Let $C$ be a simple, piecewise-smooth curve between points $A$ and $B$, as shown below:
Let $\vec{f}$ be a vector field that is defined on the curve (shown above in blue). The magnitude of $\vec{f}$ is constant everywhere; its direction is perpendicular to the tangent of the curve in the sense indicated. The integral of interest is: $$\int_{C} \vec{f}dl$$ where $dl$ is differential arclength. Note the result of the integral is a vector.
How do I show the integral is independent of path for all curves $C$, given some end points $A$ and $B$?
(Application: water pressure (simplified to 2 dimensions), for example. $\vec{f}$ can be thought of as a force per unit length on a submerged line, and the integral represents the total force on the line, which is represented by the curve $C$.)