Tensors

A tensor of type (AKA valence) \({\left(m,n\right)}\) is defined to be an element of the tensor space

\(\displaystyle V_{m,n}\equiv\left(V\otimes\dotsb\left(m\:\textrm{times}\right)\dotsb\otimes V\right)\otimes\left(V^{*}\otimes\dotsb\left(n\:\textrm{times}\right)\dotsb\otimes V^{*}\right). \)

A pure tensor (AKA simple or decomposable tensor) of type \({\left(m,n\right)}\) is one that can be written as the tensor product of \({m}\) vectors and \({n}\) 1-forms; thus a general tensor is a sum of pure tensors. The integer \({\left(m+n\right)}\) is called the order (AKA degree, rank) of the tensor, while the tensor dimension is that of \({V}\). Vectors and 1-forms are then tensors of type \({\left(1,0\right)}\) and \({\left(0,1\right)}\). The rank (sometimes used to refer to the order) of a tensor is the minimum number of pure tensors required to express it as a sum. In “tensor language” vectors \({v\in V}\) are called contravariant vectors and 1-forms \({\varphi\in V^{*}}\) are called covariant vectors (AKA covectors). A tensor of type \({\left(k,0\right)}\) is then called a contravariant tensor, with covariant tensors being of type \({\left(0,k\right)}\), and other tensor types being called mixed tensors. Scalars can be considered tensors of type \({\left(0,0\right)}\).

Δ As noted above, the meanings of tensor rank and order are often swapped in the literature. Another potential source of confusion is that a mixed tensor is not the opposite of a pure tensor.

The infinite direct sum of the tensor spaces of every type forms an associative algebra. This algebra is sometimes called the “tensor algebra,” and “tensor” sometimes refers to the general elements of this algebra, in which case tensors as defined above are called homogeneous tensors. In this book, we will always use the term “tensor” to mean homogeneous tensor, and “tensor algebra” to mean the infinite direct sum of tensor powers, not including powers of the dual space.

An Illustrated Handbook