In this problem I am given numerous vector field equations (both in 2 and 3 dimensions) and I am being asked which level curve/surface they are perpendicular to. The general process I've been using is to work backwards with the partial derivatives to attempt to recover some function which I can identify. For example one of the vector fields was defined as such;
F = $x$i + $y$j
Knowing that $x=\frac{\partial f}{\partial x} $ as well as $y=\frac{\partial f}{\partial y} $, reconstructing the level curve (from what I understand) is as simple as finding the corresponding antiderivatives which yield these partials, thus
$0 = \frac{x^{2}}{2}+\frac{y^{2}}{2}+C$ (Clearly a family of circles)
My issue lies in this vector field:
F = $-y$i + $x$j
I can't find antiderivatives that satisfy this equation and therefore can't identify the level curve the field is meant to be intersecting perpendicularly (Although when graphing it looks a lot like lines which rotate about the origin). Any help would be greatly appreciated!