Contents

2.4. Component expansions

2.4.1. Alternating forms

Some \(\varphi \in \Lambda^p V\) may be expanded as an ordered sum

\[\varphi = \sum_{i_1 < \ldots < i_p} \varphi_{i_1 \ldots i_p} \beta^{i_1} \wedge \ldots \wedge \beta^{i_p},\]

where \(\beta_i\) is a basis of the dual space \(V^\ast = \Lambda^1V\). Here \(\varphi_{i_1 \ldots i_p} \in \mathbb{R}\), and the summations run to \(n = \text{dim} \, V\).

Alternatively, we can allow \(\varphi_{i_1 \ldots i_p}\) to be defined for all indices, in which case it is a completely asymmetric tensor with values in \(\mathbb{R}\) (i.e. the codomain)

Notation

Often a short-hand is used in the literature, denoting

\[i_1 \ldots i_p = \mathcal{I},\]

when used as indices.

\[\varphi = \sum_{i_1 \ldots i_p} \varphi_{i_1 \ldots i_p} \beta^{i_1} \wedge \ldots \wedge \beta^{i_p},\]

with the transformation property

\[\varphi'_{i_1 \ldots i_p} = \sum_{j_1 \ldots j_p} \varphi_{j_1 \ldots j_p} (\gamma^{-1})^{j_1}_{\hphantom{j_1}i_1} \ldots (\gamma^{-1})^{j_p}_{\hphantom{j_p}i_p}.\]

2.4.1.1. \(n\)-forms

For the special case of \(\omega \in \Lambda^nV\), where \(n = \text{dim} \, V\), the above formulation simplifies to

\[\omega = k \beta^1 \wedge \ldots \beta^n,\]

since there is only one possible ordering of the incides. This can be rewritten as

\[\omega = \frac{k}{n!} \sum_{i_1 \ldots i_n} \varepsilon_{i_1 \ldots i_n} \beta^{i_1} \wedge \ldots \beta^{i_n}.\]

As usual, \(\varepsilon_{i_1 \ldots i_n}\) is the totally antisymmetric tensor Levi-Civita symbol.

2.4.2. Inner derivative

The inner derivative \(\iota_v\) on some \(\varphi \in \Lambda^pV\) may be expressed in components by

\[\iota_v \varphi = \frac{1}{(p-1)!} \sum_{i_1 \ldots i_{p-1}} \left( \sum_j v^j \varphi_{j i_1 \ldots i_{p-1}} \right) \beta^{i_1} \wedge \ldots \wedge \beta^{i_{p-1}},\]

for some \(v \sum_j v^j b_j \in V\), where \(b_i\) is, as usual, a basis for \(V\) and the summations run to \(n = \text{dim} \, V\).