Let be a differentiable manifold, a vector field on , and the Lie derivative along . The contraction of a vector field and a -form is denoted in modern literature by (should be lrcorner instead of rfloor LaTeX command, but it does not work in iTeX) or or .
Then the Cartan’s infinitesimal homotopy formula, nowdays called simply Cartan’s homotopy formula or even Cartan formula, says
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