on the tensor bundle a connection is needed, except for the special case of the exterior derivative of antisymmetric tensors. A tensor bundle is a fiber...

In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space...

metric field on M consists of a metric tensor at each point p of M that varies smoothly with p. A metric tensor g is positive-definite if g(v, v) > 0 for...

universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. The tensor product of an algebra and...

mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor. It is therefore a tensor field of order three....

differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing...

act as a multilinear operator on vector fields, or on other tensor fields. The tensor bundle is not a differentiable manifold in the traditional sense,...

vector bundle endowed with a bundle metric and its dual. Given a (0, 2) tensor X = Xij ei ⊗ ej, we define the trace of X through the metric tensor g by...

relationship between the Ricci tensor and the matter content of the universe. Like the metric tensor, the Ricci tensor assigns to each tangent space of...

notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern...

F is a vector bundle E ⊕ F over X whose fiber over x is the direct sum Ex ⊕ Fx of the vector spaces Ex and Fx. The tensor product bundle E ⊗ F is defined...

differential geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is a bilinear map of two input vectors...

the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product...

differentiation in general vector bundles which were, in contrast to the classical bundles of interest to geometers, not part of the tensor analysis of the manifold...

two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span V ⊗ W {\displaystyle V\otimes W} in the sense...

the dual vector bundle E ∗ {\displaystyle E^{*}} , tensor powers E ⊗ k {\displaystyle E^{\otimes k}} , symmetric and antisymmetric tensor powers S k E ...

algebra bundle is a vector bundle. Examples include the tensor-algebra bundle, exterior bundle, and symmetric bundle associated to a given vector bundle, as...

electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a...

mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear...

mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the...

In mathematics, and particularly topology, a fiber bundle (Commonwealth English: fibre bundle) is a space that is locally a product space, but globally...

tensor product of vector bundles E, F (over the same space X) is a vector bundle, denoted by E ⊗ F, whose fiber over each point x ∈ X is the tensor product...

In mathematics, a symmetric tensor is an unmixed tensor that is invariant under a permutation of its vector arguments: T ( v 1 , v 2 , … , v r ) = T (...

\mathbf {R} } be a multilinear form on W (also known as a tensor – not to be confused with a tensor field – of rank (0, s), where s is the number of factors...

manifold M {\displaystyle M} and the metric tensor is given as a covariant, second-degree, symmetric tensor on M {\displaystyle M} , conventionally denoted...

(electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), and general relativity (stress–energy tensor, curvature tensor, ...). In...

of tensor theory. For expositions of tensor theory from different points of view, see: Tensor Tensor (intrinsic definition) Application of tensor theory...

independent of any metric tensor and coordinate system. Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms...

one grand object called the tensor bundle. A tensor field is then defined as a map from the manifold to the tensor bundle, each point p {\displaystyle...

stress-energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor field quantity...