux of F through each of the three faces sharing a corner at O is equal to 0. (Only extremely simple surface integrals are needed) (c) Using a symmetry argument, verify that the values of flux of F through each of the three square sharing a corner at P are all equal to each other. Compute that common value. (No integral at all is needed)
Consider the vector field F(x, y, z) = (5xyz, 5xyz, 5xyz), the cube D = = {(x, y, z) | 0 ≤ x ≤ 4,0<≤ y ≤ 4,0 ≤ y ≤ 4} = [0,4] × [0,4] × [0,4] and the opposite pair of vertices = (0,0,0) and P = (4, 4, 4) of D. The boundary of the cube D consists of six square faces: three faces sharing a corner at O: one face on the plane z = 0 (the xy coordinate plane); - another on the plane y = 0 (the xz coordinate plane); - the third on the plane x = = 0 (the yz coordinate plane). three faces sharing a corner at P: - - - one on the plane z = 4; another on the plane y = 4; the third on the plane x = 4. To visualize all of this, pick up any cube such as a six sided game die, and imagine that O and P are opposite corners of that cube. (a) Calculate the flux of ♬ through the boundary of D by applying the divergence theorem. (A triple integral calculation is needed). (b) Using a separate calculation, verify that the flux of F through each of the three faces sharing a corner at O is equal to 0. (Only extremely simple surface integrals are needed) (c) Using a symmetry argument, verify that the values of flux of F through each of the three square sharing a corner at P are all equal to each other. Compute that common value. (No integral at all is needed)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CM: Cumulative Review
Problem 12CM
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![Consider the vector field F(x, y, z) = (5xyz, 5xyz, 5xyz), the cube
D =
= {(x, y, z) | 0 ≤ x ≤ 4,0<≤ y ≤ 4,0 ≤ y ≤ 4} = [0,4] × [0,4] × [0,4]
and the opposite pair of vertices = (0,0,0) and P = (4, 4, 4) of D. The
boundary of the cube D consists of six square faces:
three faces sharing a corner at O:
one face on the plane z = 0 (the xy coordinate plane);
- another on the plane y = 0 (the xz coordinate plane);
-
the third on the plane x = = 0 (the yz coordinate plane).
three faces sharing a corner at P:
-
-
-
one on the plane z =
4;
another on the plane y = 4;
the third on the plane x = 4.
To visualize all of this, pick up any cube such as a six sided game die, and
imagine that O and P are opposite corners of that cube.
(a) Calculate the flux of ♬ through the boundary of D by applying the
divergence theorem. (A triple integral calculation is needed).
(b) Using a separate calculation, verify that the flux of F through each of
the three faces sharing a corner at O is equal to 0. (Only extremely
simple surface integrals are needed)
(c) Using a symmetry argument, verify that the values of flux of F
through each of the three square sharing a corner at P are all equal
to each other. Compute that common value. (No integral at all is
needed)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1d06251-9cb7-4f37-8e8f-dbbf0e17b22e%2F9cc5d24a-52b0-486d-80d6-93696b4ae674%2Fq1shv_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the vector field F(x, y, z) = (5xyz, 5xyz, 5xyz), the cube
D =
= {(x, y, z) | 0 ≤ x ≤ 4,0<≤ y ≤ 4,0 ≤ y ≤ 4} = [0,4] × [0,4] × [0,4]
and the opposite pair of vertices = (0,0,0) and P = (4, 4, 4) of D. The
boundary of the cube D consists of six square faces:
three faces sharing a corner at O:
one face on the plane z = 0 (the xy coordinate plane);
- another on the plane y = 0 (the xz coordinate plane);
-
the third on the plane x = = 0 (the yz coordinate plane).
three faces sharing a corner at P:
-
-
-
one on the plane z =
4;
another on the plane y = 4;
the third on the plane x = 4.
To visualize all of this, pick up any cube such as a six sided game die, and
imagine that O and P are opposite corners of that cube.
(a) Calculate the flux of ♬ through the boundary of D by applying the
divergence theorem. (A triple integral calculation is needed).
(b) Using a separate calculation, verify that the flux of F through each of
the three faces sharing a corner at O is equal to 0. (Only extremely
simple surface integrals are needed)
(c) Using a symmetry argument, verify that the values of flux of F
through each of the three square sharing a corner at P are all equal
to each other. Compute that common value. (No integral at all is
needed)
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