What is the derivative of a basis vector?

The covariant derivative (w.r.t. the theta covariant basis vector) is said to be the result of parallel transporting the vector v′=V(p′) along the direction of a short curve to point p and then subtracting the vectors v′||−v where v′|| is the transported vector v′ at point p.

What does the covariant derivative do?

A covariant derivative introduces an extra geometric structure on a manifold that allows vectors in neighboring tangent spaces to be compared: there is no canonical way to compare vectors from different tangent spaces because there is no canonical coordinate system.

Is the covariant derivative of a vector a tensor?

The covariant derivative of this vector is a tensor, unlike the ordinary derivative.

What is the difference between Lie derivative and covariant derivative?

Covariant derivative is the analogue of directional derivative in R^n case. So if we fix a connection and assign a direction to a point, the covariant derivative at that point is well-defined. But for Lie derivative, one direction is not enough. We have to point out the vector field.

What is the covariant divergence?

It may be called covariant divergence. b. The divergence of a given covariant tensor results like for the preceding case, a, but in this case, b, the contraction must be performed for each pair of concerned indices placed in the same position, both subscript. It may also be called covariant divergence.

Is Christoffel symbols a tensor?

It is important to note, however, the Christoffel symbol is not a tensor. Its elements do not transform like the elements of a tensor.

Is Lie derivative a tensor?

The Lie derivative of a tensor field The Lie derivative is the speed with which the tensor field changes under the space deformation caused by the flow.

Are covariant derivatives commutative?

Unlike ordinary partial derivatives, covariant derivatives do not commute.

Is the connection a tensor?

This is why we are not so careful about index placement on the connection coefficients; they are not a tensor, and therefore you should try not to raise and lower their indices. There is no way to “derive” these properties; we are simply demanding that they be true as part of the definition of a covariant derivative.