Stokes and Divergence Theorem

[Baron]

I tried solving the question below in two ways, using Stokes and then using the Divergence Theorem. But I got different answers.

Q. Let \(\displaystyle S\) be the bucket shaped surface consisting of the cylindrical surface \(\displaystyle y^2\, +\, z^2\, =\, 9\) between \(\displaystyle x\, =\, 0\) and \(\displaystyle x\, =\, 5,\) and the disk inside \(\displaystyle yz\)-plane of radius \(\displaystyle 3\) centered at the origin. (The bucket \(\displaystyle S\) has a bottom, but no lid.) Orient \(\displaystyle S\) in such a way that the unit normal points outward. Compute the flux of the vector field \(\displaystyle \triangledown\, \times\, \vec{G}\) through \(\displaystyle S,\) where \(\displaystyle \vec{G}\, =\, \langle\, x,\, -z,\, y\, \rangle.\)

What is wrong?

1) Using Stokes

Take the capping surface as the disk at x = 5 with radius 3 centered on the x-axis. The curve is
counter-clockwise as the normal vector points outward. Normal vector is <1,0,0>

curl G = <2,0,0>

Flux is
double integral of curl G dot n dS = double integral of <2,0,0> dot <1,0,0< dS

= 2(area of disk) = 2(9pi) = 18pi


2) Using the divergence theorem

Close the boundary with the disk at x = 5 with radius 3 centered on x-axis. Call this surface S1
S2 is the surface we want. S1 is oriented outwards because of the divergence theorem, so its normal vector is <1,0,0>
S = S1 U S2 is closed

Flux through S = triple integral of div (curl G) dV = 0 as the div(curl G) = 0 for any "well-behaved" vector field

Flux through S1:
double integral of curl G dot n dS = double integral of <2,0,0> dot <1,0,0< dS

= 2(area of disk) = 2(9pi) = 18pi

Flux through S = flux through S1 + flux through S2
0 = 18pi + flux through S2

flux through S2 = -18pi

I got different answers. I think my orientation of one of the surfaces is wrong. Why?

 

[HallsofIvy]

The "divergence theorem" gives the integral of divergence of a vector function over a volume as the integral of the vector function itself over the surface bounding that volume. Here, you are trying to find the integral of the curl of a function over a surface so I don't see how the "divergence theorem" would apply.

Stokes theorem give the integral of the curl of a vector function over a surface as the integral of the function around the boundary of the surface. But you did not "use" Stokes theorem. You appear, rather, to have integrated the curl of the vector function, directly over the surface- but the wrong surface- instead of the "bucket" shaped surface, you have integrated over the non-existent open end of the bucket.

Here, the boundary is just the circle with radius 3 in the plane x= 5 with center at (5, 0, 0).
Parametric equations for that circle would be \(\displaystyle x= 5\), \(\displaystyle y= 5 cos(t)\), \(\displaystyle z= 5 sin(t)\). Integrate G around that circle.

 

[Baron]

The "divergence theorem" gives the integral of divergence of a vector function over a volume as the integral of the vector function itself over the surface bounding that volume. Here, you are trying to find the integral of the curl of a function over a surface so I don't see how the "divergence theorem" would apply.

I was just trying to see how many ways I could solve the problem and I thought it would be good practice to use the divergence theorem. So I tried bounding the surface with the disk at x = 5, finding the flux through that bounded surface using the divergence theorem, and then subtracting the flux through the disk at x = 5.

Stokes theorem give the integral of the curl of a vector function over a surface as the integral of the function around the boundary of the surface. But you did not "use" Stokes theorem. You appear, rather, to have integrated the curl of the vector function, directly over the surface- but the wrong surface- instead of the "bucket" shaped surface, you have integrated over the non-existent open end of the bucket.

Isn't another application of Stokes theorem that the surfaces (S1 and S2) which bound the same curve have the same flux flowing through the curl of the vector field?

i.e double integral S1 of curl F dot n dS = double integral S2 of curl F dot n dS

That was what I was trying to do by changing the boundary of the surface to the open end of the bucket

Here, the boundary is just the circle with radius 3 in the plane x= 5 with center at (5, 0, 0).
Parametric equations for that circle would be \(\displaystyle x= 5\), \(\displaystyle y= 5 cos(t)\), \(\displaystyle z= 5 sin(t)\). Integrate G around that circle.

Doing the integral as you suggested

r(t) = <5, 3cos(t), 3 sin (t) >
r'(t) = <0, -3 sin t, 3 cos t>

= integral of <x,-z,y> dot <0, -3 sin t, 3 cos t> dt from 0 to 2pi
= integral of <5,-3sin t,3 cos t> dot <0, -3 sin t, 3 cos t> dt from 0 to 2pi
= integral of 9 from 0 to 2pi
= 18pi

So that matches what I got originally using Stokes theorem. How come my application of divergence theorem didn't work?