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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 ...
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 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.
In general relativity and tensor calculus, the contracted Bianchi identities are: [1] = where is the Ricci tensor, the scalar curvature, and indicates covariant differentiation.
The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant antisymmetric tensor whose entries are B-field quantities. [1] = (/ / / / / /) and the result of raising its indices is = = (/ / / / / /), where E is the electric field, B the magnetic field, and c the speed of light.
The collection of tensors on a vector space and its dual forms a tensor algebra, which allows products of arbitrary tensors. Simple applications of tensors of order 2 , which can be represented as a square matrix, can be solved by clever arrangement of transposed vectors and by applying the rules of matrix multiplication, but the tensor product ...