Ad
related to: lie derivative vs covariant derivative practice problems calc 2 quiz
Search results
Results From The WOW.Com Content Network
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]
In fact in the above expression, one can replace the covariant derivative with any torsion free connection ~ or locally, with the coordinate dependent derivative , showing that the Lie derivative is independent of the metric. The covariant derivative is convenient however because it commutes with raising and lowering indices.
There are many ways to understand the gauge covariant derivative. The approach taken in this article is based on the historically traditional notation used in many physics textbooks. [1] [2] [3] Another approach is to understand the gauge covariant derivative as a kind of connection, and more specifically, an affine connection.
The covariant derivative of a vector field with components is given by: ; = ) = + and similarly the ...
In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form.
The Lie derivative is another derivative that is covariant under basis transformations. Like the exterior derivative, it does not depend on either a metric tensor or a connection. The Lie derivative of a type (r, s) tensor field T along (the flow of) a contravariant vector field X ρ may be expressed using a coordinate basis as [20]
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