Contravariant tensor

of valency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c0259702.png" />

A tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c0259703.png" />, i.e. an element of the tensor product

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c0259704.png" />

of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c0259705.png" /> copies of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c0259706.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c0259707.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c0259708.png" /> is a vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c0259709.png" /> with respect to the operations of addition of contravariant tensors of the same valency and multiplication of them by a scalar. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597010.png" /> be a finite-dimensional vector space with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597011.png" />. Then the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597012.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597013.png" />; one possible basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597014.png" /> is given by all possible contravariant tensors of the form

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597015.png" />

Any contravariant tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597016.png" /> can be represented in the form

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597017.png" />

The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597018.png" /> are called the coordinates or components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597019.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597021.png" />. On changing to a new basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597022.png" /> according to the formulas

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597023.png" />

the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597024.png" /> change according to the so-called contravariant law

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597025.png" />

When the valency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597026.png" /> equals 1, a contravariant tensor is the same as a vector, that is, an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597027.png" />; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597028.png" />, a contravariant tensor can be related in an invariant way with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597029.png" />-linear mapping into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597030.png" /> of the direct product

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597031.png" />

of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597032.png" /> copies of the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597033.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597034.png" />. For this it suffices to take as the components of the contravariant tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597035.png" /> the values of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597036.png" />-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597037.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597038.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597039.png" /> are the basis elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597040.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597041.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597042.png" />), and conversely. For this reason contravariant tensors are sometimes directly defined as multilinear functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597043.png" />.

Comments

More generally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597044.png" /> be a commutative ring with unit element and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597045.png" /> a unitary module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597046.png" />. Then the elements of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597047.png" />-fold tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597048.png" /> are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597050.png" />-contravariant tensors or contravariant tensors of valency or order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597051.png" />. The phrase "contravariant tensor of order r" is also used to denote a contravariant tensor field of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597052.png" /> over a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597053.png" />; cf. Tensor bundle. Such a field assigns to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597054.png" /> an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597055.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597056.png" />-fold tensor product of the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597057.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597058.png" />. In the setting of rings and modules such a tensor field is simply an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597059.png" />-contravariant tensor of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597060.png" /> of sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597061.png" /> (i.e. the vector fields) over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597062.png" /> of smooth functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597063.png" />.

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)