Differential Forms as Matrices

[Xittenn]

What are the more common theorems used to derive, for example a sphere in differential form, as a matrix representation? What are the more common expressions used to describe differential forms as matrices?

[ajb]

Are you asking about the volume form on the 2-sphere?

 

In general a two form you can think of as a matrix, not that I tend to do that.

 

We have [math]\omega = \frac{1}{2}dx^{i} \wedge dx^{j}\omega_{j i}(x)[/math],

 

and given the antisymmetric property of the wedge you can think of [math]\omega_{j i}(x)[/math] as a completely antisymmetric matrix, i.e.

 

[math]\omega_{ij} = - \omega_{ji}[/math].

 

Is this what you are asking about?

[Xittenn]

Yes, that got me there. Thanks ajb!

 

 

Leimkuhler, B., and G.W. Patrick. "A Symplectic Integrator For Riemannian Manifolds." Journal Of Nonlinear Science 6.4 (1996): 367. Academic Search Premier. Web. 28 Apr. 2012.

[ajb]

Symplectic integration is a numerical scheme for integration, that preserves geometric (symplectic) properties of the exact flow of a differential equation. This is not something I know much about.

[Xittenn]

More of an engineering thing . . . .

[ajb]

More of an engineering thing . . . .

 

People who have mentioned symplectic integrators to me are applied mathematicians, so not a million miles away from engineering. I forget why this came up in a conversation.