nLab
Cartan's homotopy formula

Let $M$ be a differentiable manifold, $X$ a vector field on $M$, and ${ℒ}_X$ the Lie derivative along $X$. The contraction of a vector field and a $k$-form $\omega$ is denoted in modern literature by $X\rfloor \omega$ (should be lrcorner instead of rfloor LaTeX command, but it does not work in iTeX) or ${\iota }_X\omega$ or $\iota (X)(\omega )$.

Then the Cartan’s infinitesimal homotopy formula, nowdays called simply Cartan’s homotopy formula or even Cartan formula, says

$${ℒ}_X\omega =d\iota (X)\omega +\iota (X)d\omega$$\mathcal{L}_X \omega = d \iota(X)\omega + \iota(X) d\omega

The word “homotopy” is used because it supplies a homotopy operator for some manipulation with chain complexes in de Rham cohomology. Cartan’s homotopy formula is part of Cartan calculus.

Regarding that the Cartan’s formula can be viewed as a formula about the de Rham complex, which has generalizations, one can often define the Cartan’s formulas for those generalizations. For example

• Masoud Khalkhali, On Cartan homotopy formulas in cyclic homology, Manuscripta mathematica 94:1, pp 111-132 (1997) doi