7. Using the divergence theorem compute the outward flux of the vector field F(x, y, z) = x°i+ y³j+10k over the boundary of the region between the upper hemispheres of radius 1 and 2 centered at the origin with bases in the xy-plane.

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7. Using the divergence theorem compute the outward flux of the vector field
3
F(x, y, z) = x³i + y°j+10k
over the boundary of the region between the upper hemispheres of radius 1 and 2 centered
at the origin with bases in the xy-plane.
Transcribed Image Text:7. Using the divergence theorem compute the outward flux of the vector field 3 F(x, y, z) = x³i + y°j+10k over the boundary of the region between the upper hemispheres of radius 1 and 2 centered at the origin with bases in the xy-plane.
Expert Solution
Step 1

step:-1

Divergence Theorem:- 

F.ndS=divF dV=divF  ρ2 sinϕ dρ dθ dϕ 

Given that

 F=x3i +y3j+10kdiv.F=xi+xj + xkx3i +y3j+10kdiv.F=3x2+3y2=3x2+y2div.F=3x2+y2

Step:-2

Using spherical co-ordinates, so take

x=ρ sinϕ sinθ, y=ρ sinϕ cosθ, z=ρ cosϕ x2+y2=ρ2 sin2ϕ sin2θ+ρ2 sin2ϕ cos2θ=ρ2 sin2ϕsin2θ+cos2θx2+y2=ρ2 sin2ϕas sin2θ+cos2θ=1

Advanced Math homework question answer, step 1, image 1

In this picture, we can see the upper hemisphere forms circle in x y- plane, so

0θ2π

As given hemisphere of radius 1 and 2 center at origin so, 

1ρ2

and 0ϕπ2 as hemisphere,

Step:-3

Using above divergence theorem we have

F.ndS=divF  ρ2 sinϕ dρ dθ dϕF.ndS=302π0π212ρ4 sin3ϕ dρ dϕ dθ   using equation (1) F.ndS=302π02πsin3ϕ255-155dϕdθ=3×315×402π3-13dθ  =3×315×4×83×2πF.ndS=124π5

 

 

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