= Metric volume tensor
{title2=$\epsilon_{a_1\ldots a_d}$}
On an oriented $d$-dimensional <pseudo-Riemannian manifold>, the metric volume tensor is the totally antisymmetric <tensor> with $\epsilon_{1\ldots d}=\sqrt{|\det g|}$ in positively oriented coordinates. It supplies the components of the <volume form>. It differs from the coordinate <Levi-Civita symbol>, whose entries are just $0,1,-1$. Raising indices with a <Lorentzian metric> can change signs; the metric, dimension and orientation must be specified.
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