Consider a twodimensional triangle T sitting in R3 with vert

Consider a two-dimensional triangle T sitting in R^3 with vertices p, p + a, and p + b. In class we found that the surface integral of a constant scalar field f(x, y, z) = c over T was c//a times b//. Suppose instead we wanted to find the surface integral of a constant vector field F(x, y, z) = c over T. Show that this is equal to 1/2 c.a times b (or the negative of that, depending on the orientation).

Solution

Follows from the result stated for scalar field : Set c = c[1]i+c[2]j+c[3]k , where c[1],c[2],c[3] are scalars.

Then the surface integral is 1/2c.(axb), by linearity