= Solution
Let $(\theta^1,\ldots,\theta^n)$ be the <dual basis> of the positively oriented <orthonormal basis> $(v_1,\ldots,v_n)$. By the definition of the <Riemannian volume form>,
$$
\omega_X=\theta^1\wedge\cdots\wedge\theta^n.
$$
The <interior product of a differential form> with the outward unit normal is
$$
\iota_{v_1}\omega_X=\theta^2\wedge\cdots\wedge\theta^n.
$$
The vectors $(v_2,\ldots,v_n)$ form a positive orthonormal frame of $T\partial X$ by the <outward-normal-first boundary orientation>. Consequently the pullback of the last display is the positive unit boundary volume form:
$$
\boxed{\omega_{\partial X}=F^*(\iota_{v_1}\omega_X).}
$$
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