Surface Integrals

We can consider these much like we did line integrals. Instead of integrating along a curve, we know integrate along a surface.

Let f(x,y,z) be a continuous vector field on a smooth surface
z=S(x,y), where (x,y) is in some plane region.

For line integral of g(x,y) over the curve r(t) we had
Integral of g(r(t)).dr, where dr is a portion of arc length.

For our surface integral, we have Integral over the region of f(x,y,S(x,y))dsigma
where dsigma is a portion of surface area, that is,
dsigma=Sqrt[1+(DxS)^2+(DyS)^2] dx dy

Note: some texts use dS or dA. We will use dsigma as above or dS for short.

Here capital sigma is a piecewise smooth surface z=S(x,y) with (x,y) in D.

Example 1

Example 2

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