A vector field can do work along a path when its component in the direction of motion is accumulated along that path.

Let a curve $C$ be parametrized by

$$\mathbf r(t),\qquad a\le t\le b,$$

and let $\mathbf F$ be a vector field. The line integral of $\mathbf F$ along $C$ is

$$\int_C\mathbf F\cdot d\mathbf r

\int_a^b\mathbf F(\mathbf r(t))\cdot\mathbf r'(t),dt.$$

Work

If $\mathbf F$ is a force field and $C$ is the path of an object, then

$$W=\int_C\mathbf F\cdot d\mathbf r$$

is the work done by the force along the path.

Only the component of the field tangent to the motion contributes to the integral.

Orientation matters

Reversing the path changes $d\mathbf r$ to its negative, so

$$\int_{-C}\mathbf F\cdot d\mathbf r =-\int_C\mathbf F\cdot d\mathbf r.$$

This differs from a scalar line integral with respect to $ds$, which does not depend on orientation.

Circulation

When $C$ is a closed curve, the notation

$$\oint_C\mathbf F\cdot d\mathbf r$$

is often used. It measures the circulation of the field around the loop.

Line integrals of vector fields connect a field distributed through space with its accumulated tangential effect along a chosen oriented path.