Trying to get my head around some basic exterior calculus concepts

[BWV]

Trying to get my head around some basic exterior calculus concepts


So #1 - 1-form or covector the grad is a covector / 1-form defined as something that combined with a vector takes it to a scalar, for example divergence is a scalar which is the dot-product of a function with its grad (e.g. a 1-form or covector) - correct?


#2 - is there an correspondance between curl and a wedge product? can curl be written as a wedge product?

 

[tiny-tim]

Hi BWV!

#1 - 1-form or covector the grad is a covector / 1-form defined as something that combined with a vector takes it to a scalar, for example divergence is a scalar which is the dot-product of a function with its grad (e.g. a 1-form or covector) - correct?

#2 - is there an correspondance between curl and a wedge product? can curl be written as a wedge product?

#1 - yes!

#2 - a cross-product of two three-dimensional vectors is a pseudovector, not a vector, and it's a dual 1-form, the dual of the wedge product 2-form …

same with curl (so it can be written as a wedge product starred)