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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 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 terminology is justified because under a change of basis,
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 explicit form of a covariant transformation is best introduced with the transformation properties of the derivative of a function. Consider a scalar function f (like the temperature at a location in a space) defined on a set of points p, identifiable in a given coordinate system , =,, … (such a collection is called a manifold).
A covariant derivative in differential geometry is a linear differential operator which takes the directional derivative of a section of a vector bundle in a covariant manner. It also allows one to formulate a notion of a parallel section of a bundle in the direction of a vector: a section s is parallel along a vector X {\displaystyle X} if ∇ ...
The material derivative is defined for any tensor field y that is macroscopic, with the sense that it depends only on position and time coordinates, y = y(x, t): +, where ∇y is the covariant derivative of the tensor, and u(x, t) is the flow velocity.
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 ...
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.