Pure significance of line integrals of vector fields

Question

I can understand how the line integral of a scalar function has pure-mathematical significance, representing an area beneath a curve. But is there a pure significance to the line integral of a vector field? The only understanding I have of this is the physical example of work done on a moving particle. Thanks for any insight.

Answer 1

In general, the vector line integral of a vector field $\mathbf F : \mathbb{R}^n \to \mathbb{R}^n$ along a smooth oriented curve $C$ is a measure of how much the field $\mathbf F$ "flows" along the curve $C$. That is, if $\int_C \mathbf F \cdot d\mathbf s > 0$, the field $\mathbf F$ has a net flow with $C$, while if $\int_C \mathbf F \cdot d\mathbf s < 0$, the field $\mathbf F$ has a net flow against $C$. This is a physical interpretation you can give to the line integral of a vector field.

Answer 2

We may perform line integrals of vector fields around closed loops to gather information about sources of those fields. This is true in electrostatics, where the line integral of the magnetic field about a closed loop produces an electric current within that loop. In electrodynamics, line integrals of electric and magnetic fields about a closed loop produces the time rate of change of magnetic and electric flux, respectively, through that loop.

Answer 3

Taking a line integral of a vector field along a curve is the same as integrating the corresponding $1$-form along that curve. So I guess it's an example of something quite pure, although this alone doesn't demonstrate significance.