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Let S be the part of the plane 2x + 1y + z = 2 which lies in the first octant, oriented upward. Find the flux of the vector field F = 1i + 1j + 3k across the surface S.



Answer :

The vector field F = 1i + 1j + 3k has a flux of 2 across the surface S. We use Green's Theorem to determine the vector-flux field.

Explain the Green's Theorem?

  • The simple formula for Green's theorem is the link between the total amount of microscopic circulation inside of curve C and the macroscopic circulation revolving around it.
  • Green's Theorem in flux form for the specified vector-field:

φ = ∫ F.n ds

φ = ∫∫ F. divG.dA

In which, G is equivalent to a part of given plane = 2x + 1y + z = 2.

And,  F = 1i + 1j + 2k

Thus,

div G = div(2x + 1y + z = 2) = 2i + 4j + k

Flux = ∫(1i + 1j + 3k) (2x + 1y + z).dA

φ = ∫ (2 + 4 + 2).dA

φ = 8∫dA

A = 1/2 XY (on this given x-y plane)

2x+4y =2

For, x = 0, y = 1/2

y = 0, x = 1

1/2 (1*1/2) = 1/4

Thus,  flux = 8*1/4 = 2

φ = 2.

There are several uses for Green's theorem. Solving two-dimensional flow integrals, whose assert that its sum of fluid outflows from just a volume equals the sum of outflows around an enclosed area, is one way to accomplish this.

To know more about the Green's Theorem, here

https://brainly.com/question/23265902

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The complete question is-

Let S be the part of the plane 2x + 1y + z = 2 which lies in the first octant, oriented upward. Find the flux of the vector field F = 1i + 1j + 3k across the surface S. F = 1i + 1j + 3k across the surface s2x + 1y + z = 2 which lies in the first octant, oriented upward. Find the flux of the vector field.

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