Proof the Levi-Civita symbol is a tensor

If $R^i_j$ are the components of the matrix of an orthogonal linear transformation in Euclidean $3$-space, then the general transformation rule for an affine tensor $\epsilon_{ijk}$ should read $$\epsilon'_{ijk} = R^p_iR^q_j R^r_k \epsilon_{pqr}.$$

Since here $\epsilon$ is the Levi-Civita symbol, whose values depend not on the coordinate system but only on the numerical indices $i,j,k$ (i.e., $\epsilon' = \epsilon$), this is the same thing as $$\epsilon_{ijk} = R^p_iR^q_j R^r_k \epsilon_{pqr}.$$

Note that since $\epsilon_{pqr}$ vanishes on degenerate multi-indices, the right-hand side only consists of six terms (one for each proper multi-index) and looks like $R^1_iR^2_jR^3_k - R^2_iR^1_jR^3_k + \cdots$. To evaluate this, there are three cases to consider:

• If any two of $i,j,k$ share the same value, then the terms cancel each other pairwise and we get $0$.
• If $ijk$ is an even permutation, then this is just the Leibniz formula for the determinant, and the right-hand side is $\det(R)$.
• If $ijk$ is an odd permutation, then we may swap components pairwise in every term to recover $-\det(R)$ from the previous computation.

Therefore: if $R$ is a proper ($\det = 1$) orthogonal transformation, we find that the right-hand side coincides with the left-hand side, and $\epsilon$ behaves like a honest affine tensor of rank $3$. On the other hand, if $R$ is improper ($\det = -1$), we find that $$R^p_iR^q_j R^r_k \epsilon_{pqr} = -\epsilon_{ijk}.$$ This says that the Levi-Civita symbol is not a proper affine tensor but rather a pseudotensor.