Converting surface integral to line

[Niles]

Homework Statement

I have the following integral
<br /> \int_{ABC}{\mathbf{v}\cdot \nabla f_id\sigma}<br />
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
<br /> \mathbf v\cdot \int_{AB}{f_id\mathbf l} + I_s,<br />
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.