diff --git a/docs/en/mathematics/linear-algebra/tensors/tensor-formalism.md b/docs/en/mathematics/linear-algebra/tensors/tensor-formalism.md index c47e97e..431a4d1 100644 --- a/docs/en/mathematics/linear-algebra/tensors/tensor-formalism.md +++ b/docs/en/mathematics/linear-algebra/tensors/tensor-formalism.md @@ -53,9 +53,9 @@ $$ > > for all $(x,y) \in X \times Y$. -The outer product is associative, distributive with respect to addition and scalar multiplication, but not commutative. +The outer product is associative and distributive with respect to addition and scalar multiplication, but not commutative. -Note that although the same symbol is used for the outer product and the denotion of a tensor space, these are not equivalent. But are closely related. +Note that although the same symbol is used for the outer product and the denotion of a tensor space, these are not equivalent. For the following statements we take $p=q=r=s=1$ without loss of generality. diff --git a/docs/en/mathematics/linear-algebra/tensors/tensor-symmetries.md b/docs/en/mathematics/linear-algebra/tensors/tensor-symmetries.md index e69de29..25f4e94 100644 --- a/docs/en/mathematics/linear-algebra/tensors/tensor-symmetries.md +++ b/docs/en/mathematics/linear-algebra/tensors/tensor-symmetries.md @@ -0,0 +1,30 @@ +# Tensor symmetries + +We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n$, a pseudo inner product $\bm{g}$ on $V$ and a corresponding dual space $V^*$ with a basis $\{\mathbf{\hat e}^i\}.$ + +## Symmetric tensors + +> *Definition 1*: let $\pi = [\pi(1), \dots, \pi(k)]$ be any permutation of labels $\{1, \dots, k\}$, then $\mathbf{T} \in \mathscr{T}^0_q(V)$ is a symmetric covariant tensor if for all $\mathbf{v}_1, \dots, \mathbf{v}_q \in V$ we have +> +> $$ +> \mathbf{T}(\mathbf{v}_{\pi(1)}, \dots, \mathbf{v}_{\pi(q)}) = \mathbf{T}(\mathbf{v}_1, \dots, \mathbf{v}_q), +> $$ +> +> with $k = q$. +> +> Likewise $\mathbf{T} \in \mathscr{T}^p_0(V)$ is called a symmetric contravariant tensor if for all $\mathbf{\hat u}_1, \dots, \mathbf{\hat u}_p \in V^*$ we have +> +> $$ +> \mathbf{T}(\mathbf{\hat u}_{\pi(1)}, \dots, \mathbf{\hat u}_{\pi(p)}) = \mathbf{T}(\mathbf{\hat u}_1, \dots, \mathbf{\hat u}_p), +> $$ +> +> with $k = p$. + +This symmetry implies that the ordering of the (co)vector arguments in a tensor evaluation do not affect the outcome. + +> *Definition 2*: the vector space of symmetric covariant $q$-tensors is denoted by $\bigvee_q(V) \subset \mathscr{T}^0_q(V)$ and the vector space of symmetric contravariant $p$-tensors is denoted by $\bigwedge^p(V) \subset \mathscr{T}^p_0(V).$ + +Alternatively one may write $\bigvee_q(V) = V^* \otimes_s \cdots \otimes_s V^*$ and $\bigwedge^p(V) = V \otimes_s \cdots \otimes_s V$. + +## Antisymmetric tensors + diff --git a/docs/en/stylesheets/extra.css b/docs/en/stylesheets/extra.css index 4de843b..77b57a2 100644 --- a/docs/en/stylesheets/extra.css +++ b/docs/en/stylesheets/extra.css @@ -1,4 +1,4 @@ -[dir="ltr"] .md-typeset blockquote { +.md-typeset blockquote { border-left: .2rem solid rgba(68, 138, 255, 1); background-color: rgba(34, 44, 63, 0.6); color: rgb(240, 240, 240);