Pseudo vector rotation

Question

I originally posted this in the math stack exchange but after no one answered I remembered that tensors are way more used in physics than in math.

in my textbook, it says that if we have 2 vectors, a and b, $c=a\times b$ is a pseudovector. Then, the rotation transformation over c is $c^\prime_i=det(R)R_{ij}c_j$. Then it says that this is because we have a Levi-Civita in the definition of c, $c_i=\epsilon_{ijk}a_jb_k$. R is the rotation matrix. Then, in good math book fashion, it leaves the "trivial" prove to the reader. But for me, it's far from trivial I don´t even know how to start the prove of the rotation property. I think it has to do with

$\epsilon_{ijk}^\prime=det(R)R_{il}R_{jm}R_{km}\epsilon_{lmn}$

and/or that a pseudovector can be written as

$c_i=\frac{1}{2}\epsilon_{ijk}C_{jk}$

But I havent manage to do any progress. if someone can show me how to go from $c_i^\prime=R_{ij}c_j=R_{ij}\epsilon_{jkl}a_kb_l$ to $c^\prime_i=det(R)R_{ij}c_j$ I will appreciate it a lot.

Answer 1

I manage to prove it here's how I did it.

First of all, we need to remember that the following properties.

1. The Levi-Civita symbol doesn't change under rotations (prime quantities are rotated tensors) and the following relationship holds \begin{equation} \epsilon^\prime_{ijk}=\epsilon_{ijk}=det(R)\epsilon_{olm}R_{io}R_{jl}R_{km} \end{equation}

2. The rotation matrix R is orthogonal, this means that $R^{-1}_{ij}=R^T_{ij}$, therefore \begin{equation} R_{ki}R_{kj}=R_{ik}R_{jk}=\delta_{ij} \end{equation}

Then, we hypothesize that a and b are vectors, then the rotations of them are \begin{equation} A^\prime_i=R_{ij}A_j\quad B^\prime_i=R_{ij}B_j \end{equation} Then, the rotation of c must be \begin{equation} C_i^\prime=\epsilon^\prime_{ijk}A^\prime_jB^\prime_k \end{equation} Then, substituting and using the above properties we have \begin{equation} \begin{split} &C_i^\prime=\epsilon_{ijk}(R_{jp}A_p)(R_{kq}B_q)\\ &C_i^\prime=det(R)\epsilon_{olm}R_{io}R_{jl}R_{km}(R_{jp}A_p)(R_{kq}B_q)\\ &C_i^\prime=det(R)\epsilon_{olm}\delta_{lp}\delta_{mq}R_{io}A_pB_q\\ &C_i^\prime=det(R)\epsilon_{olm}R_{io}A_lB_m\\ &C_i^\prime=det(R)R_{io}(A\times B)_o\\ &C_i^\prime=det(R)R_{io}C_o\\ \end{split} \end{equation} proving the rotation of the psuedovector propierty