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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.
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.
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 .
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
I'm feeling optimistic in part because I think of the transformation that's possible with AI. But I also think there's more awareness right now of the perils and I hope, hope, hope that the ...
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.