I asked a question on physics stack exchange a few days ago here:
and got some really good answers. The one I accepted seemed like it made sense at first glance, but at a closer look it's puzzling me. The answerer uses the properties $(i,j=u,v)$
$$ \oint f \phi_i du dv = - \oint f_i \phi du dv $$ $$ \oint f \phi_{ij} du dv = \oint f_{ij} \phi du dv $$
which seems sort of odd. He's using a line integral to denote a double integral (physicists, you know... I do it too) and I'm just very confused as to where this property came from. As far as I know, integration by parts in 2D doesn't look like that. Does the claim make any sense? Any help is appreciated :)