Interior contraction of the Riemannian volume form
= Interior contraction of the Riemannian volume form
{title2=$\iota_X\omega_g=\sum_i(-1)^{i-1}f_i\theta_i$}
In a positive <Riemannian orthonormal frame> $X=\sum_i f_iE_i$ with dual coframe $\omega_i$, the <Riemannian volume form> is $\omega_1\wedge\cdots\wedge\omega_n$. Inserting $E_i$ in its first slot omits $\omega_i$ and produces sign $(-1)^{i-1}$, giving the displayed <interior product> formula.