Multivariable calculus and vector analysis

Surface area calculation

Let’s say you are given a surface parametrized by $\mathbf{\Phi }\left(u,v\right)$ for $\left(u,v\right)$ in some region $D$. The function $\mathbf{\Phi }$ 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 $\mathbf{\Phi }\left(u,v\right)$.

The function $\mathbf{\Phi }$ 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 $\left(u,v\right)$, then $\mathbf{\Phi }$ maps the lower-left corner to the point $\mathbf{\Phi }\left(u,v\right)$ on the “curvy rectangle.”

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

What is area of the “curvy rectangle” (the image of the small rectangle under $\mathbf{\Phi }$)?

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

One side of the parallalogram is

$$\begin{array}{llll}\hfill \mathbf{\Phi }\left(u+\Delta u,v\right)-\mathbf{\Phi }\left(u,v\right)& =\frac{\mathbf{\Phi }\left(u+\Delta u,v\right)-\mathbf{\Phi }\left(u,v\right)}{\Delta u}\Delta u& \hfill & \\ \hfill & \approx \frac{\partial \mathbf{\Phi }}{\partial u}\left(u,v\right)\Delta u& \hfill & \end{array}$$

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

$$\begin{array}{lll}\hfill \mathbf{\Phi }\left(u,v+\Delta v\right)-\mathbf{\Phi }\left(u,v\right)\approx \frac{\partial \mathbf{\Phi }}{\partial v}\left(u,v\right)\Delta v.& & \hfill \end{array}$$

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

$$\begin{array}{lll}\hfill ∥\frac{\partial \mathbf{\Phi }}{\partial u}\left(u,v\right)\times \frac{\partial \mathbf{\Phi }}{\partial v}\left(u,v\right)∥\Delta u\Delta v& & \hfill \end{array}$$

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

$$\begin{array}{lll}\hfill A={\iint }_D∥\frac{\partial \mathbf{\Phi }}{\partial u}\left(u,v\right)\times \frac{\partial \mathbf{\Phi }}{\partial v}\left(u,v\right)∥dudv& & \hfill \end{array}$$

as promised.

Contents by topic

1. Geometry in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

4a. Vector field basics

4b. Parametrized curves

4c. Parametrized surfaces

• Parametrized surfaces
• Parametrized surface examples
• Surface area of parametrized surfaces
• Surface area calculation
• Normal vector of parametrized surfaces
• Orienting surfaces
• A surface that is not orientable
• Surface area example

4d. Path integrals and line integrals

4e. Surface integrals

5. The fundamental theorems of vector calculus

6. Review material