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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).
The covariant derivative of a vector field with components is given by: ; = = + and ... Note that this transformation formula is for the mean curvature vector, ...
Because the components of the linear functional α transform with the matrix A, these components are said to transform covariantly under a change of basis. The way A relates the two pairs is depicted in the following informal diagram using an arrow. A covariant relationship is indicated since the arrows travel in the same direction:
The transformations between frames are all arbitrary (invertible and differentiable) coordinate transformations. The covariant quantities are scalar fields, vector fields, tensor fields etc., defined on spacetime considered as a manifold. Main example of covariant equation is the Einstein field equations.
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.
In probability theory and statistics, the covariance function describes how much two random variables change together (their covariance) with varying spatial or temporal separation. For a random field or stochastic process Z ( x ) on a domain D , a covariance function C ( x , y ) gives the covariance of the values of the random field at the two ...
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These ...
Yoneda's lemma implies that every natural transformation between Hom functors is of this form. In other words, the Hom functors give rise to a full and faithful embedding of the category C into the functor category Set C op (covariant or contravariant depending on which Hom functor is used).