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.