- #1
KamYi
- 2
- 0
I just discovered this forum: very very nice!
And here's my first question:
An exterior p-form is a multilinear antisymmetric map from p copies of a vector space (in particular, a tangent space located at some point P of a manifold) to the reals.
Now what could it mean to have an integral of a p-form over a submanifold of dimension p?? If I think of the integral as a sum of p-forms at different points P along the submanifold, then what would the argument of the sum of the p-forms be? At each point it should be a *different* argument, so how can you add p-forms at different points??
Edit: Is this the answer? By definition of the integral, the p-form [tex] \alpha^p [/tex] in any coordinate patch such that [tex] \alpha=a_idx^i [/tex], is to be evaluated by taking as arguments the corresponding basis tangent vectors [tex]\partial/\partial x^i. [/tex] That indeed makes the integral independent of reparametrization.
Thanks,
S.
And here's my first question:
An exterior p-form is a multilinear antisymmetric map from p copies of a vector space (in particular, a tangent space located at some point P of a manifold) to the reals.
Now what could it mean to have an integral of a p-form over a submanifold of dimension p?? If I think of the integral as a sum of p-forms at different points P along the submanifold, then what would the argument of the sum of the p-forms be? At each point it should be a *different* argument, so how can you add p-forms at different points??
Edit: Is this the answer? By definition of the integral, the p-form [tex] \alpha^p [/tex] in any coordinate patch such that [tex] \alpha=a_idx^i [/tex], is to be evaluated by taking as arguments the corresponding basis tangent vectors [tex]\partial/\partial x^i. [/tex] That indeed makes the integral independent of reparametrization.
Thanks,
S.
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