 Let "X" and "Y" be vector fields on an N-dimensional real manifold "M" and let "ξ" and "η" be "p"-forms.Then "X+ξ" and "Y+η" are sections of the direct sum of the tangent bundle and the bundle of "p"-forms.The Courant bracket of "X+ξ" and "Y+η" is defined to be

where formula_2 is the Lie derivative along the vector field "X", "d" is the exterior derivative and "i" is the interior product.