Just a quick question to do with how tangential surfaces.
Consider the two surfaces $S_f$ and $S_g $ that are defined geometrically as:
$S_f : f(r) = f_0 $ and $S_g : g(r) = g_0 $,
where $f(r)$ and $g(r)$ are scalar fields, and $f_0$ and $g_0$ are constants. The intersection of the surfaces $S_f$ and $S_g$ is a curve, called C.
Why must $\nabla f \times \nabla g$ be tangential to C?
$\nabla f(r)$ is normal to $T_{r}S_f$ and $\nabla g(r)$ is normal to $T_{r}S_g$, for $r$ in $S_f$, resp. $S_g$.
$\nabla f(r)\times\nabla g(r)$ is orthogonal to both $\nabla f(r)$ and $\nabla g(r)$, hence it is tangent to both surfaces, for $r\in C$.