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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).
A system of n quantities that transform oppositely to the coordinates is then a covariant vector (or covector). This formulation of contravariance and covariance is often more natural in applications in which there is a coordinate space (a manifold ) on which vectors live as tangent vectors or cotangent vectors .
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.
The covariant derivative of a vector field with components is given by: ; = = + and ... Note that this transformation formula is for the mean curvature vector, ...
A covariant (invariant theory) is a bihomogeneous polynomial in x, y, ... and the coefficients of some homogeneous form in x, y, ... that is invariant under some group of linear transformations. Covariance and contravariance of vectors, properties of how vector coordinates change under a change of basis
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 linear algebra, linear transformations can be represented by matrices.If is a linear transformation mapping to and is a column vector with entries, then there exists an matrix , called the transformation matrix of , [1] such that: = Note that has rows and columns, whereas the transformation is from to .