Table of Contents
What is covariant derivative of metric tensor?
Formal definition A covariant derivative is a (Koszul) connection on the tangent bundle and other tensor bundles: it differentiates vector fields in a way analogous to the usual differential on functions.
Why is there no contravariant derivative?
But while upper indices are often used in conjunction with the covariant derivative operator for notational convenience, it is not usually called the “contravariant derivative”. That is because the actual differentiation still takes place with respect to tangent vectors, whether or not indices are raised afterwards.
Is the covariant derivative of a vector a tensor?
The intesting property about the covariant derivative is that, as opposed to the usual directional derivative, this quantity transforms like a tensor, i.e. it is independant of the manner in which it is expressed in a coordinate system.
What is the covariant derivative used for?
, which is a generalization of the symbol commonly used to denote the divergence of a vector function in three dimensions, is sometimes also used. (Weinberg 1972, p. 104).
Is derivative of a tensor A tensor?
According to what I’ve read, the derivative of a tensor is not in general a tensor (according to Steven Weinberg).
How do you find the covariant derivative?
Starts here50:10Tensor Calculus 6b: The Covariant Derivative – YouTubeYouTube
What is the difference between contravariant and covariant?
In differential geometry, the components of a vector relative to a basis of the tangent bundle are covariant if they change with the same linear transformation as a change of basis. They are contravariant if they change by the inverse transformation.
What is the difference between Lie derivative and covariant derivative?
Hopefully this illustrates the big differences between the two derivatives: the covariant derivative should be used to measure whether a tensor is parallel transported, while the Lie derivative measures whether a tensor is invariant under diffeomorphisms in the direction of the vector ξa.
Why is partial derivative of tensor not a tensor?
The derivative is independent of coordinates if the basis elements are constant (you’re in Euclidean space), but on a general manifold, you have to use the product rule, but the derivative of a tensor only differentiates the components and thus ignores half of the answer.
Is partial derivative a tensor?
Tensor Calculus. where we have taken the special case of a contravariant vector . We now show explicitly that the partial derivative of a contravariant vector cannot be a tensor.
What is the difference between covariant tensor and Contravariant tensor?
Originally Answered: What is physical & general difference between contravariant and covariant tensor? A contravariant tensor (in other words a vector), transform ‘oppositely’ (contra) to the way basis vectors transform, while a covariant tensor (or dual vector) transforms in he same way as basis vectors.
What is tensor variant and covariant tensor?
A covariant tensor, denoted with a lowered index (e.g., ) is a tensor having specific transformation properties. In general, these transformation properties differ from those of a contravariant tensor.