is completely anti-symmetric. Then the exterior covariant derivative assigns to each a multilinear map given by the formula where are arbitrary tangent vectors at which are extended to smooth locally-defined vector fields . The legitimacy of this definition depends on the fact that the above expression depends only on , and not on the choice of extension. This can be verified by the Leibniz rule for covariant differentiation and for the Lie bracket of vector fields. The pattern established in the above formula in the case can be directly extended to define the exterior covariant derivative for arbitrary . The exterior covariant derivative may be characterized by the axiomatic property of defining for each a real-linear map which for is the covariant derivative and in general satisfies the Leibniz rule for any differential -form and any vector-valued form . This may also be viewed as a direct inductive definition. For instance, for any vector-valued differential 1-form and any local frame of the vector bundle, the coordinates of are locally-defined differential 1-forms . The above inductive formula then says that In order for this to be a legitimate definition of , it must be verified that the choice of local frame is irrelevant. This can be checked by considering a second local frame obtained by an arbitrary change-of-basis matrix; the inverse matrix provides the change-of-basis matrix for the 1-forms . When substituted into the above formula, the Leibniz rule as applied for the standard exterior derivative and for the covariant derivative cancel out the arbitrary choice. A vector-valued differential 2-form may be regarded as a certain collection of functions assigned to an arbitrary local frame of over a local coordinate chart of . The exterior covariant derivative is then defined as being given by the functions The fact that this defines a tensor field valued in is a direct consequence of the same fact for the covariant derivative. The further fact that it is a differential 3-form valued in asserts the full anti-symmetry in and is directly verified from the above formula and the contextual assumption that is a vector-valued differential 2-form, so that . The pattern in this definition of the exterior covariant derivative for can be directly extended to larger values of .This definition may alternatively be expressed in terms of an arbitrary local frame of but without considering coordinates on . Then a vector-valued differential 2-form is expressed