Comparing two definitions of determinant of coherent sheaves

Question

Let $f:X \to S$ be a smooth, projective morphism of $k$-schemes for some field $k$. Let $\mathcal{F}$ be a coherent sheaf on $X$ flat over $S$. We know (by Proposition $2.1.10$ of Huybrechts-Lehn, Geometry of the moduli space of sheaves) that $\mathcal{F}$ has a finite locally free resolution, say $$0 \to \mathcal{E}_n \to ... \to \mathcal{E}_1 \to \mathcal{F} \to 0.$$ We can define determinant of $\mathcal{F}$ as $\otimes_{i=1}^n (\mbox{det}(\mathcal{E}_i))^{(-1)^i}$. There is another definition of determinant of $\mathcal{F}$ as the top exterior power (exterior product with itself takes rank $\mathcal{F}$ number of times, the definition of rank coming from the leading coefficient of the Hilbert polynomial of $\mathcal{F}$). Is there any relation between these two definitions? When are they the same?