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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:
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 covariance principle states the laws of physics should transform from one frame to another covariantly, that is, according to a representation of the covariance group. Special relativity considers observers in inertial frames , and the covariance group consists of rotations , velocity boosts , and the parity transformation .
Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. This is an artifact of the way in which one must compose the morphisms. The pair of functors Hom(A, –) and Hom(–, B) are related in a natural manner.
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.
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Fig. 3 – Transformation of local covariant basis in the case of general curvilinear coordinates Consider the one-dimensional curve shown in Fig. 3. At point P , taken as an origin , x is one of the Cartesian coordinates, and q 1 is one of the curvilinear coordinates.