In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector bundle with a connection. Definition Let G be a Lie group and be a principal G-bundle on a smooth manifold M. Suppose there is a connection on P; this yields a natural direct sum decomposition of each tangent space into the horizontal and vertical subspaces. Let be the projection to the horizontal subspace. If ϕ is a k-form on P with values in a vector space V, then its exterior covariant derivative Dϕ is a form defined by where vi are tangent vectors to P at u. Suppose that is a representation of G on a vector space V. If ϕ is equivariant in the sense that where , then Dϕ is a tensorial -form on P of the type ρ: it is equivariant and horizontal (a form ψ is horizontal if .) By abuse of notation, the differential of ρ at the identity element may again be denoted by ρ: Let be the connection one-form and the representation of the connection in That is, is a -valued form, vanishing on the horizontal subspace. If ϕ is a tensorial k-form of type ρ, then where, following the notation in , we wrote Unlike the usual exterior derivative, which squares to 0, the exterior covariant derivative does not. In general, one has, for a tensorial zero-form ϕ, where is the representation in of the curvature two-form Ω. The form F is sometimes referred to as the field strength tensor, in analogy to the role it plays in electromagnetism. Note that D2 vanishes for a flat connection (i.e. when ). If , then one can write where is the matrix with 1 at the -th entry and zero on the other entries. The matrix whose entries are 2-forms on P is called the curvature matrix. For vector bundles Given a smooth real vector bundle with a connection and rank , the exterior covariant derivative is a real-linear map on the vector-valued differential forms which are valued in : The covariant derivative is such a map for . The exterior covariant derivatives extends this map to general . There are several equivalent ways to define this object: Suppose that a vector-valued differential 2-form is regarded as assigning to each a multilinear map which