Integral of Triangular SurfaceDate: 8/9/96 at 12:27:0 From: Anonymous Subject: Integral of Triangular Surface Is it possible to numerically integrate the following? Integral over S { 1/3 (x^3 i + y^3 j + 0 k) . n} dS where n is the unit normal to the surface which is available. The surface S is a triangle in a plane. Date: 9/1/96 at 0:36:8 From: Doctor Jerry Subject: Re: Integral of Triangular Surface It looks like the integral you have written is a surface integral in standard form. The vector n would be a constant since S is a triangle. I interpret the 1/3 as a scalar multiple of the vector (x^3,y^3,0). I think you don't need to do numerical integration. It would be quite easy to give a parametrization of the triangle and then do an easy exact integration. It appears, therefore, that your question is: how to parametrize the triangle S. Suppose a, b, and c are vectors specifying the vertices of the triangle. Then b-a and c-a are vectors from a and lying along the sides of the triangle. You could take n equal to b-a cross c-a, afterwards dividing by the length of the cross product. The parametrization of the triangle is r(u,v) = u(b-a) + v(c-a) where (u,v) varies over the triangle in the (u,v)-plane bounded by (0,0), (1,0), and (0,1). Is this understandable? Please write back if my comments aren't clear. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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