Timeline for Geometric imagination of differential forms
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 29, 2015 at 22:50 | comment | added | Richard Montgomery | See also W Burke's `Applied Differential Geometry' for wonderful pictures. Burke's pictures are basically carefully thought out, clarified, pared-down versions of those in Misner-Thorne-Wheeler. | |
| Sep 21, 2013 at 15:00 | comment | added | Tim Campion | Can't say fairer! | |
| May 8, 2013 at 3:36 | comment | added | Deane Yang | Tim, you make some good points. I shouldn't have said just "speed". It's a "directional speed" but in the following sense: Imagine a bunch of evenly spaced ordered set of parallel planes in space and an object traveling through the planes. The orientation of the planes represents the "direction" in an affine invariant way (i.e., without using the notion of angle or the inner product). Then you can measure how quickly the object passes through the parallel planes. And since the parallel planes are ordered, you can also assign a sign to this speed. | |
| May 8, 2013 at 2:22 | comment | added | Tim Campion | Moreover, the most important subclass of 1-forms are the closed ones, whose integrals along curves are locally path-independent. If you had a notion of "speed" with this property, then its integral should be a distance. Then a "closed speed" gives you distances which are locally path-independent. Very weird. | |
| May 8, 2013 at 2:21 | comment | added | Tim Campion | A 1-form certainly assigns a number to a vector, but to me it seems kind of misleading to think of this number as the "speed" of the vector because "speed" shouldn't be linear in the vector. Sure, the speed should scale linearly with the vector. But it shouldn't be additive the way a 1-form is. Two vectors pointing in opposite directions should have the same speed, not the opposite speed. A "speed" should not have a codimension-1 hyperplane of vectors in its kernel. And so on. | |
| Jun 4, 2010 at 22:22 | comment | added | Deane Yang | Well, the 2-dimensional analogue of a vector field is a field of infinitesimal parallelograms. So a 2-form is an instrument for measuring the area of such an infinitesimal parallelogram. | |
| Jun 4, 2010 at 19:04 | comment | added | Mircea | very interesting explanation.. so a 2-form is [something giving] a coordinate-free way to measure the speed of "diffusion of particles" along a [given] 2-plane distribution, right? | |
| Jun 4, 2010 at 18:56 | vote | accept | Mircea | ||
| Jun 4, 2010 at 18:56 | |||||
| Jun 3, 2010 at 17:49 | history | answered | Deane Yang | CC BY-SA 2.5 |