In the following equation, we have

$\displaystyle u=u(x,y),~v=v(x,y)$ are scalar functions (solutions to a PDE).

The directional derivative I'm familiar with is the following:

The dir. deriv of $\displaystyle u(x,y)$ in the unit direction $\displaystyle \hat{v}$ is

$\displaystyle \nabla{u}\cdot \hat{v}=u_x v_1 + u_y v_2$.

So if our vector $\displaystyle \vec{v} = (\cos(\theta), \sin(\theta))$ then I get the second step. But what does $\displaystyle u_\sigma\vert_\theta$ mean? And how did they get $\displaystyle \displaystyle\frac{dx}{d\sigma}$?