Don has asked us how we would explain to a student why we do not correctly
obtain the area of a sphere by adding the areas of what naively appear to
be infinitesimal triangles with vertices at a pole and infinitesimal bases
along the equator. This is a really good question as it concerns one of
the critical elements in setting up integrals which is, in turn, one of
the hallmarks abilities of a good physicist.
Clearly, as has already been discussed, the reason the "triangle method"
doesn't work is that the "triangles" aren't triangles. Nevertheless,
understanding this still leaves the original question unanswered.
I'd *like* to be able to say something like, "the effects of curvature
don't become negligible since the area is only singly infinitesimal." In
other words, relate the error to the fact that even the infinitesimal
triangle "samples" a large enough portion of the sphere's surface to
retain artifacts of its curvature. An argument like this would seem to
suffice for explaining why we can get the surface area by adding up the
Euclidean areas of flattened out, *doubly* infinitesimal patches even
though larger patches don't work. But then there is the fact that singly
infinitesimal latitudinal strips *do* work.
At any rate, I still have no particularly good answer to the original
question, but I do hope to stimulate more consideration of this
interesting and, I think, important question.
John
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A. John Mallinckrodt email: mallinckrodt@csupomona.edu
Professor of Physics voice: 909-869-4054
Cal Poly Pomona fax: 909-869-5090
Pomona, CA 91768 office: Building 8, Room 223
web: http://www.sci.csupomona.edu/~mallinckrodt/