# I Differential forms as a basis for covariant antisym. tensors

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1. May 8, 2017

### Physics_Stuff

In a text I am reading (that I unfortunately can't find online) it says:

"[...] differential forms should be thought of as the basis of the vector space of totally antisymmetric covariant tensors. Changing the usual basis $$dx^{\mu_1} \otimes ... \otimes dx^{\mu_n}$$ with $$dx^{\mu_1} \wedge ... \wedge dx^{\mu_n}$$ of some covariant tensor we can extract its totally antisymmetric part
$$T= \frac{1}{n!}T_{\mu_1 ... \mu_n}\hspace{1pt} d x^{\mu_1} \wedge ... \wedge d x^{\mu_n}= \frac{1}{n!}T_{[\mu_1 ... \mu_n]}\hspace{1pt} d x^{\mu_1} \wedge ... \wedge d x^{\mu_n}."$$

What is the point here? Is T an arbitrary tensor with n covariant components, or must T already be antisymmetric in order for this expression to hold? In order to know the components $$T_{\mu_1 ... \mu_n}$$ of T, so we can use the expression on the RHS above, we must already know what the tensor T looks like? Then, what is the point of such a decomposition?

2. May 8, 2017

### Staff: Mentor

$T$ is an arbitrary tensor. Until now, this doesn't say anything more than $T$ is a multi-dimensional scheme of numbers. In the first step, you say, these numbers represent coordinates. So the question is, according to which basis? As you answer "covariant multilinear forms", it means $T$ is interpreted according to a basis $dx^{\mu_1} \otimes \ldots \otimes dx^{\mu_n}$. It is still the same scheme of numbers. Now you say "but my multilinear forms are alternating differential forms". This means you pass from the tensor algebra $\mathcal{T}(V^*)$ onto the homomorphic image of its Graßmann algebra $\Lambda(V^*)$. It means, the basis vectors are now alternating differential forms and $T_{\mu_1 \ldots \mu_n}$ the coordinates of $T$ according to this basis. It is still the same scheme of numbers, however, interpreted as an element of the algebra of alternating differential forms (with a normalization factor).

Your question is as if you had asked, whether $(1,2)$ is a point, a line, a tangent, a slope, a linear mapping or a differential form. It is whatever you want it to be. The usual way to get there is of course the opposite direction: given an alternating differential form $T$, what are its coordinates according to the basis $dx^{\mu_1} \wedge \ldots \wedge dx^{\mu_n}\;$?