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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]
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 anti-symmetric.
The covariant derivative of a function ... The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero:
The second equation, called the Codazzi equation or Codazzi-Mainardi equation, states that the covariant derivative of the second fundamental form is fully symmetric. It is named for Gaspare Mainardi (1856) and Delfino Codazzi (1868–1869), who independently derived the result, [ 3 ] although it was discovered earlier by Karl Mikhailovich ...
The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space). Since the Levi-Civita connection is torsion-free, its curvature can also be expressed in terms of the second covariant derivative [3]
In tensor analysis, a covariant vector varies more or less reciprocally to a corresponding contravariant vector. Expressions for lengths, areas and volumes of objects in the vector space can then be given in terms of tensors with covariant and contravariant indices.
When the torsion tensor is zero, so that [,] =, we may use this fact to write Riemann curvature tensor as [2] (,) =,,. Similarly, one may also obtain the second covariant derivative of a function f as
(Covariant derivative) The standard derivative () in Euclidean contexts satisfies certain dependencies on and , the most fundamental being linearity. A covariant derivative is defined to be any operation ( v , X ) ↦ ∇ v X {\displaystyle (v,X)\mapsto \nabla _{v}X} which mimics these properties, together with a form of the product rule .