11. Unitary and rotation tensors#
Some notes in the introduction to classical mechanics: Rotations: Tensor formalism for rotations.
todo Move the mathematical treatment here?
todo Discuss vector operations that are invariant under 3-dimensional rotation and other unitary tensor applications (i.e. reflections). These properties are useful to prove the general expression of isotropic tensors.
11.1. Invariant operations under rotations and unitary transformations#
Proof that/if:
the only independent invariant operations producing a scalar with two vectors \(\mathbf{u}\), \(\mathbf{v}\) are
\[\begin{split}\begin{aligned} & \mathbf{u} \cdot \mathbf{u} = |\mathbf{u}|^2 \\ & \mathbf{v} \cdot \mathbf{v} = |\mathbf{v}|^2 \\ & \mathbf{u} \cdot \mathbf{v} \end{aligned}\end{split}\]the only independent invariant operations producing a scalar with three vectors \(\mathbf{u}\), \(\mathbf{v}\), \(\mathbf{w}\) are
\[\begin{aligned} & \mathbf{u} \times \mathbf{v} \cdot \mathbf{w} \end{aligned}\]the only independent invariant operations producing a scalar with four vectors \(\{ \mathbf{u}^{(i)} \}_{i=1:4}\) are
\[\begin{aligned} \mathbf{u}^{(k_1)} \cdot \mathbf{u}^{(k_2)} \ \mathbf{u}^{(k_3)} \cdot \mathbf{u}^{(k_4)} \ , \end{aligned}\]with every index \(k_j\) independently ranging from \(1\) to \(4\).