Rotation of a pseudovector 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.
 A: I manage to prove it, here's how I did it.
First of all, we need to remember that the following properties

*

*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}


*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
