Assume I have a vector field expressed as:
$F=f(x,y,z)i + g(x,y,z)j +h(x,y,z)k$
Assume I have a level surface (spheres, ellipsoids, hyperboloid...) $S$.
What does it mean for $F$ to be perpendicular to $S$? In other words, what mathematical property must be satisfied?
When we say some vector is perpendicular to some curve/surface at $p$, what we really mean is that vector is perpendicular to the tangent space of that curve/surface. In the case of the curve, the tangent space is a line. In the case of a surface, the tangent space is a plane.
In this case it means $F$ is perpendicular to the tangent plane of $S$ (at some point $p$). Which means that $F$ is orthogonal to all tangent vectors to $S$ at $p$.