# Wedge product of a 2-form with a 1-form

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• GR191511
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.
GR191511
Let ##\omega## be 2-form and ##\tau## 1-form on ##R^3## If X,Y,Z are vector fields on a manifold,find a formula for ##(\omega\bigwedge\tau)(X,Y,Z)## in terms of the values of ##\omega## and ##\tau ## on the vector fields X,Y,Z.
I have known how to deal with only one vector field.But there are three vector fields,I have no idea.

GR191511 said:
I have known how to deal with only one vector field.
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.

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

GR191511 said:
Thank you.
I know ##\omega (X) = (a_idx^i)(b^j\frac{\partial }{\partial x^j})=a_ib^i##
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?

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?
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.

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.
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.

Orodruin 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.
Thanks.But ##v_1,v_2,v_3## is vector from ##R^1## while X,Y,Z are three vector fields
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.

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

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.

Orodruin said:
The question don't mention that. I'm not sure about it either.

GR191511 said:
The question don't mention that. I'm not sure about it either.
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.

Orodruin 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.
Is that "alternating n-linear function on a vector space"? I have seen it...it confused me

## 1. What is the wedge product of a 2-form with a 1-form?

The wedge product of a 2-form with a 1-form is a mathematical operation that combines two differential forms to create a new differential form. It is denoted by the symbol ∧ and is also known as the exterior product.

## 2. How is the wedge product of a 2-form with a 1-form calculated?

The wedge product of a 2-form with a 1-form is calculated by multiplying the two forms together and then taking the anti-symmetric part of the resulting product. This means that the order of the factors does not matter and any terms that are repeated with a different sign are cancelled out.

## 3. What is the purpose of the wedge product in mathematics?

The wedge product is used in mathematics to define the exterior algebra, a mathematical structure that generalizes the concept of vectors and matrices to higher dimensions. It is also used in differential geometry to describe geometric objects such as curves and surfaces.

## 4. What are some properties of the wedge product?

The wedge product has several important properties, including associativity, distributivity, and the fact that it is anti-commutative. It also satisfies the graded-commutative property, which means that the product of two forms of different degrees is equal to the product of their degrees in reverse order.

## 5. How is the wedge product related to the cross product?

The wedge product is related to the cross product in three dimensions, but it is a more general operation that can be applied to any number of dimensions. In three dimensions, the cross product can be thought of as a special case of the wedge product, where the resulting form is a 3-form. However, in higher dimensions, the wedge product is a more powerful tool for describing geometric objects.

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