Use the Divergence Theorem to calculate the surface integral S F · dS; that is, calculate the flux of F across S. F(x, y, z) = x2yi + xy2j + 3xyzk, S is the surface of the tetrahedron bounded by the planes x = 0, y = 0, z = 0, and x + 2y + z = 2.

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Step-by-step explanation:

- Divergence theorem, expresses an explicit way to determine the flux of a force field ( F ) through a surface ( S ) with the help of "del" operator ( D ) which is the sum of spatial partial derivatives of the force field ( F ).

- The given force field as such:

Where,

         i, j, k are unit vectors along the x, y and z coordinate axes, respectively.

- The surface ( S ) is described as a tetrahedron bounded by the planes:

- The divergence theorem gives us the following formulation:

- We will first apply the del operator on the force field as follows:

- Now, we will define the boundaries of the solid surface ( Tetrahedron ).

- The surface ( S ) is bounded in the z - direction by plane z = 0 and the plane [ z = 2 - x - 2y ]. The limits of integration for " dz " are as follows:

                      dz: [ z = 0 - > 2 - x - 2y ]

- Now we will project the surface ( S ) onto the ( x-y ) plane. The projection is a triangle bounded by the axes x = y = 0 and the line: x = 2 - 2y. We will set up our limits in the x- direction bounded by x = 0 and x = 2 - 2y. The limits of integration for " dx " are as follows:

                     dx: [ x = 0 - > 2 - 2y ]

- The limits of "dy" are constants defined by the axis y = 0 and y = -2 / -2 = 1. Hence,

                    dy: [ y = 0 - > 1 ]

- Next we will evaluate the triple integral as follows:

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