Examples

Vector line integral - flux across a path

Vector line integral: flux across a path

Problem: Suppose $\mathbf{F}=(e^y,2x-1)$. Compute the flux of $\mathbf{F}$ across the curve $y=x^2$ for $x\in [0,1]$ (oriented left to right).
Solution: Use $\mathbf{r}(t)=(t,t^2)$ for $t\in [0,1]$. Then ${\mathbf{r}}^{\prime }(t)=(1,2t)$ and $\mathbf{n}(t)=(2t,-1)$. At $\mathbf{r}(t)$, $\mathbf{F}$ has the value $(e^{t^2},2t-1)$. So $\mathbf{F}\cdot \mathbf{n}(t)=2t{e}^{t^2}-2t+1$, and ${\int }_{\mathcal{C}}\mathbf{F}\cdot {\mathbf{e}}_{\mathbf{n}}ds={\int }_0^{1}2t{e}^{t^2}-2t+1dt=e-1$.