Let's say you are given a surface parametrized by
(u, v) for
(u, v) in some region D. The function
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
(u, v).
The function
maps the small rectangle show in green (on the
left) to the "curvy rectangle" floating in space (on the right).
Let
u and
v be width and height of the small
rectangle. If the lower-left corner of the rectangle is at position
(u, v), then
maps the lower-left corner to the point
(u, v) on the "curvy rectangle."
The lower-right corner of the rectangle is the point
(u +
u, v),
so it gets mapped to
(u +
u, v). Similarly, the
upper-left corner of the rectangle is mapped to
(u, v +
v).
What is area of the "curvy rectangle" (the image of the small
rectangle under
)?
Approximate it as a parallelogram (okay for
u and
v small).
One side of the parallalogram is
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The area of a parallegram spanned by these vectors is the magnitude of their cross product. The area of the "curvy rectangle" is approximately
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
u,
v
0, then the Riemann sum converges
to a double integral, and we find that the total surface area is
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