# Meaning of integrating exterior forms

#### KamYi

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 $$\alpha^p$$ in any coordinate patch such that $$\alpha=a_idx^i$$, is to be evaluated by taking as arguments the corresponding basis tangent vectors $$\partial/\partial x^i.$$ That indeed makes the integral independent of reparametrization.

Thanks,

S.

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#### KamYi

precondition said:
Oh darn, I thought someone answered my question