let $(\frac{\partial w}{\partial s})_u =\nabla w \cdot \frac{dr}{ds}= \nabla w \cdot \vec{u}$
I get why we let u be constant because we are intersecting some plane in the direction of u, but I don't understand why we let ds vary what is ds?
I think ds is $ds = \sqrt{dx^2+dy^2}$