Converting surface integral to line

[Niles]

Homework Statement

I have the following integral
$$\int_{ABC}{\mathbf{v}\cdot \nabla f_id\sigma}$$
where $d\sigma$ is an area element, $\mathbf v$ is a velocity vector and $f_i$ some function. The integral is performed across a triangle ABC and it is assumed that f is linear.

In my book this integral becomes
$$\mathbf v\cdot \int_{AB}{f_id\mathbf l} + I_s,$$
where $I_s$ is the flux across $BC$ and $AC$. Can someone explain to me how this integral is solved?

 

[HallsofIvy]

Use "Stoke's theorem" that you should remember from Calculus:
$$\oint_C \vec{f}\cdot d\vec{r}= \int_\sigma\int \nabla\times\vec{f}dS$$
where C is the boundary of surface $\sigma$.

 

[Niles]

Use "Stoke's theorem" that you should remember from Calculus:
$$\oint_C \vec{f}\cdot d\vec{r}= \int_\sigma\int \nabla\times\vec{f}dS$$
where C is the boundary of surface $\sigma$.

Hi HallsofIvy

Thanks, I remember that. But the RHS is in terms of a rotation operatir, not the gradient as I have.