The transformation equation of a mixed tensor Tᵢˆₖ of rank 3 involves the components changing under a change of basis using the transformation matrix A and its inverse, following the summation convention.
The transformation equation of a mixed tensor Tᵢˆₖ of rank 3 expresses how the components of the tensor change under a change of basis. When the basis vectors are transformed by a matrix Aⁿᵣ, the components of the tensor transform according to the rule:
Tᵢᵣₖ' = Aᵢˢ Aₙᵣ (A⁻¹)ₖˡ Tᵠᵡₒ
In this equation, A⁻¹ represents the inverse of the matrix A, and summation over repeated indices is implied following Einstein's summation convention. This expression gives us the new components Tᵢᵣₖ' of the tensor in the transformed basis.