My textbook defines the divergence of a vector field in a non orthogonal constant basis the following way: $$div(\vec{u})=\vec{a}^i\cdot\frac{\partial \vec{a}_ku^k}{\partial x^i}=\frac{\partial u^i}{\partial x^i}$$ Can you geometrically explain why do we take the dot product with the DUAL basis vector $\vec{a}^i$. My first thought was that we wanted to take the projection of the partial derivative in the direction orthogonal to the parallelogram (if you think in $\mathbb{R}^2$) and as we know $\vec{a}^k$-th dual basis is orthogonal to all regular basis vectors except $\vec{a}_k$. However the problem i can't seem to understand is that it is not a unit vector so it is not exactly the projection but has additional scaling. Where does that scaling come from?
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