The Problem
Let and $\vec{v}$ and $\vec{w}$ be two vectors and $u$ be a function such that $$u_{a}-u_{b}=-\int\limits_b^a\vec{v}\cdot{d\vec{w}}$$ I need a pure mathematical proof of "$u$ decreases along the vector $\vec{v}$."
Information/s that can help
Maybe this could be done with its differential form i.e. $du=-\vec{v}\cdot{d\vec{w}}$ as it becomes simpler. But I can't go any further.