Let $X$ be a complex manifold and $D$ a normal crossing divisor. Let $U = X - D$ and $j: U \rightarrow X$ the natural map. Voisen observes that there is a natural inclusion $$\Omega^k_X(\log D) \subset j_*\Omega_U^k$$ Why is this so? Certainly, by adjointness of $j^{-1}, j_*$ we get a natural map $\Omega^k_X(\log D) \rightarrow j_*\Omega_U^k$. It's not obvious to me that this map is injective at the level of stalks. Basically I have two questions:

1.) Is it infact obvious that the natural map produced by adjointness is injective at the level of stalks? (Does this follow from more general "sheaf theory" theorems"?)

2.) Are you able to see in a more obvious way that a map $\Omega^k_X(\log D) \rightarrow j_*\Omega_U^k$ exists, and is an inclusion of sheaves "geometrically", that is, without using adjointness of $j^{-1}, j_*$?