# Isotropic tensors #turbulence #completed Isotropic tensors refer to tensors which are invariant under rotations. It is known from the [[Invariant theory|theory of invariants]] that any invariant function of any number of vectors $\xi_i, a_i, b_i, \dots$ can be expressed in terms of the fundamental invariants of the following two types: 1. Scalar products $(\vec{\xi}\cdot\vec{a})=\xi_i a_i$ for any two vectors including $\vec{\xi}\cdot\vec{\xi}$. 2. Determinants $[\vec{\xi}\vec{a}\vec{b}] = \epsilon_{ijk}\xi_i a_j b_k$ ![[General expressions of isotropic tensors#Rank 2]] Following a similar procedure, we can also derive expressions for [[General expressions of isotropic tensors#Rank 1|isotropic vectors]], and [[General expressions of isotropic tensors#Rank 3|isotropic tensors of rank 3]]. We can also consider taking [[Derivative of isotropic tensors|derivatives]] and performing various vector calculus operations on these tensors, such as taking the [[Divergence of isotropic tensors|divergence]] and [[Curl of isotropic tensors |curl]], and even the [[Laplacian of isotropic tensors|Laplacian]]. If we relax the constraint that the tensor is invariant under reflections, we can obtain what is known as [[Skew-isotropic tensors|skew-isotropic tensors]]. ## References 1. Robertson, H. P. (1940, April). The invariant theory of isotropic turbulence. In _Mathematical Proceedings of the Cambridge Philosophical Society_ (Vol. 36, No. 2, pp. 209-223). Cambridge University Press.