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I have known how to deal with only one vector field.But there are three vector fields,I have no idea.

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In summary, the conversation discusses the calculation of the wedge product of a 2-form and a 1-form on a manifold, when given three vector fields. The formula for this calculation is provided as well as a discussion on the anti-symmetry of n-forms. The conversation concludes with a suggestion to seek additional resources on this topic.

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I have known how to deal with only one vector field.But there are three vector fields,I have no idea.

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- #2

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Please expand on what you mean by this. The wedge product of a 2-form and a 1-form is a 3-form and so must take 3 vector arguments.GR191511 said:I have known how to deal with only one vector field.

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Thank you.Orodruin said:Please expand on what you mean by this. The wedge product of a 2-form and a 1-form is a 3-form and so must take 3 vector arguments.

I know ##\omega (X) = (a_idx^i)(b^j\frac{\partial }{\partial x^j})=a_ib^i## but ##(\omega\bigwedge\tau)(X,Y,Z)=?##

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In the case you have presented in the OP, ##\omega## is a 2-form so this is not true. What you have written here is true if it is a 1-form. It feels like you may have skipped some reading regarding how to go from 1-forms to higher order (0,n) tensors in general and n-forms in particular?GR191511 said:Thank you.

I know ##\omega (X) = (a_idx^i)(b^j\frac{\partial }{\partial x^j})=a_ib^i##

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Thank you!I'm learning from 《An introduction to manifolds》Loring W.Tu...All I know now is that:Orodruin said:In the case you have presented in the OP, ##\omega## is a 2-form so this is not true. What you have written here is true if it is a 1-form. It feels like you may have skipped some reading regarding how to go from 1-forms to higher order (0,n) tensors in general and n-forms in particular?

##\omega\bigwedge\tau = a_Ib_Jdx^I\bigwedge dx^J## and ##(f\bigwedge g)(v_1,v_2,v_3)=f(v_1,v_2)g(v_3)-f(v_1,v_3)g(v_2)+f(v_2,v_3)g(v_1)## ...I don't know what I should do next.

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The first of those relations is the wedge between two 1-forms. The second describes a two-form ##f## with a one-form ##g## - which also happens to be the case you are asked about.GR191511 said:Thank you!I'm learning from 《An introduction to manifolds》Loring W.Tu...All I know now is that:

##\omega\bigwedge\tau = a_Ib_Jdx^I\bigwedge dx^J## and ##(f\bigwedge g)(v_1,v_2,v_3)=f(v_1,v_2)g(v_3)-f(v_1,v_3)g(v_2)+f(v_2,v_3)g(v_1)## ...I don't know what I should do next.

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Thanks.But ##v_1,v_2,v_3## is vector from ##R^1## while X,Y,Z are three vector fieldsOrodruin said:The first of those relations is the wedge between two 1-forms. The second describes a two-form ##f## with a one-form ##g## - which also happens to be the case you are asked about.

Maybe the answer is##(\omega\bigwedge\tau)(X,Y,Z)=\omega(X,Y)\tau(Z)-\omega(X,Z)\tau(Y)+\omega(Y,Z)\tau(X)##But It always feels like something is wrong.

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No, they are not.GR191511 said:But v1,v2,v3 is vector from

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Is your result fully anti-symmetric?GR191511 said:Maybe the answer is(ω⋀τ)(X,Y,Z)=ω(X,Y)τ(Z)−ω(X,Z)τ(Y)+ω(Y,Z)τ(X)But It always feels like something is wrong.

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The question don't mention that. I'm not sure about it either.Orodruin said:Is your result fully anti-symmetric?

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Again, this seems to imply that you are missing information fundamental to n-forms (namely that they are fully anti-symmetric (0,n) tensors). If this is not covered by your textbook, I would suggest looking up another textbook that does.GR191511 said:The question don't mention that. I'm not sure about it either.

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Is that "alternating n-linear function on a vector space"? I have seen it...it confused meOrodruin said:Again, this seems to imply that you are missing information fundamental to n-forms (namely that they are fully anti-symmetric (0,n) tensors). If this is not covered by your textbook, I would suggest looking up another textbook that does.

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Yes, that is just another way of putting it. See https://en.wikipedia.org/wiki/Alternating_multilinear_mapGR191511 said:Is that "alternating n-linear function on a vector space"? I have seen it...it confused me

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...I'm in China,I can't open wikipedia.Thank you all the same.Orodruin said:Yes, that is just another way of putting it. See https://en.wikipedia.org/wiki/Alternating_multilinear_map

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