Addded tensor symmetries.

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.
 

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
+

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);