Author 
Message 
Lynn Kurtz science forum Guru
Joined: 02 May 2005
Posts: 603

Posted: Wed Jul 19, 2006 6:12 pm Post subject:
Re: Vector field flow problem  help?



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^(xyz) + y, e^(xyz)  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 zaxis.)
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.
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 

Back to top 


Daniel Nierro science forum beginner
Joined: 19 Jul 2006
Posts: 2

Posted: Wed Jul 19, 2006 10:28 am Post subject:
Vector field flow problem  help?



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^(xyz) + y, e^(xyz)  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 zaxis.)
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.
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 

Back to top 


Google


Back to top 



The time now is Thu Jan 24, 2019 12:56 am  All times are GMT

