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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]
In the math branches of differential geometry and vector calculus, the second covariant derivative, or the second order covariant derivative, of a vector field is the derivative of its derivative with respect to another two tangent vector fields.
The components v i [f] are the contravariant components of the vector v in the basis f, and the components v i [f] are the covariant components of v in the basis f ...
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.
Then the derivative of f(v) ... Covariant derivative; Ricci calculus; References This page was last edited on 18 April 2024, at 03:06 (UTC). Text is ...
The covariant derivative in general relativity is a special example of the gauge covariant derivative. It corresponds to the Levi Civita connection (a special Riemannian connection ) on the tangent bundle (or the frame bundle ) i.e. it acts on tangent vector fields or more generally, tensors.
This "universal lift" then immediately induces lifts to vector bundles associated with F and hence allows the covariant derivative, and its generalisation to forms, to be recovered. If σ is a representation of K on a finite-dimensional vector space V , then the associated vector bundle F x K V over M has a C ∞ ( M )-module of sections that ...
Of course, the Jacobian matrix of the composition g ° f is a product of corresponding Jacobian matrices: J x (g ° f) =J ƒ(x) (g)J x (ƒ). This is a higher-dimensional statement of the chain rule. For real valued functions from R n to R (scalar fields), the Fréchet derivative corresponds to a vector field called the total derivative.