Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 12B (Vector Calculus)

A tensor of rank $n$ is a multilinear map of $n$ vector or covector arguments, equivalently an element of an $n$-fold tensor product. In components it obeys one change-of-basis factor for each of its $n$ indices.

A tensor is totally antisymmetric tensor if interchanging any two indices changes its sign. Consequently every component with repeated indices vanishes. In ${\mathbb{R}}^3$:

These are all possibilities.

A tensor is isotropic tensor if its components are unchanged under every proper orthogonal change of basis. The general isotropic rank-four tensor iswhere $a,b,c$ are scalars.

Both sides are alternating in $i,j,k$ and in $l,m,n$. If either triple repeats an index, both sides vanish. Otherwise each triple is a permutation of $(1,2,3)$; the determinant of the corresponding permutation matrix is exactly the product of the two permutation signs. HenceThere are $3!=6$ nonzero components of ${ϵ}_{ijk}$, each equal to $\pm 1$, so

Contracting two antisymmetric tensors gives the nonzero isotropic rank-four tensorA nonzero isotropic rank-five tensor is, for example,

More generally, one obtains the rank-five isotropic building blocks by choosing which two of the five indices belong to the Kronecker delta and putting the remaining three into the Levi-Civita symbol. There aresuch placements:Thus the unrestricted isotropic rank-five ansatz requires ten scalar component coefficients, one for each delta-epsilon placement.