A recap of the derivation of the surface area element *dS*. Set
through the following animation to see the thought process behind
finding the surface area of a surface. Each step is explained in a
pop-up window:
Click here to open the pop-up
window for the current frame of the demonstration.

So, the key here is to find the vectors pointing along two sides of
the near-parallelogram *dS* and take their cross product.
Finding the vectors is shown in the steps given above, which arrive at
the vectors

and, similarly for

(The third component of these can best be seen by thinking of the
equation of a line in point-slope form:
*y*-*y*_{0} =
*m* (*x*-*x*_{0}).
So the change in *y* (in our case, *z*) is just the slope
times the change in *x* -- which is *dx*, as shown above.)

Then the area *dS* = |**a** x **b**|. The
cross product is

= sqrt( (

And, Voila!, we have the expression we want for *dS*!

*Note! Go back and be sure that you understand this. In
particular, make sure that the whole area-of-a-parallelogram thing
makes sense, make sure that you see where the vectors a and
b came from, do the cross-product, and make sure that you can
find the magnitude to figure out what dS is!*

surface areas

Last modified: Fri Mar 19 12:54:25 EST 2004

Comments to:glarose@umich.edu

©2004 Gavin LaRose, UM Math Dept.