# Flux integrals - verifying Stokes Theorem

• Aug 19th 2009, 07:38 PM
Zharac
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!
• Aug 19th 2009, 08:28 PM
Calculus26
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
• Aug 19th 2009, 09:41 PM
Zharac
Gah so my line integral result is incorrect?
• Aug 19th 2009, 10:02 PM
Calculus26
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 ?
• Aug 19th 2009, 10:24 PM
Zharac

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

F.dr = -sin^2(t) + 8cos(t)
• Aug 19th 2009, 10:30 PM
Calculus26
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
• Aug 19th 2009, 11:20 PM
Zharac
How did you go about doing the first integral, ie "we obtain using the parameterization". In the word document i meant
• Aug 20th 2009, 08:58 AM
Calculus26
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