Can Vectors Only Associate Pseudo-Forms in Non-Cartesian Coordinates?

[JonnyMaddox]

In Frankel's book he writes that in R^{3} with cartesian coordinates, you can always associate to a vector \vec{v} a 1-form v^{1}dx^{1}+v^{2}dx^{2}+v^{3}dx^{3} and a two form v^{1}dx^{2}\wedge dx^{3}+v^{2}dx^{3}\wedge dx^{1} +v^{3}dx^{1} \wedge dx^{2}, now in general this is not possible and you have to convert the components of a vector with a metric to get covariant components, for example like this v_{i}=g_{ij}v^{j}

Then he asks what is in general the associated 2-form for \vec{v} ? He then proofs that to a vector \vec{v} one does associate a pseudo-2-form \beta^{2}:= \iota_{\vec{v}}vol^{3} Later when he discusses the cross product he writes that one would like to say that v^{1} \wedge \omega^{1} is the 2-form associated to the vector \vec{v} \times \vec{w}, but we only have a pseudo-2-form associated to a vector thus the pseudovector \vec{v} \times \vec{w} is associated to the 2-form v^{1} \wedge \omega^{1} (which is just a flip flop of words I think).

Now is it true that if we have other coordinates than cartesian, one can only associate pseudo-forms to a vector? Because in the text he calls the forms in cartesian coordiantes just forms, but in general he says pseudo-forms.

For example (everything in cartesian coord.) when I have a simple vector field v=3x \partial_{x}+4x \partial_{y} then according to the formula \beta^{1}= \iota_{v}vol^{2}= 3xdy-4ydx is the associated pseudo-1-form. Now what about the form \gamma^{1}=3x dx + 4y dy isn't that the 1-form associated to the vector field? And \gamma^{2}= 3x dy \wedge dz+4y dz \wedge dx should be the associated 2-form, but at the same time there should also be an associated pseudo-2-form 3x dx\wedge dy -4y dx \wedge dz according to the formula and if I calculated right. Now does this in general mean (none-cart.) that I have only pseudo-forms associated to vectorfields ??

 

[Greg Bernhardt]

I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?

 

[Incnis Mrsi]

Possibly my answer will be too abstract, but you do not need Euclidean structure in three dimensions to have a correspondence between vectors and (pseudo)-2-forms. You need a weaker thing. Depending on how it deals with orientation, it may be called either volume form (that identifies vectors with 2-forms) or density (that identifies vectors with pseudo-2-forms). Euclidean/Riemannian metric induces a density, but one can’t recover the metric from a density only.

In other words, if in certain coordinates you see a familiar correspondence between vectors and pseudo-2-forms, you can’t be sure that these coordinates are not skew.

 

[WWGD]

AFAIK, a pseudometric is enough to define a natural isomorphism between a space--say vectors here--and their duals, being the forms.

 

[Incnis Mrsi]

WWGD: you completely miss the point. You think about lowering an index: ω_k = g_kℓ X^ℓ. It makes an 1-form from a vector, and ι_X ω = 1. It corresponds to scalar product (dot product in Euclidean space, or whatever, not necessarily positive-definite).

Original poster asks about ω_jk = ε_jkℓ X^ℓ, a different thing making a 2-form from a vector. It corresponds to vector product in 3 dimensions and gives ι_X ω = 0.