Using Green's theorem to Reduce a Surface Integration to a Path Integral

Suppose there is a uniformly charged surface (σ≡charge/area=1) occupying an equilaterial triangle in the z=0 plane:

The electrical potential goes like , therefore the potential of a unit charge located at (x,y,z) from a small surface patch at (ξ,η,0) is =

Try to convert the integral over the region to an integral over its boundary. We may be able to find a closed form—and even if not the numerical integration should be much more efficient...

We will use Green's Theorem in the Plane:

∫∫(-)dxdy = ∫(dx +dy)

Find a function, that when we take the derivative with respect to x, we get 1/r.

We will let be the function we find, and set =0

To use Green's theorem in the plane =

Try to find a vector function (, ) that matches the integrand over the domain. Let = 0, then to find which when differentiated by η gives 1/r

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The third (horizontal) boundary of the triangle patch looks like the easiest, let's see if an integral can be found over that patch:

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The following attempt to find a closed-form integral seems to take "forever" to evaluate and I ended up aborting it, then continuing to evaluate the remainder of the notebook. This command has been commented out to avoid evaluation

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The following attempt to find a closed-form integral seems to take "forever" to evaluate and I ended up aborting it, then continuing to evaluate the remainder of the notebook. This command has been commented out to avoid evaluation

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Created by Wolfram Mathematica 6.0 (28 September 2007) |