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In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold.Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection.
The covariant derivative of a vector field with components is given by: ; = = + and similarly ... The divergence of an antisymmetric tensor field of type ...
Equivalently, some authors define the divergence of a mixed tensor by using the musical isomorphism ♯: if T is a (p, q)-tensor (p for the contravariant vector and q for the covariant one), then we define the divergence of T to be the (p, q − 1)-tensor
In general relativity and tensor calculus, the contracted Bianchi identities are: [1] = where is the Ricci tensor, the scalar curvature, and indicates covariant differentiation.
Contravariant vectors are often just called vectors and covariant vectors are called covectors or dual vectors. The terms covariant and contravariant were introduced by James Joseph Sylvester in 1851. [3] [4] Curvilinear coordinate systems, such as cylindrical or spherical coordinates, are often used in physical and geometric problems ...
The operator can be extended to operate on tensors as the divergence of the covariant derivative. Alternatively, the operator can be generalized to operate on differential forms using the divergence and exterior derivative. The resulting operator is called the Laplace–de Rham operator (named after Georges de Rham).
After the final Divergent film’s 2016 debut, many of the franchise’s stars have gone on — or continued — to have very successful acting careers. Based on Veronica Roth’s book series of ...
The covariant divergence of every Killing vector field vanishes. If X {\displaystyle X} is a Killing vector field and Y {\displaystyle Y} is a harmonic vector field , then g ( X , Y ) {\displaystyle g(X,Y)} is a harmonic function .