Use the Divergence Theorem to calculate the surface integral int int_S F · dS, that is, calculate the flux of F across S. F(x,y,z) = (cos(z)+xy^2) i + xe^-z j + (sin(y)+x^2z) k S is the surface of the solid bounded by the paraboloid z = x^2+y^2 and the plane z = 9.

necessaryh 2020-12-06 Answered
Use the Divergence Theorem to calculate the surface integral SF·dS, that is, calculate the flux of F across S.
F(x,y,z)=(cos(z)+xy2)i+xezj+(sin(y)+x2z)k
S is the surface of the solid bounded by the paraboloid z=x2+y2 and the plane z = 9.
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Expert Answer

Tasneem Almond
Answered 2020-12-07 Author has 91 answers

Step 1
Here,
F=<cosz+xy2,xez,siny+x2z>
Step 2
Find grad F.
F=Fxx+Fyy+Fzz
=y2+0+x2
=x2+y2
Step 3
Assume, (x,y,z)=(rcos0,rsin0,z) where
0 0<0<2π
0<r<z
Apply divergence theorem.
SFds=EFdV
=E(x2+y2)dV
=0902π0z(r2cos20+r2sin20)rdrd0dz
=0902π0zr3drd0dz
=0902π[r44]0zd0dz
=0902π[z440]d0dz
=1409z2[0]02πdz
=1409z2[2π0]dz
=2π409z2dz
=π2[9330]
=243π2
Step 4
Result:

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