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Vector field flow problem - help?
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Daniel Nierro
science forum beginner


Joined: 19 Jul 2006
Posts: 2

PostPosted: Wed Jul 19, 2006 10:28 am    Post subject: Vector field flow problem - help? Reply with quote

Hi!
I'm struggling with some calculations of what the flow of a vector field
through a surface is.
The vector field is F(x,y,z) = (e^(x-y-z) + y, e^(x-y-z) - z, x)
and the surface (S) is the part of the paraboloid z = x^2 + y^2 where z <=
x - y + 2.
(The surface S is oriented so it's normal points away from the z-axis.)

I think I know how to attack this problem; by completing the surface S with
a "lid" (L) so it forms a closed object (O), and then the value of the
sought value of the surface integral becomes equal to the triple integral
over the solid closed O object minus the surface integral over the lid L.
I think the proper lid L is z = x - y + 2, but then I get kind of stuck in
my calculations after a while. Sad
Am I attacking this in the wrong way?
I've been struggeling with this for quite some time now and becoming a bit
desperate, does anyone know what the solution of this is?

Very grateful for any feedback,
Daniel Nierro
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Lynn Kurtz
science forum Guru


Joined: 02 May 2005
Posts: 603

PostPosted: Wed Jul 19, 2006 6:12 pm    Post subject: Re: Vector field flow problem - help? Reply with quote

On Wed, 19 Jul 2006 12:28:45 +0200, "Daniel Nierro"
<dani4965@student.uu.se> wrote:

Quote:
Hi!
I'm struggling with some calculations of what the flow of a vector field
through a surface is.
The vector field is F(x,y,z) = (e^(x-y-z) + y, e^(x-y-z) - z, x)
and the surface (S) is the part of the paraboloid z = x^2 + y^2 where z <=
x - y + 2.
(The surface S is oriented so it's normal points away from the z-axis.)

I think I know how to attack this problem; by completing the surface S with
a "lid" (L) so it forms a closed object (O), and then the value of the
sought value of the surface integral becomes equal to the triple integral
over the solid closed O object minus the surface integral over the lid L.
I think the proper lid L is z = x - y + 2, but then I get kind of stuck in
my calculations after a while. Sad
Am I attacking this in the wrong way?
I've been struggeling with this for quite some time now and becoming a bit
desperate, does anyone know what the solution of this is?

Very grateful for any feedback,
Daniel Nierro


Let's use n for the unit normal directed outwards from the volume. So
you want the surface integral Int_S F dot n dS and you have observed

(Int_S + Int_L) F dot dS = Int_O del dot F dV

You have probably noticed that del dot F is zero so if you can
calculate the integral over the lid: Int_L F dot dS, you are home
free. So far, so good, and no, I don't think you are attacking the
problem the wrong way.

Expressing your lid parametrically using x and y as the parameters:

R(x,y) = < x, y, x - y + 2 >
R_x = < 1, 0 1 > and R_y = < 0, 1, -1 >
R_x cross R_y = < -1, 1, 1 >

so your ndS vector becomes < -1, 1, 1 > dy dx.

The surface integral over your lid is:

int_L < exp( x - y - z) + y, exp( x - y - z) - z, x > dot < -1, 1, 1 >
dy dx

What you need to do now is substitute z = x - y + 2 to get everything
in terms of x and y, simplify, and get the appropriate limits for the
intersection of the plane and the paraboloid. I haven't worked the
rest of the details out. You may find a switch to polar coordinates
helpful.

--Lynn
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