I have been learning about differential forms, but do not understand exactly why a differential $dx$ forms a dual basis to the basis $\frac{\partial}{\partial x}$.
For example, expand a vector $\vec{w}$ as $\vec{w}=w^je_j$, $i=1,2,\ldots, n$ on an $n$-dimensional manifold $M$ with coordinates $x^i$. Why is it that $$dx^i\vec{w}=dx^iw^je_j=w^i$$ In other words, why is is true that $$dx^ie_j=\delta^i_j$$ I am used to thinking of the differential $dx$ as a differential displacement in the $x$ direction, and not as a dual basis. Is it possible to reconcile the two notions?
By definition, for any real-valued function $f$, the differential form $df$ is a map that takes a derivative operator to that derivative of $f$.
So $dx^i\frac{\partial}{\partial x^j}=\frac{\partial x^i}{\partial x^j}$ by definition.