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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).
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:
Download as PDF; Printable version; ... The covariant derivative of a function ... Note that this transformation formula is for the mean curvature vector, ...
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.
The relationship between general covariance and general relativity may be summarized by quoting a standard textbook: [3] Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior geometry" and for a geometric, coordinate-independent formulation of physics.
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 ...
In physics, general covariant transformations are symmetries of gravitation theory on a world manifold. They are gauge transformations whose parameter functions are vector fields on X {\displaystyle X} .
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).