Vectors and associated 2-forms

1. Aug 19, 2014

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 formular and if I calculated right. Now does this in general mean (none-cart.) that I have only pseudo-forms associated to vectorfields ??

Last edited: Aug 19, 2014
2. Aug 27, 2014

Greg Bernhardt

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

3. Sep 3, 2014

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.

4. Sep 3, 2014

WWGD

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

5. Sep 3, 2014

Incnis Mrsi

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

Original poster asks about ωjk = εjkℓX, a different thing making a 2-form from a vector. It corresponds to vector product in 3 dimensions and gives ιX ω = 0.