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valid for any vector fields X and Y and any tensor field T.. Considering vector fields as infinitesimal generators of flows (i.e. one-dimensional groups of diffeomorphisms) on M, the Lie derivative is the differential of the representation of the diffeomorphism group on tensor fields, analogous to Lie algebra representations as infinitesimal representations associated to group representation ...
The covariant derivative is a generalization of the directional derivative from vector calculus. As with the directional derivative, the covariant derivative is a rule, , which takes as its inputs: (1) a vector, u, defined at a point P, and (2) a vector field v defined in a neighborhood of P. [7]
Generally the convective derivative of the field u·∇y, the one that contains the covariant derivative of the field, can be interpreted both as involving the streamline tensor derivative of the field u·(∇y), or as involving the streamline directional derivative of the field (u·∇) y, leading to the same result. [10]
The covariant derivative is such a map for k = 0. The exterior covariant derivatives extends this map to general k. There are several equivalent ways to define this object: [3] Suppose that a vector-valued differential 2-form is regarded as assigning to each p a multilinear map s p: T p M × T p M → E p which is completely anti-symmetric.
Another important tensorial derivative is the Lie derivative. Unlike the covariant derivative, the Lie derivative is independent of the metric, although in general relativity one usually uses an expression that seemingly depends on the metric through the affine connection.
Connections (gauge connection) define this principal bundle, yielding a covariant derivative ∇ in each associated vector bundle. If a local frame is chosen (a local basis of sections), then this covariant derivative is represented by the connection form A, a Lie algebra-valued 1-form, which is called the gauge potential in physics. This is ...
The covariant derivatives (also called "tangential derivatives") of Tullio Levi-Civita and Gregorio Ricci-Curbastro provide a means of differentiating smooth tangential vector fields. Given a tangential vector field X and a tangent vector Y to S at p , the covariant derivative ∇ Y X is a certain tangent vector to S at p .
The gauge covariant derivative is often assumed to satisfy additional conditions making additional structure "constant" in the sense that the covariant derivative vanishes. For example, if we have a Hermitian product on the fields (e.g. the Dirac conjugate inner product ¯ for spinors) reducing the gauge group to a unitary group, we can impose ...