Surface Integrals (click here for discussion of surface integrals with oriented surfaces)
In a Nut Shell: Two questions are relevant. What is a surface integral and how do
you evaluate a surface integral?
Recall the line integral, I, provides the value of a function, f(x,y,z), evaluated along
a curve, C, in space. Here the line integral is I = ∫ f(x, y, z) ds
where ds is the arc length along the curve, C. The surface integral, Is, is analogous
to the line integral in that it provides the value of a function, f(x,y,z), evaluated over a
“smooth” surface, S, in space. Here the surface integral is Is = ∫ ∫ f(x, y, z) dS
where dS is the element of surface area on the spatial surface.
How do you evaluate a surface integral?
Usually the surface, S, in space is somewhat complicated. So it is usually convenient
to transform the element of surface, dS, from the x-y-z space into a parallelogram
dA = du dv in the u-v plane.
where r(u,v) is the parametric representation of the surface, S.
r(u,v) = < x(u,v), y(u,v), z(u,v) > is the position vector to point on surface, S
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