# Thread: Flux integrals - verifying Stokes Theorem

1. ## Flux integrals - verifying Stokes Theorem

Ok, I've been stuck on this problem for hours now and its really irritating, so I need help!

Here is the question:

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Verify Stokes Theorem for the vector field

F = yi + 2zj + xzk

and the surface S defined by

x^2 + y^2 + z^2 = 25 & Z>=4

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Right, I'm quite new to these so please explain clearly if possible, cheers!

So far I've computed the line integral and got a result of -Pi

Now I'm stuck on the flux integral. I've worked out that Curl F = -2i -zj -k but I'm not sure what the normal vector is or the limits.

Any help much appreciated!

2. The surface is the portion of the hemi -sphere x^2 +y^2 +z^2 =25

above z= 4

z= sqrt(25-x^2 -y^2)

N = -dz/dx i -dz/dy j + K

N = x/sqrt(25-x^2) i + y /sqrt(25-y^2) j + k

The region of integration is the circle x^2 + y^2 = 9

(using z=4)

See the attachment for the calculation of the line integral using both Stokes Theorem and the definition of line integral

3. Gah so my line integral result is incorrect?

4. you're computing the line integral around the circle of radius 3 , 4units

above the x-y plane.

How'd you compute the line integral ?

What was your parameterization and F*dr/dt ?

My parameterization was r(t) = cos(t)i +sin(t)j +4k for 0<t<2Pi

F.dr = -sin^2(t) + 8cos(t)

6. x^2 + y^2 +z^2 = 25

if z = 4 x^2 + y^2 = 9

Which gives x=3cos(t) y = 3sin(t) z = 4

r ' = -3sin(t) i +3cos(t) j

F = 3sin(t) i - 4 j + 12cos(t) k

As in the attachment. you then end up with -9pi which is the - 28.274 in the attachment

7. How did you go about doing the first integral, ie "we obtain using the parameterization". In the word document i meant

8. The first integral is the integral curl F*N

Then using the pararmaterization for the bounding curve we obtain the second integral of F*dr/dt