Tensor (intrinsic definition)

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 multilinear concept. Their 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 extensively in abstract algebra and homological algebra, where tensors arise naturally.

Definition via tensor products of vector spaces

Given a finite set {V1, ..., Vn} of vector spaces over a common field F, one may form their tensor product V1 ⊗ ... ⊗ Vn, an element of which is termed a tensor.

A tensor on the vector space V is then defined to be an element of (i.e., a vector in) a vector space of the form:

V

⊗

⋯

⊗

V

⊗

V

∗

⊗

⋯

⊗

V

∗

{\displaystyle V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*}}

where V∗ is the dual space of V.

If there are m copies of V and n copies of V∗ in our product, the tensor is said to be of type (m, n) and contravariant of order m and covariant of order n and of total order m + n. The tensors of order zero are just the scalars (elements of the field F), those of contravariant order 1 are the vectors in V, and those of covariant order 1 are the one-forms in V∗ (for this reason, the elements of the last two spaces are often called the contravariant and covariant vectors). The space of all tensors of type (m, n) is denoted

T

n

m

(

V

)

=

V

⊗

⋯

⊗

V

⏟

m

⊗

V

∗

⊗

⋯

⊗

V

∗

⏟

n

.