# Optical flow Horn & Schunck smoothness error

I was reading about the Horn & Schunck method and they define the smoothness error as (Robot Control 1988 (SYROCO'88): Selected Papers from the 2nd IFAC Symposium p. 348):

$$E_{sm} = \iint{\left(u_x^2+u_y^2+v_x^2+v_y^2\right)}dx\:dy$$

, where $$(u,v)$$ is the optical flow velocity vector and the indexes are shorthand for the $$x$$ and $$y$$ derivatives.

But since $$u$$ is already in the $$x$$ direction, shouldn't $$u_y$$ be zero? Same for $$v_x$$.