Timeline for Why is it important that partial derivatives commute?
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| Feb 23, 2013 at 16:52 | comment | added | Deane Yang | I have two responses: Designing an intrinsic geometric structure modeled on submanifolds of Euclidean space does not mean we are restricting our attention to submanifolds of Euclidean space, notwithstanding the Nash isometric embedding theorem. Nor does it mean that the geometric structure is not co-ordinate-free. The study of Riemannian geometry does not depend on co-ordinates. Of course, even the study of submanifolds in Euclidean space does not, too. | |
| Feb 23, 2013 at 16:09 | comment | added | horse with no name | Maybe it's circular in this setting, but there is Nash's embedding theorem. | |
| Feb 23, 2013 at 15:59 | comment | added | R S | Thanks. This makes sense, although in a way it kind of cancels the entire point of differential geometry as I see it so far (which is not a lot): I thought the motivation is to define calculus again in an intrinsic, coordinate-free way on general smooth manifolds; specifically, since we believe our universe is modeled well by one. Reducing it all back to the Euclidian setting when we stumble along some complication does not seem to follow the same spirit. Perhaps this means we should think our universe is some complicated embedding in a larger Euclidian space, unreachable to us? | |
| Feb 23, 2013 at 15:51 | history | answered | Deane Yang | CC BY-SA 3.0 |