Definition:Tensor

Definition

Let $V$ and $V^*$ be a vector space and its dual.

Then a (mixed) tensor $F$ of type $\tuple {k, l}$ is a multilinear map such that:

$\ds F : \underbrace {V \times \ldots \times V}_{\text {$k$ times}} \times \underbrace { {V^*} \times \ldots \times {V^*} }_{\text {$l$ times}} \to \R$

Cartesian Tensor

A cartesian tensor is a tensor on $\R^3$ that transforms appropriately under rotations.

Also known as

A tensor of type $\tuple {k, l}$ is also known as a $k$-times covariant and $l$-times contravariant tensor.

Also see

• Results about tensors can be found here.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.): Appendix $\text B$. Review of Tensors