What Defines the Components of a 2-form?

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In summary, the conversation discusses confusion about the definition of the components of a form. The author introduces an argument using the wedge product and finds an extra factor of 2, which is later found to be due to an error in manipulating vector fields. The conversation then references a source where 'p-vectors' are introduced and the argument is mentioned again. The conclusion is that the argument is fallacious.
  • #1
center o bass
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Hi I'm a bit confused about what actually define the components of a form. I just saw an argument where one found that

[tex] \underline{d \omega}^\rho (\vec e_\mu \wedge e_\nu) = - c_{\mu \nu}^\rho [/tex]

and then the author wrote that this implied that

[tex] \underline{d \omega}^\rho = - \frac{1}2 c_{\mu \nu}^\rho \underline{\omega}^\mu \wedge \underline{\omega}^\nu [/tex]

so if the components of a p-form is defined as

[tex] \underline{\alpha} = \frac{1}{p!} \alpha_{\mu_1 \ldots \mu_p} \underline{\omega}^{\mu_1} \wedge \ldots \wedge \underline{\omega}^{\mu_p} [/tex]

where [itex]\alpha_{\mu_1 \ldots \mu_p}[/itex] are the components, it seems the argument above implies that one can find these by applying the p-form to the basis p-vectors.

However I tried this with a two form [itex]\underline{\alpha} = 1/2 \alpha_{\mu \nu} \underline{\omega}^\mu \wedge \underline{\omega}^\nu [/itex] using the definition of the wedge product

[tex] \underline{\alpha}(\vec{e}_\alpha \wedge \vec{e}_\beta) = \frac{1}{2} \alpha_{\mu \nu} \underline{\omega}^\mu \wedge \underline{\omega}^\nu (\vec{e}_\alpha \wedge \vec{e}_\beta) \\
= \frac{1}{2} \alpha_{\mu \nu} 4 \underline \omega^{[\mu} \underline \omega^{\mu ]}( \vec e_{[\alpha} \vec e_{\beta]}) \\
= 2\alpha_{\mu \nu} \delta^{[\mu}_{[\alpha} \delta^{\nu ]}_{\beta]} \\
= 2 \alpha_{\alpha \beta} [/tex]

Where I have used that [itex] \underline \omega ^\mu \underline \omega^\nu = 2! \underline \omega^\mu \underline \omega^\nu[/itex]. But should I not get [itex]\alpha_{\alpha \beta}[/itex] here? Is the caculation wrong or is my assumption of what defines the components wrong?
 
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  • #2
center o bass said:
Hi I'm a bit confused about what actually define the components of a form. I just saw an argument where one found that

[tex] \underline{d \omega}^\rho (\vec e_\mu \wedge e_\nu) = - c_{\mu \nu}^\rho [/tex]

Your notation is a bit confusing, which might be the reason for the extra factor of 2 that you get later. The [itex]e_\mu[/itex] are a basis for vector fields, so we do not take a wedge product of them. It's typical to write that expression as

[tex] \underline{d \omega}^\rho ( e_\mu , e_\nu) = - c_{\mu \nu}^\rho, [/tex]

where it's understood that

[tex](\alpha \wedge \beta) (v,w) = \alpha(v) \beta(w) - \beta(w) \alpha(v)[/tex]

pointwise. The pair of terms here is of course responsible for the factor of 2. However it's clear that in your expression

[tex] \underline{\alpha}(\vec{e}_\alpha \wedge \vec{e}_\beta) = \frac{1}{2} \alpha_{\mu \nu} \underline{\omega}^\mu \wedge \underline{\omega}^\nu (\vec{e}_\alpha \wedge \vec{e}_\beta) \\
= \frac{1}{2} \alpha_{\mu \nu} 4 \underline \omega^{[\mu} \underline \omega^{\mu ]}( \vec e_{[\alpha} \vec e_{\beta]}) \\
= 2\alpha_{\mu \nu} \delta^{[\mu}_{[\alpha} \delta^{\nu ]}_{\beta]} \\
= 2 \alpha_{\alpha \beta} [/tex]

you're introducing an extra factor of 2 from the erroneous manipulation of the vector fields.
 
  • #3
fzero said:
Your notation is a bit confusing, which might be the reason for the extra factor of 2 that you get later. The [itex]e_\mu[/itex] are a basis for vector fields, so we do not take a wedge product of them. It's typical to write that expression as

[tex] \underline{d \omega}^\rho ( e_\mu , e_\nu) = - c_{\mu \nu}^\rho, [/tex]

where it's understood that

[tex](\alpha \wedge \beta) (v,w) = \alpha(v) \beta(w) - \beta(w) \alpha(v)[/tex]

pointwise. The pair of terms here is of course responsible for the factor of 2. However it's clear that in your expression



you're introducing an extra factor of 2 from the erroneous manipulation of the vector fields.


Check out page 59 in

http://www.google.no/url?sa=t&rct=j...uSWMaT_Vs9fO_AucIQGIMOA&bvm=bv.43287494,d.Yms

Here 'p-vectors' are introduced where wedge products of the vector basis is taken.
The argument I referred to is at page 131. Check out the equations (6.175) and (6.176).

The that argument then fallacious?
 

Related to What Defines the Components of a 2-form?

What is a 2-form?

A 2-form is a mathematical object used in multivariable calculus and differential geometry to describe the local behavior of a vector field on a smooth surface. It is a type of differential form that assigns a value to each point on the surface and is represented as a linear combination of basis 2-forms.

What are the components of a 2-form?

The components of a 2-form are the coefficients that represent the basis 2-forms used to construct it. These coefficients are typically denoted as a and b in the form a dx ∧ dy + b dy ∧ dz, where dx, dy, and dz are the basis 1-forms and ∧ represents the exterior product.

What is the purpose of the exterior product in a 2-form?

The exterior product in a 2-form is used to construct a new form from two existing forms. In the case of a 2-form, the exterior product is used to combine two basis 1-forms into a single basis 2-form. This allows us to define a 2-form as a linear combination of basis 2-forms, making it easier to work with in calculations.

How are 2-forms used in physics?

In physics, 2-forms are used to describe physical quantities, such as electric and magnetic fields, which have both magnitude and direction. The components of a 2-form can represent the strength and direction of these fields at a given point in space. They are also used in theories such as general relativity to describe the curvature of spacetime.

What is the relationship between a 2-form and a vector field?

A 2-form and a vector field are closely related, as a 2-form can be used to describe the behavior of a vector field on a surface. The components of a 2-form can be interpreted as the dot product between the vector field and the surface's normal vector. This relationship allows us to use 2-forms to calculate flux and other properties of vector fields on surfaces.

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