I'm trying to understand why the line integral over vector fields are defined as such.
From Wikipedia article: "the line integral is viewed as the trajectory of a particle (in red) along a curve inside a vector field. Starting from $a$, the particle traces the path $C$ along the vector field $F$. The dot product (green line) of its tangent vector (red arrow) and the field vector (blue arrow) defines an area under a curve, which is equivalent to the path's line integral."
So my question is why this is relevant? Why do mathematicians define line integral over vector fields in this way? Is it because this line integral is the work done by the particle?