Surface Integrals

Surface Area

There are two types of surface integrals: those that involve scalars and those that involve vectors, known as flux. When doing scalars, this is equivalent to finding the mass of a 2D surface. Refer back to my surface area post.

Back when we did surface area, we took the sum of the local plane at each point. Now, imagine each point has a different density, we are finding that density * the area, to find the mass at the point. The surface integral, or mass of the surface equals: $$\iint f(r(u,v))|r_u \times r_v|\,du\, dv$$

Flux

Imagine a storm grate  in a lake. The lake has currents, as the water isn't perfectly still. The flux measures how much water flows through the cage per unit time. A positive flux means that water is flowing out (it's a source), while a negative flux means that water is flowing in (it's a sink).

Now, imagine that the current goes parallel to the surface of the cage. The flux must be 0, because the current does not go into the region. So, if the current is perpendicular to the normal, the flux must be 0. Also, if the current goes directly in or out, it should have the greatest magnitude. The ideal function for this is the dot product!

$$\Phi (\text{ Flux}) = \iint F \cdot n \, dS$$ The normal is the cross product of two tangents. $$= \iint F \cdot \frac{r_u \times r_v}{|r_u \times r_v|} \, dS$$ dS is equal to $|r_u \times r_v|dudv$, so the integral just equals $$\iint F \cdot (r_u \times r_v)\, dS$$