Let $D$ be a normal crossings divisor on a smooth projective algebraic variety $X$. Put $U=X -D$ and $j: U \hookrightarrow X$.

One usually defines the sheaf of logarithmic differentials along $D$ as the subsheaf $\Omega^p_X(\log D)$ of $j_\ast \Omega^p_U$ such that, for any open $V \subset X$, $\Omega^p_X(\log D)(V)$ consists of those $\omega \in j_\ast \Omega^P_U(V)$ such that both $\omega$ and $d\omega$ have at worst simple poles along $D \cap V$.

My question concerns the precise, rigurous definition of "having simple poles". I've seen the following:

**Definition**. $\omega \in j_\ast \Omega^p_U(V)$ has at worst simple poles along $D \cap V$ if, for any affine open $W \subseteq V$ such that the ideal of $D \cap W$ in $\mathcal{O}_X(W)$ is principal and for any generator $f$ of this ideal,

the section $f \cdot \omega_{| U \cap W}$ lies in the image of the restriction $\Omega^p_X(W) \to \Omega^p_X(U \cap W)$.

Q1: Can anybody explain to me why this definition agrees with the intuitive idea of simple poles.

Q2: Is there a simpler definition?

For instance, if we see $j_\ast \Omega^p_U$ as the direct limit $$ \varinjlim_{n} \Omega^p_X(n D) $$ can one somehow say that $\omega \in j_\ast \Omega^p_U(V)$ has at worst simple poles if it "belongs" to $\Omega^p_X(D)(V)$?

Thanks for your help :)