Contravariant tensor
of valency
A tensor of type , i.e. an element of the tensor product
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of copies of a vector space
over a field
. The space
is a vector space over
with respect to the operations of addition of contravariant tensors of the same valency and multiplication of them by a scalar. Let
be a finite-dimensional vector space with basis
. Then the dimension of
is
; one possible basis in
is given by all possible contravariant tensors of the form
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Any contravariant tensor can be represented in the form
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The numbers are called the coordinates or components of
with respect to
in
. On changing to a new basis in
according to the formulas
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the components of change according to the so-called contravariant law
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When the valency equals 1, a contravariant tensor is the same as a vector, that is, an element of
; when
, a contravariant tensor can be related in an invariant way with an
-linear mapping into
of the direct product
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of copies of the dual space
to
. For this it suffices to take as the components of the contravariant tensor
the values of the
-linear mapping
at
(where
are the basis elements in
dual to
, that is,
), and conversely. For this reason contravariant tensors are sometimes directly defined as multilinear functionals on
.
Comments
More generally, let be a commutative ring with unit element and
a unitary module over
. Then the elements of the
-fold tensor product
are called
-contravariant tensors or contravariant tensors of valency or order
. The phrase "contravariant tensor of order r" is also used to denote a contravariant tensor field of order
over a smooth manifold
; cf. Tensor bundle. Such a field assigns to each
an element of
, the
-fold tensor product of the tangent space to
at
. In the setting of rings and modules such a tensor field is simply an
-contravariant tensor of the module
of sections of
(i.e. the vector fields) over the ring
of smooth functions on
.
References
[a1] | N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Multilinear algebra" , Addison-Wesley (1974) pp. Chapt. 3 (Translated from French) |
[a2] | M. Marcus, "Finite dimensional multilinear algebra" , 1 , M. Dekker (1973) |
[a3] | B. Spain, "Tensor calculus" , Oliver & Boyd (1970) |
Contravariant tensor. I.Kh. Sabitov (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contravariant_tensor&oldid=19302