Contravariant tensor

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of valency

A tensor of type , i.e. an element of the tensor product

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

Any contravariant tensor can be represented in the form

The numbers are called the coordinates or components of with respect to in . On changing to a new basis in according to the formulas

the components of change according to the so-called contravariant law

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

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 .


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 .


[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)
How to Cite This Entry:
Contravariant tensor. I.Kh. Sabitov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098