View source for 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

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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

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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

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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

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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

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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

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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|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Algebra: Algebraic structures. Multilinear algebra" , Addison-Wesley  (1974)  pp. Chapt. 3  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Marcus,   "Finite dimensional multilinear algebra" , '''1''' , M. Dekker  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Spain,   "Tensor calculus" , Oliver &amp; Boyd  (1970)</TD></TR></table>