Covariant tensor

of valency $ s \geq 1 $

A tensor of type $ ( 0, s) $, an element of the tensor product $ T _ {s} ( E) = E ^ {*} \otimes \dots \otimes E ^ {*} $ of $ s $ copies of the dual space $ E ^ {*} $ of the vector space $ E $ over a field $ K $. The space $ T _ {s} ( E) $ is itself a vector space over $ K $ with respect to the addition of covariant tensors of the same valency and multiplication of them by scalars. Let $ E $ be finite dimensional, let $ e _ {1} \dots e _ {n} $ be a basis of $ E $ and let $ e ^ {1} \dots e ^ {n} $ be the basis dual to it of $ E ^ {*} $. Then $ \mathop{\rm dim} T _ {s} ( E) = n ^ {s} $ and the set of all tensors of the form $ e ^ {i _ {1} } \otimes \dots \otimes e ^ {i _ {s} } $, $ 1 \leq i _ {1} \dots i _ {s} \leq n $, forms a basis for $ T _ {s} ( E) $. Any covariant tensor can be represented in the form $ t = t _ {i _ {1} \dots i _ {s} } e ^ {i _ {1} } \otimes \dots \otimes e ^ {i _ {s} } $. The numbers $ t _ {i _ {1} \dots i _ {s} } $ are called the coordinates, or components, of the covariant tensor relative to the basis $ e _ {1} \dots e _ {n} $ of $ E $. Under a change of a basis of $ E $ according to the formulas $ e _ {j} ^ \prime = a _ {j} ^ {i} e _ {i} $ and the corresponding change of the basis of $ T _ {s} ( E) $, the components of the covariant tensor $ t $ are changed according to the so-called covariant law

$$ t _ {j _ {1} \dots j _ {s} } ^ \prime = \ a _ {j _ {1} } ^ {i _ {1} } \dots a _ {j _ {s} } ^ {i _ {s} } t _ {i _ {1} \dots i _ {s} } . $$

If $ s = 1 $, the covariant tensor is called a covariant vector; when $ s \geq 2 $ a covariant tensor corresponds in an invariant way with an $ s $- linear mapping from the direct product $ E ^ {s} = E \times \dots \times E $( $ s $ times) into $ K $ by taking the components of the covariant tensor $ t $ relative to the basis $ e _ {1} \dots e _ {n} $ as the values of the $ r $- linear mapping $ \widetilde{t} $ at the basis vectors $ ( e _ {i _ {1} } \dots e _ {i _ {s} } ) $ in $ E ^ {s} $, and conversely; for this reason a covariant tensor is sometimes defined as a multilinear functional on $ E ^ {s} $.

For references see Covariant vector.