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2d
comment Metric equivalence
I assume this is for Riemannian metrics? For the semi-Riemannian case this is known to be quite tough in general. Given a metric expressed in a certain set of coordinates, it's nontrivial to answer even certain very basic questions about its coordinate-independent characteristics. E.g., you might think to test using curvature invariants, but in >=3 dimensions and semi-Riemannian signature, curved spaces exist such that all the curvature invariants vanish.
Feb
8
comment Differential geometry
How about writing a more descriptive title?
Feb
7
comment What is a foliation and why should I care?
In general relativity you see foliations over and over as ways of discussing Cauchy problems and the time evolution of systems. These are foliations using spacelike surfaces. Because GR doesn't have a preferred time coordinate or a preferred notion of simultaneity, a foliation in spacelike surfaces is the next best thing.
Jan
18
comment What is the exterior derivative intuitively?
The visualizations you refer to in Misner, Thorne, and Wheeler were originated by J.A. Schouten, and first presented in Ricci-Calculus: An Introduction to Tensor Analysis and its Geometrical Applications amazon.com/… . William L. Burke has given more concise and modern presentations, e.g., in Applied Differential Geometry.
Jan
8
comment Does a classical wave detect compact dimensions?
The question is about classical physics. When you refer to momentum, is that really interchangeable with wavenumber (inverse wavelength)?
Jan
8
comment What is the modern consensus on the difficulty of infinitesimals?
The motivation for the thread on math.SE seems to have been pedagogical: if one is teaching freshman calculus using an approach similar to Keisler's, how does one describe this issue to the students? In that context, I think the answer is clear. The students are learning a body of practices for manipulating infinitesimals, and these practices have been standardized and in use by scientists and engineers without interruption since Leibniz and Newton. In that body of standard practices, we never distinguish an individual infinitesimal.
Jan
6
awarded  Necromancer
Dec
30
awarded  Necromancer
Dec
27
comment Algorithmic complexity of formal proof verification?
The interpretation of this would seem to me to be that real-world proof assistants don't have the right theoretical properties to be good embodiments of whatever we mean theoretically by computationally effective proof. Similarly, a real-world computer doesn't have the right theoretical properties to be a good embodiment of a Turing machine. I could fill a room with monkeys banging on typewriters and set them the task of proving the Riemann hypothesis, with their output checked by grad students. This would be a proof assistant, albeit one with worse than polynomial performance.
Dec
23
comment Do mathematical objects disappear?
@Michael: I don't think we're in disagreement. Nothing in your comment contradicts anything in my answer. Quaternions used to be used for what we would nowadays do using div, grad, and curl. Nobody uses quaternions for that task anymore.
Dec
23
comment Do mathematical objects disappear?
@CarlMummert: Infinitesimals never disappeared from engineering and the sciences.
Dec
22
answered Do mathematical objects disappear?
Dec
22
comment Do mathematical objects disappear?
I don't think this is an accurate depiction of the history. Newton's notation and Leibniz's notation were isomorphic to each other. Newton's $o$ was equivalent to Leibniz's $dt$. (Newton had a convention about omitting the $o$ in certain contexts, which confused things somewhat. See Boyer, p. 201 ) Also, Leibniz did not wholeheartedly endorse the idea that his notation was a notation for infinitesimals. He sometimes preferred to describe differentials in the same way we would today describe differential forms (Boyer, p. 210). Leibniz notation won because it was more expressive.
Dec
22
comment Do mathematical objects disappear?
This would be better for hsm.stackexchange.com .
Dec
21
comment Find the axis of symmetry in a point cloud
I would try computing the quadrupole moment tensor Q and then diagonalizing it. Any basis in which Q is diagonal is likely to coincide with the symmetry axes of the cloud.
Dec
14
comment A “better” rational approximation of pi?
This doesn't answer the question.
Dec
11
comment “You can't push a rope”
It doesn't matter whether it's doing work or not. The idea is simply that a rope can sustain tension but not compression.
Nov
22
comment Location of Archimedes' grave in Syracuse (math/archaelogy trivia)
This would have been perfect for hsm.SE.
Nov
10
comment Intermediate value for a vector-valued function
I see. But if Jordan-Brouwer was proved in 1911, and this is the only deep issue involved, it still seems a little surprising that the Poincare-Miranda conjecture wasn't proved until 1940.
Nov
10
comment Intermediate value for a vector-valued function
@user78588: Hmm...OK, to be more explicit, how about this. If there is a continuous curve C connecting p to the origin, such that C does not intersect the image of the surface, then the sign is positive. (We don't need to appeal to an orientation of the image surface.)