Let F = [x^2,0,z^2], and S the surface of the box abs(x) <= 1, abs(y) <= 3, 0 <= z <= 2. Evaluate the surface integral int int_S F*n dA by the divergence theorem.

Sinead Mcgee 2021-02-21 Answered
Let F=[x2,0,z2], and S the surface of the box |x|1,|y|3,0z2.
Evaluate the surface integral SFndA by the divergence theorem.
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Expert Answer

tafzijdeq
Answered 2021-02-22 Author has 92 answers

Step 1
The divergence of the function F is:
F=Px+Qy+Rz
÷F=x(x2)+y(0)+z(z2)
=2x+2z
Step 2
By the divergence theorem convert the surface integral into a triple integral:
SPdydz+Qdxdz+Rdxdy=V(Px+Qy+Rz)dxdydx
=V(2x+2z)dxdydz
Step 3
V(2x+2z)dxdydz=11dx33dy02(2x+2z)dz
=211dx33dy02(x+z)dz
=211dx33dy[(xz+z22)02]
=211dx33(2x+2)dy
=411dx[(xy+y)33]
=411dx[(3x+3)(3x3)]

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