Everything about Component-free Treatment Of Tensors totally explained
In
mathematics, the modern component-free approach to the theory of
tensors views tensors initially as
abstract objects, 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 references 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, which is fairly abstract, requires an understanding of the
tensor product of
vector spaces without chosen
bases. The notion of a tensor product generalizes to vector spaces without chosen bases, and even further, to
modules. If you find this article difficult, try reading the main
tensor article and the
classical or
intermediate level treatments first.
Definition: Tensor Product of Vector Spaces
Let
V and
W be two
vector spaces over a common
field F. Their
tensor product
»
is a vector space over the same field F together with a
bilinear map
»
which is
universal in the following sense:
for every vector space
X over the field F and every F-bilinear map
»
there is a
unique F-linear map
»
such that
»
It is easy to see that a vector space
is unique up to isomorphism if it exists, and we write "
the tensor product" instead of "
a tensor product."
All its properties, except its existence, follow from the abstract definition, although some properties are more easily understood from an explicit model.
An explicit construction is easy to give using
bases _k
In older texts this transformation rule often serves as the definition of a tensor. Formally, this means that tensors were introduced as specific
representations of the
group of all changes of coordinate systems.
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