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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).
Expressions for lengths, areas and volumes of objects in the vector space can then be given in terms of tensors with covariant and contravariant indices. Under simple expansions and contractions of the coordinates, the reciprocity is exact; under affine transformations the components of a vector intermingle on going between covariant and ...
In Newtonian mechanics the admissible frames of reference are inertial frames with relative velocities much smaller than the speed of light.Time is then absolute and the transformations between admissible frames of references are Galilean transformations which (together with rotations, translations, and reflections) form the Galilean group.
The constitutive relations between the and F tensors, proposed by Minkowski for a linear materials (that is, E is proportional to D and B proportional to H), are: = = where u is the four-velocity of material, ε and μ are respectively the proper permittivity and permeability of the material (i.e. in rest frame of material), and denotes the ...
The Kulkarni–Nomizu product is an important tool for constructing new tensors from existing tensors on a Riemannian manifold. Let A {\displaystyle A} and B {\displaystyle B} be symmetric covariant 2-tensors.
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 this more general framework, higher rank tensors can also have arbitrarily many and mixed covariant and contravariant ranks at the same time, denoted by raised and lowered indices within the Einstein summation convention.)
In general relativity and tensor calculus, the contracted Bianchi identities are: [1] = where is the Ricci tensor, the scalar curvature, and indicates covariant differentiation.