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
(The third component of these can best be seen by thinking of the equation of a line in point-slope form: y-y0 = m (x-x0). 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
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!