What is the meaning of covariant derivatives?

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. The covariant derivative generalizes straightforwardly to a notion of differentiation associated to a connection on a vector bundle, also known as a Koszul connection.

Why do we need the covariant derivative?

Covariant derivatives are also used in gauge theory: when the field is non zero, there is a curvature and it is not possible to set the potential to identically zero through a gauge transformation. They may also be purely convenient, for example when using angular parameters in a spherically symmetric potential.

Is the covariant derivative commutative?

This is called the geodesic equation. Unlike ordinary partial derivatives, covariant derivatives do not commute.

Is the covariant derivative a tensor?

The covariant derivative of this vector is a tensor, unlike the ordinary derivative. Here we see how to generalize this to get the absolute gradient of tensors of any rank. First, let’s find the covariant derivative of a covariant vector (one-form) Bi. (2) The covariant derivative obeys the product rule.

What is the covariant derivative of a scalar?

More generally, for a tensor of arbitrary rank, the covariant derivative is the partial derivative plus a connection for each upper index, minus a connection for each lower index.

What is Lie derivative?

A Lie derivative is the derivative of a vector field along the flow of a “benchmark” field, ξ in your notation. It is as though a pioneering surveyor has mapped the manifold for you in advance by laying down a field which we use to compare all other fields to. Everything is measured by its relationship with ξ.

What is a covariant derivative?

The notion of covariant derivative appears naturally when one tries to solve the following problem. Suppose that E → M is a smooth vector bundle over a smooth manifold M. For example, E could be the tangent bundle of M. We seek a notion of parallel transport that will allow us to compare vectors situated in different fibers of the bundle.

What is the Lie derivative under diffeomorphisms?

The Lie derivative is also natural under general diffeomorphisms but only as a bilinear operator, which takes one vector field and one section of a general vector bundle (for example a tensor field) as it’s entries. In particular it is a bi-differential operator, so both the vector field and the other section are differentiated.

What is the difference between exterior derivative and connection derivative?

The exterior derivative takes differential forms as inputs. Connections take sections of a vector bundle (such as tensor fields) as inputs, and differentiation is done with respect to a vector field. The Lie derivative takes tensor fields as inputs, and differentiation is done with respect to a vector field.