I’ve been transforming an operator into spherical coordinates and came across this problem: So I have to compute $\vec{e}_r \times \vec{e}_\phi$ using the Levi Civita notation this comes to $\epsilon_{i r \theta} \vec{e}_r \vec{e}_\phi $ with i then being equal to $\phi$ as it’s in spherical coordinates now $\epsilon_{\phi r \theta} = - \epsilon_{r \theta \phi}$. In the solutions they then go on to say that because of this previous statement $\vec{e}_r \times \vec{e}_\phi = -\vec{e}_\theta$, I don’t really see how they came to this last equality from the aforementioned statement, thanks in advance.
2026-04-24 07:00:28.1777014028
Levi civita cross products of spherical unit vectors
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Ok as answer I got that this was just conventional just like choosing $E_x \times E_z = -E_y$ so i'm assuming this 'proof' is then just quite wrong really.