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
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