Contraction in Coordinates

Given a manifold \(M\), the interior product is defined as the contraction of a differential form with a vector field.

\[\mathbf{i}_{X}:\quad\Lambda^{k}(M)\rightarrow\Lambda^{k-1}(M)\]

The contraction of a one-form \(\alpha=\alpha_{i}e^{i}\) with a vector field \(X=X^{i}e_{i}\) is given by

\[\begin{split}\begin{aligned} \mathbf{i}_{X}\alpha & =\langle X,\alpha\rangle\\ & =\alpha_{i}X^{j}\langle e^{i},e_{j}\rangle\\ & =\alpha_{i}X^{j}\delta_{j}^{i}\\ & =\alpha_{i}X^{i}\end{aligned}\end{split}\]

The contraction of a two-form \(\beta=\beta_{ij}e^{i}\wedge e^{j}\) with the same vector field is given by

\[\begin{split}\begin{aligned} \mathbf{i}_{X}\beta & =\beta_{ij}\mathbf{i}_{X}(e^{i}\wedge e^{j})\\ & =\beta_{ij}\left((\mathbf{i}_{X}e^{i})\wedge e^{j}-e^{i}\wedge(\mathbf{i}_{X}e^{j})\right)\\ & =\beta_{ij}\left(X^{i}e^{j}-X^{j}e^{i}\right)\end{aligned}\end{split}\]

The contraction of a three-form \(\gamma=\gamma{}_{ijk}e^{i}\wedge e^{j}\wedge e^{k}\) is going to be

\[\begin{split}\begin{aligned} \mathbf{i}_{X}\gamma & =\gamma_{ijk}\mathbf{i}_{X}(e^{i}\wedge e^{j}\wedge e^{k})\\ & =\gamma_{ijk}\left(\mathbf{i}_{X}e^{i}\wedge e^{j}\wedge e^{k}-e^{i}\wedge\mathbf{i}_{X}e^{j}\wedge e^{k}+e^{i}\wedge e^{j}\wedge\mathbf{i}_{X}e^{k}\right)\\ & =\gamma_{ijk}\left(X^{i}e^{j}\wedge e^{k}-X^{j}e^{i}\wedge e^{k}+X^{k}e^{i}\wedge e^{j}\right)\end{aligned}\end{split}\]

In 2D the explicit formula for the contractions are

\[\begin{split}\begin{aligned} \alpha & =\alpha_{x}dx+\alpha_{y}dy\\ \mathbf{i}_{X}\alpha & =X^{x}\alpha_{x}+X^{y}\alpha_{y}\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned} \beta & =\beta_{xy}\,dx\wedge dy\\ \mathbf{i}_{X}\beta & =-\beta_{xy}X^{y}dx+\beta_{xy}X^{x}dy\end{aligned}\end{split}\]

If 3D, on the other hand, the contractions become

\[\begin{split}\begin{aligned} \alpha & =\alpha_{x}dx+\alpha_{y}dy+\alpha_{z}dz\\ \mathbf{i}_{X}\alpha & =X^{x}\alpha_{x}+X^{y}\alpha_{y}+X^{z}\alpha_{z}\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned} \beta & =\beta_{xy}dx\wedge dy+\beta_{yz}dy\wedge dz+\beta_{zx}dz\wedge dx\\ \mathbf{i}_{X}\beta & =\left(\beta_{zx}X^{z}-\beta_{xy}X^{y}\right)dx+\\ & +\left(\beta_{xy}X^{x}-\beta_{yz}X^{z}\right)dy+\\ & +\left(\beta_{yz}X^{y}-\beta_{zx}X^{x}\right)dz\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned} \gamma & =\gamma_{xyz}dx\wedge dy\wedge dz\\ \mathbf{i}_{X}\gamma & =\gamma_{xyz}\left(X^{z}dx\wedge dy+X^{x}dy\wedge dz+X^{y}dz\wedge dx\right)\end{aligned}\end{split}\]