Tensor (intrinsic Definition)

Tensor (intrinsic Definition)

For an introduction to the nature and significance of tensors in a broad context, see Tensor.

In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multi-linear concept. Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.

In differential geometry an intrinsic geometric statement may be described by a tensor field on a manifold, and then doesn't need to make reference to coordinates at all. The same is true in general relativity, of tensor fields describing a physical property. The component-free approach is also used heavily in abstract algebra and homological algebra, where tensors arise naturally.

Note: This article assumes an understanding of the tensor product of vector spaces without chosen bases. An overview of the subject can be found in the main tensor article.

Read more about Tensor (intrinsic Definition):  Definition Via Tensor Products of Vector Spaces, Tensor Rank, Universal Property, Tensor Fields, Basis

Other articles related to "tensor":

Tensor (intrinsic Definition) - Basis
... given coordinate system we have a basis ei}for the tangent space V this may vary from point to point if the manifold is not linear) and a corresponding dual basis ei}for the cotangent space V*(see ... For example purposes,then,take a tensorA in the space The components relative to our coordinate system can be written ... Here we used the Einstein notation,a convention useful when dealing with coordinate equations when an index variable appears both raised and lowered on the same side of an equation,we are ...