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Sarah B. science forum beginner
Joined: 15 Dec 2005
Posts: 2

Posted: Thu Jul 20, 2006 3:15 pm Post subject:
Help with weird integral?



Hi boys & girls!
Is there anyone out there who is good at integrals of vector fields? I've
been struggeling with a hard one, or is it maybe just me that approaches it
in the wrong way?
I'm trying to calculate this curve integral: [integral_over_C] F [dot] dr
of the vector field F(x,y,z) = (e^(x^2), x^2  (1/2)z^2, xy)
where the curve C is the intersection of the surfaces z = x^2 + y^2  x  4y
and z = x^2  y^2 + 3x + 4 oriented anticlockwise seen from a point high
up at the zaxis. I know that the curve C is in a plane, so that helps a
bit.
I think that the projection of the curve C onto the xyplane is a circle
with radius 2 and center in (1,1).
But then I get stuck. I've gotten many different results from different
methods, but I don't think they're correct... I'm getting slightly
desperate, is there anyone that knows the correct value that could help me?
Please...?
Sarah 

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

Posted: Thu Jul 20, 2006 8:43 pm Post subject:
Re: Help with weird integral?



On Thu, 20 Jul 2006 17:15:54 +0200, "Sarah B."
<strangelove@dekadance.se> wrote:
Quote:  Hi boys & girls!
Is there anyone out there who is good at integrals of vector fields? I've
been struggeling with a hard one, or is it maybe just me that approaches it
in the wrong way?
I'm trying to calculate this curve integral: [integral_over_C] F [dot] dr
of the vector field F(x,y,z) = (e^(x^2), x^2  (1/2)z^2, xy)
where the curve C is the intersection of the surfaces z = x^2 + y^2  x  4y
and z = x^2  y^2 + 3x + 4 oriented anticlockwise seen from a point high
up at the zaxis. I know that the curve C is in a plane, so that helps a
bit.
I think that the projection of the curve C onto the xyplane is a circle
with radius 2 and center in (1,1).
But then I get stuck. I've gotten many different results from different
methods, but I don't think they're correct... I'm getting slightly
desperate, is there anyone that knows the correct value that could help me?
Please...?
Sarah

You are correct that the projection is that circle. Presumably you
have studied Stoke's theorem:
int_C F dot dR = int int_S nhat dot del cross F dS
where S is a surface bounded by the curve. So for example, you could
use z = x^2  y^2 + 3x + 4 as the surface. Calculate del cross F and
nhat dS for that surface (oriented properly) and do the surface
integral. You will discover the exp(x^2) term disappears. Once you
have it set up in xy you might wish to use polar coordinates
translated to the center of your circle to evaluate the integral.
Can you take it from there?
Lynn 

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

Posted: Fri Jul 21, 2006 12:54 am Post subject:
Re: Help with weird integral?



On Thu, 20 Jul 2006 20:43:35 GMT, "[Mr.] Lynn Kurtz"
<kurtzDELETETHIS@asu.edu> wrote:
Quote:  On Thu, 20 Jul 2006 17:15:54 +0200, "Sarah B."
strangelove@dekadance.se> wrote:
Hi boys & girls!
Is there anyone out there who is good at integrals of vector fields? I've
been struggeling with a hard one, or is it maybe just me that approaches it
in the wrong way?
I'm trying to calculate this curve integral: [integral_over_C] F [dot] dr
of the vector field F(x,y,z) = (e^(x^2), x^2  (1/2)z^2, xy)
where the curve C is the intersection of the surfaces z = x^2 + y^2  x  4y
and z = x^2  y^2 + 3x + 4 oriented anticlockwise seen from a point high
up at the zaxis. I know that the curve C is in a plane, so that helps a
bit.
I think that the projection of the curve C onto the xyplane is a circle
with radius 2 and center in (1,1).
But then I get stuck. I've gotten many different results from different
methods, but I don't think they're correct... I'm getting slightly
desperate, is there anyone that knows the correct value that could help me?
Please...?
Sarah
You are correct that the projection is that circle. Presumably you
have studied Stoke's theorem:
int_C F dot dR = int int_S nhat dot del cross F dS
where S is a surface bounded by the curve. So for example, you could
use z = x^2  y^2 + 3x + 4 as the surface.

For that matter, you could use the the plane that the intersection
curve is in for the surface too.
Lynn 

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