Exploring Relativity Theory - What is a "One Form"?

[Jomapil]

Hello, everyone

I began do learn Relativity Theory. I don't understand the term " One Forms " that appear in some books. Can anyone tell me what did it mean? The translation to Portuguese doesn't any sense and the explanation of the books is not very explicative for me.

Thank you very much.

 

[ManyNames]

''One forms'' are a mathematical lingo for a linear function that takes a vector and returns a scalar; it's therefore a type of tensor.

 

[dx]

1-forms are the geometric objects corresponding to the local linear behavior of functions on a manifold.

If a patch of a manifold M is described by the coordinate functions xⁱ : M → R, then for any function f on that patch, its local linear behavior can be given a coordinate representation in terms of the coordinate 1-forms dxⁱ thus: df = ∑(∂f/∂xⁱ)dxⁱ, where the numnbers ∂f/∂xⁱ are called the components of df.

For more details I recommend these books: "Applied Differential Geometry" and "Spacetime, Geometry and Cosmology" by William Burke.

 

[haushofer]

A one-form is exactly the same as what other people call "a covariant vector". :) But to understand this you need some basis in tensor analysis.

 

[Jomapil]

Good evening.

haushofer - I studied tensor calculus 35 years ago and I'm reading and exercising at the same time I'm learning Relativity Theory, so I understand your explanation - EVERY covariant tensor is an One-Form.

ManyNames - I think I understood your explanation - for example the operator divergence [ div(vctor) --> escalar] is an One-Forms.

dx - I'm going soon to read about manifolds - anyway I also understood your help.

Thank you everyone and a good new week.