Covariant tensor

of valency

A tensor of type , an element of the tensor product of copies of the dual space of the vector space over a field . The space is itself a vector space over with respect to the addition of covariant tensors of the same valency and multiplication of them by scalars. Let be finite dimensional, let be a basis of and let be the basis dual to it of . Then and the set of all tensors of the form , , forms a basis for . Any covariant tensor can be represented in the form . The numbers are called the coordinates, or components, of the covariant tensor relative to the basis of . Under a change of a basis of according to the formulas and the corresponding change of the basis of , the components of the covariant tensor are changed according to the so-called covariant law

If , the covariant tensor is called a covariant vector; when a covariant tensor corresponds in an invariant way with an -linear mapping from the direct product ( times) into by taking the components of the covariant tensor relative to the basis as the values of the -linear mapping at the basis vectors in , and conversely; for this reason a covariant tensor is sometimes defined as a multilinear functional on .

For references see Covariant vector.