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#Actualmax343

Posted 29 November 2012 - 08:15 AM

x and x0 are points in R^3 and not coordinates.
As for the second question:

F*dS is a two-form defined over the boundary of the volume, while <F,n>*dA is the corresponding two-form defined over subset of R^2.
F*dS = <F,n>*dA = (<x,n> - <x0,n>)/3*dA = (d - <x0,n>)/3*dA
Now, d - <x0,n>/3 is a constant within the triangle, so the integral is: A*(d - <x0,n>)/3.

I've been abusing notation here a bit. It makes little sense to do an inner product between 1-forms and vectors, but it should be fairly obvious what I meant in <~,n>.

Does this make more sense?

#1max343

Posted 29 November 2012 - 08:14 AM

x and x0 are points in R^3 and not coordinates. So the vector field is something lik
As for the second question:

F*dS is a two-form defined over the boundary of the volume, while <F,n>*dA is the corresponding two-form defined over subset of R^2.
F*dS = <F,n>*dA = (<x,n> - <x0,n>)/3*dA = (d - <x0,n>)/3*dA
Now, d - <x0,n>/3 is a constant within the triangle, so the integral is: A*(d - <x0,n>)/3.

I've been abusing notation here a bit. It makes little sense to do an inner product between 1-forms and vectors, but it should be fairly obvious what I meant in <~,n>.

Does this make more sense?

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