Surface area calculation

Let's say you are given a surface parametrized by $\Phi$(u, v) for (u, v) in some region D. The function $\Phi$ 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 $\Phi$(u, v).

The function $\Phi$ maps the small rectangle show in green (on the left) to the "curvy rectangle" floating in space (on the right).

Let $\Delta$u and $\Delta$v be width and height of the small rectangle. If the lower-left corner of the rectangle is at position (u, v), then $\Phi$ maps the lower-left corner to the point $\Phi$(u, v) on the "curvy rectangle."

The lower-right corner of the rectangle is the point (u + $\Delta$u, v), so it gets mapped to $\Phi$(u + $\Delta$u, v). Similarly, the upper-left corner of the rectangle is mapped to $\Phi$(u, v + $\Delta$v).

What is area of the "curvy rectangle" (the image of the small rectangle under $\Phi$)?

Approximate it as a parallelogram (okay for $\Delta$u and $\Delta$v small).

One side of the parallalogram is

$$\Phi \Delta \Phi {\frac{{\mathbf{\Phi}(u+\Delta u,v)-\mathbf{\Phi}(u,v)}}{{\Delta u}}} \Delta$$

$$\approx {\frac{{\partial \mathbf{\Phi}}}{{\partial u}}} \Delta$$

when $\Delta$u is small. Similarly, when $\Delta$v is small, the other side of the parallelogram is approximatly

$$\Phi \Delta \Phi \approx {\frac{{\partial \mathbf{\Phi}}}{{\partial v}}} \Delta$$

The area of a parallegram spanned by these vectors is the magnitude of their cross product. The area of the "curvy rectangle" is approximately

$$\left\Vert\vphantom{ \frac{\partial \mathbf{\Phi}}{\partial u}(u,v) \times \frac{\partial \mathbf{\Phi}}{\partial v}(u,v) }\right. {\frac{{\partial \mathbf{\Phi}}}{{\partial u}}} {\frac{{\partial \mathbf{\Phi}}}{{\partial v}}} \left.\vphantom{ \frac{\partial \mathbf{\Phi}}{\partial u}(u,v) \times \frac{\partial \mathbf{\Phi}}{\partial v}(u,v) }\right\Vert \Delta \Delta$$

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 $\Delta$u,$\Delta$v$\to$ 0, then the Riemann sum converges to a double integral, and we find that the total surface area is

$$\iint_{D}^{} \left\Vert\vphantom{ \frac{\partial \mathbf{\Phi}}{\partial u}(u,v) \times \frac{\partial \mathbf{\Phi}}{\partial v}(u,v) }\right. {\frac{{\partial \mathbf{\Phi}}}{{\partial u}}} {\frac{{\partial \mathbf{\Phi}}}{{\partial v}}} \left.\vphantom{ \frac{\partial \mathbf{\Phi}}{\partial u}(u,v) \times \frac{\partial \mathbf{\Phi}}{\partial v}(u,v) }\right\Vert$$

as promised.