105 - Surface Integrals

When we have a vector that maps two variables into a surface in three dimensions, where and form , the parameter space of this surface, we can compute the area of this surface by using a double integral known as the Surface Integral, which is calculated as follows

1 - Divide Parameter Surface T into infinitesimal pieces to obtain parallelogram, each base of the parallelogram is the partial derivative with respect to that parameter

Infinitesimal steps along both parameters yield movement following a parallelogram on the surface.

2 - Find the area of the parallelogram defined as the magnitude of the cross product of the two bases

An infinitesimal step along this surface we're integrating for then is determined as

3 - Integrate to sum all these infinitesimal steps

We end up with the double integral over the parameter space for this infinitesimal step along the surface, and we have, finally
Which is our surface integral.

WARNING

Calculating surface integrals can get very complicated. It's usually mostly done by computers.

106 - Flux in Three Dimensions