Calculation of the surface area of a parametrized surface

A parametrized surface is a mapping by a function $\dlsp: \R^2 \to \R^3$ (confused?) of a planar region $\dlr$ onto a surface floating in three dimensions. To calculate the area of this surface, we chop up the region $\dlr$ into small rectangles, as displayed below for the function \begin{align*} \dlsp(\spfv,\spsv) = (\spfv\cos \spsv, \spfv\sin \spsv, \spsv). \end{align*}.

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A parametrized helicoid with surface area elements. The function $\dlsp(\spfv,\spsv) = (\spfv\cos \spsv, \spfv\sin \spsv, \spsv)$ parametrizes a helicoid when $(\spfv,\spsv) \in \dlr$, where $\dlr$ is the rectangle $[0,1] \times [0, 2\pi]$. The region $\dlr$ is shown as the green rectangle floating above the helicoid, and it is divided into small rectangles. As you drag the green point in $\dlr$ to specify both $\spfv$ and $\spsv$, a surrounding small rectangle is outlined in green. This rectangle is mapped into the small region of the helicoid that is outlined in red and surrounds the red point $\dlsp(\spfv,\spsv)$. The area of the red region on the helicoid depends on how $\dlsp$ stretches or shrinks the small green rectangle as it maps it on the surface.

The function $\dlsp$ maps each small rectangle to a “curvy rectangle” on the surface. Here's another view of this mapping, where you have to imagine that the blue surface on the right is floating in space.

Let $\Delta \spfv$ and $\Delta \spsv$ be width and height of the small rectangle. If the lower-left corner of the rectangle is at position $(\spfv,\spsv)$, then $\dlsp$ maps the lower-left corner to the point $\dlsp(\spfv,\spsv)$ on the “curvy rectangle.”

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

To calculate the area of the “curvy rectangle”, i.e., the image of the small rectangle under $\dlsp$, we approximate it as a parallelogram. This approximation is okay when $\Delta \spfv$ and $\Delta \spsv$ small.

One side of the parallalogram is \begin{align*} \dlsp(\spfv+\Delta \spfv,\spsv)-\dlsp(\spfv,\spsv) &= \frac{\dlsp(\spfv+\Delta \spfv,\spsv)-\dlsp(\spfv,\spsv)}{\Delta \spfv} \Delta \spfv\\ &\approx\pdiff{\dlsp}{\spfv}(\spfv,\spsv)\Delta \spfv \end{align*} when $\Delta \spfv$ is small. Similarly, when $\Delta \spsv$ is small, the other side of the parallelogram is approximatly \begin{align*} \dlsp(\spfv,\spsv+\Delta \spsv) - \dlsp(\spfv,\spsv) \approx \pdiff{\dlsp}{\spsv}(\spfv,\spsv)\Delta \spsv. \end{align*}

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{align*} \Delta A=\left\|\pdiff{\dlsp}{\spfv}(\spfv,\spsv) \times \pdiff{\dlsp}{\spsv}(\spfv,\spsv)\right\| \Delta \spfv \Delta \spsv. \end{align*}

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 $\Delta \spfv, \Delta \spsv \to 0$, then the Riemann sum converges to a double integral, and we find that the total surface area is \begin{align*} A =\left.\iint_\dlr \left\| \pdiff{\dlsp}{\spfv}(\spfv,\spsv) \times \pdiff{\dlsp}{\spsv}(\spfv,\spsv) \right\|\right. d\spfv\,d\spsv \end{align*} as promised.

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