Re: Integrating a Circular Shaped Loop in a Vector Field

[Juan Pablo Carbajal]

On Sat, Dec 14, 2013 at 1:16 AM, astaton <address@hidden> wrote:
> Sir..sorry for not quickly responding,
>
> It is a line integral in the shape of a circle circumference.
>
> I found the below code snippet that uses the trapz function to numerically
> integrate a vector:
>
> x=0:pi/357:pi;
> y=??????
> area=pi/357*trapz(y);
>
> The problem is I do not have a function to integrate..per say.  It is a
> vector field of 3 dimensions with only values and no function.
>
> I suppose I could simplify and only integrate the 'z' component?  Maybe
> B_field= |Z|*sin (theta)?  That creates another problem because I still need
> to vary the circle radius?
>
> Any help is still greatly appreciated.
>
> -AS
>
>
>
> --
> View this message in context: 
> http://octave.1599824.n4.nabble.com/Integrating-a-Circular-Shaped-Loop-in-a-Vector-Field-tp4660038p4660058.html
> Sent from the Octave - General mailing list archive at Nabble.com.
> _______________________________________________
> Help-octave mailing list
> address@hidden
> https://mailman.cae.wisc.edu/listinfo/help-octave

I understand then that you vector field is data based and you do not
know its mathematical formulation.
Question:
a. Is your vector sampled on a uniform 3d mesh?

If yes, can:
1. Interpolate the vector field to get values that lay in your curve,
e.g. using trilinear interpolation[1] (I have tried to use interp3
many times, but it isn't that useful), that is find the cube that
encloses each point on your curve and interpolate the vector to that
point.
2. Calculate the nodal dot product between interpolated vector field
and tangent of your curve, this gives you samples of a scalar
function: the integrand.
3. Parametrize your curve with the length of arc in [0,1].
4. Find the interpolating polynomial for the integrand, e.g. using
interp1 or data2fun in signal package.
5. Use any scalar integration formula to find the integral along your
curve, e.g. *quad or trpz or whatever.

The validity of this approach is highly dependent on the nature of
your vector field and your curve, if everything is smooth or at least
continuous, I think the results should be reasonably good. Do check it
against an exact result (i.e. write your own mathematical toy problem
and solve it exactly).

[1] http://en.wikipedia.org/wiki/Trilinear_interpolation