Timeline for How do I make the conceptual transition from multivariable calculus to differential forms?
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| Jan 5, 2010 at 15:07 | comment | added | Deane Yang | Heck, a lot of my research was about or on manifolds, and I rarely used differential forms. My thesis was actually about exterior differential systems (systems of equations defined by exterior differential forms), and even there I used very little of the formalism of differential forms! Use differential forms only if the formalism makes your life easier and not harder. Also, for me a lot of things on $R^n$ make a lot more sense and are much easier to work with, when I see that they do not require the use of a global co-ordinate system or inner product. | |
| Jan 5, 2010 at 8:03 | comment | added | Joel Fine | It's true that there is a natural coordinate system on R<sup>n</sup>, but you might well want to use a different one depending on the situation, e.g., polar coordinates for a spherically symmmetric problem. When you change coordinates there is a formula telling you how integrals behave. One way to think of "coordinate independence" of differential forms is just that the way differential forms change under change of coordinates neatly encodes the behaviour of integrals. | |
| Jan 5, 2010 at 3:25 | comment | added | David Corwin | The main thing I've never understood about differential forms is their "coordinate independence." I know what the general definition of a differentiable manifold is, so I see why it might be interesting there (topology might obstruct the existence of a global coordinate patch), but I don't understand why differential forms are taught for R^n. Since in R^n there is a natural inner product (by which you mean on the tangent space?) and coordinate system, why do we care? In what way are they "coordinate independent"? Is the coordinate change just if you want to change to polar or something? | |
| Jan 4, 2010 at 14:04 | history | edited | Deane Yang | CC BY-SA 2.5 |
added 580 characters in body
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| Jan 3, 2010 at 13:52 | history | answered | Deane Yang | CC BY-SA 2.5 |