line integral of a vector field (work)

The integral $\int_C \mathbf{F}\cdot d\mathbf{r}$ , adding up the component of a vector field along a curve's direction of travel — physically, work.

With $C$ parametrized by $\mathbf{r}(t)$ , $t\in[a,b]$ , the line integral of a vector field is $\int_C \mathbf{F}\cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot \mathbf{r}'(t)\,dt$ . Only the part of $\mathbf{F}$ pointing along the curve's own direction of travel contributes; a force perpendicular to motion does no work. Unlike the scalar line integral, this one depends on orientation: reversing $C$ 's direction negates $\mathbf{r}'(t)$ at every point and hence negates the whole integral, $\int_{-C}\mathbf{F}\cdot d\mathbf{r} = -\int_C\mathbf{F}\cdot d\mathbf{r}$ .

Examples

For $\mathbf{F}(x,y)=(y,x)$ along $\mathbf{r}(t)=(t,\,t^2)$ , $0\le t\le1$ : $\int_C \mathbf{F}\cdot d\mathbf{r} = 1$ .