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I understand that, if both arguments are vectors (or vector fields) and we're in euclidean space, the computation is exactly as if I were doing a dot product. However, if we're in a manifold (Lets say... On the surface of a unit sphere in [tex]\Re^3[/tex]) how would the computation be done?

What if the first argument is a 1-form?

From what I've read, I've found lots of helpful information concerning properties of inner products, their usefulness as metrics, and nice identities with them. But when it comes to finding the value of one, I am lost.

Thanks!