Timeline for How do I make the conceptual transition from multivariable calculus to differential forms?
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| Nov 5, 2013 at 14:46 | comment | added | Tim Campion | There are "things you can integrate" more general than differential forms, such as arc length or surface area. A fairly general notion of something you can integrate is a density in the sense of Gelfand. To amplify Harald's point on the importance of Stokes' Theorem, according to this MO question, imposing the linearity condition on a density is equivalent to asking that Stokes' Theorem holds. | |
| Jan 4, 2010 at 0:18 | comment | added | Ryan Budney | IMO the answer to your question as to why we choose forms to be linear goes back to a far simpler observation. That the determinant is the unique alternating multi-linear function on square matrices that takes value $1$ on the identity. This says alternating multi-linear objects measure (signed) volume. A form is just a linear combination of projections followed by determinants, so forms are precisely the objects you need to measure signed volume when you have positive co-dimension. | |
| Jan 3, 2010 at 22:17 | comment | added | Harald Hanche-Olsen | Of course forms have to be signed (indeed, alternating) objects. Think of the classical Stokes' theorem, the direction of the normal vector and the associated direction of the boundary. If you reverse one you have to reverse the other, and the integrals change sign. As for linearity, think of integration over a very small surface – say, a parallelogram. If you double one side, the integral should double too (asymptotically). Similarly, if the sides are parallel, the integral vanishes. From ω(X,X)=0 and linearity you get the alternating property. Oh, and remember the Jacobian determinant? | |
| Jan 3, 2010 at 19:13 | history | edited | Ilya Grigoriev | CC BY-SA 2.5 |
added 168 characters in body; added 16 characters in body
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| Jan 3, 2010 at 19:04 | history | answered | Ilya Grigoriev | CC BY-SA 2.5 |