I've been reading in my calc book that the gradient vector is always orthogonal to the surface.
So for a surface in space described by the level surface $f(x,y,z) = k$ where $k$ is a constant, $\nabla f$ is orthogonal to the surface at every point because the gradient is the normal vector of the surface at every point.
Then later I read about parametric surfaces where a surface is described by vector valued function $r(u,v) = <x(u,v), y(u,v), z(u,v)>$ and a normal vector $r_u \times r_v$ or $r_v \times r_u$
How are $r_u \times r_v$ and $\nabla f$ related here? I am referring to James Stewart's Text.
Also a last comment I want to make is, what about a normal vector to a surface that doesn't need to be described by a level surface? For example $f(x,y) = z = x^2 + y^2$? How would I go finding the normal vector at any point without rewriting it as $z - x^2 - y^2 = 0$ or parametrizing it?
A final Remark: I've been confusing the notion of the gradient vector being tangent to a surface instead of normal to it. There is this rather confusing picture I have which seems to suggests that the gradient vector really is tangent to a surface rather than normal because the gradient is formed by the vector sum of $\partial/\partial x$ and $\partial/\partial y$ and according to the picture I have, both $\partial/\partial x$ and $\partial/\partial x$ are "flat" and their sum should also be "flat" and not "pointing up"