Surface area calculation
Let’s say you are given a surface parametrized by Φ(u,v) for (u,v) in some region D. The function Φ maps D onto the surface.
As discussed in the previous reading, we can chop up the region D into small rectangles and find the image of one small rectangle under Φ(u,v).
The function Φ maps the small rectangle show in green (on the left) to the “curvy rectangle” floating in space (on the right).
Let Δu and Δv be width and height of the small rectangle. If the lower-left corner of the rectangle is at position (u,v), then Φ maps the lower-left corner to the point Φ(u,v) on the “curvy rectangle.”
The lower-right corner of the rectangle is the point (u + Δu,v), so it gets mapped to Φ(u + Δu,v). Similarly, the upper-left corner of the rectangle is mapped to Φ(u,v + Δv).
What is area of the “curvy rectangle” (the image of the small rectangle under Φ)?
Approximate it as a parallelogram (okay for Δu and Δv small).
One side of the parallalogram is
| Φ(u + Δu,v) - Φ(u,v) | =  Δu | ||
≈ (u,v)Δu |
Φ(u,v + Δv) - Φ(u,v) ≈ (u,v)Δv. |
The area of a parallegram spanned by these vectors is the magnitude of their cross product. The area of the “curvy rectangle” is approximately
 ΔuΔv |
The area of the whole surface is a Riemann sum over all the small rectangles. Each term is of the above form. If we take the limits Δu, Δv → 0, then the Riemann sum converges to a double integral, and we find that the total surface area is
A =  dudv |
