Parametric Surfaces & Surface Area

Parametrize r(u,v) and integrate | r_u × r_v| for surface area.

The idea

A parametric surface is the image of a function of two variables, $\mathbf{r}(u,v) = \langle x(u,v),\ y(u,v),\ z(u,v) \rangle$, as $(u,v)$ ranges over a flat region $D$ — just as a curve is the image of a function of one variable, $\mathbf{r}(t)$. Holding $v$ fixed and varying $u$ traces one curve across the surface; holding $u$ fixed and varying $v$ traces another.

To measure the surface's area, follow a small rectangle in $D$ with sides $du$ and $dv$. Increasing $u$ by $du$ moves the image point by about $\mathbf{r}_{u}\,du$, and increasing $v$ by $dv$ moves it by about $\mathbf{r}_{v}\,dv$, where $\mathbf{r}_{u} = \partial \mathbf{r}/\partial u$ and $\mathbf{r}_{v} = \partial \mathbf{r}/\partial v$ are tangent to the surface. The image of the rectangle is therefore a small parallelogram with those two edges, and its area is the length of their cross product, $|\mathbf{r}_{u} \times \mathbf{r}_{v}|\,du\,dv$. Adding the patches gives the area of the whole surface: $A = \iint_{D} |\mathbf{r}_{u} \times \mathbf{r}_{v}| \, dA.$

The factor $|\mathbf{r}_{u} \times \mathbf{r}_{v}|$ measures how much a unit of area in the parameter region stretches on its way onto the surface — the same role the Jacobian plays in a change of variables.

Ways to work on it

• Walkthrough. Tangent vectors, the normal, and the area element on a cylinder.
• Practice. Surface area of a plane graph via √1 + f_x^2 + f_y^2.
• Hardest. Sphere area from the parametrization, where the area element varies.

Not sure where to start? Take the ten-question placement test.