Let $M$ be a differentiable manifold, $X$ a vector field on $M$, and $\mathcal{L}_X$ 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

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

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