Gradient as normal vector.

Question

We know that gradient of a scalar valued function $f$ gives the normal vector to level surfaces $f=const.$

My question: Is gradient $\nabla f$ always gives outward normal vector and $-\nabla f$ gives inward normal vector to closed level surfaces of $f$ ?

Answer 1

No. Specifically, since $\nabla(-f) = -\nabla f$, we have that if $\nabla f$ gives an outward normal vector, $\nabla(-f)$ gives an inward normal vector.

Answer 2

No.

First of all, the concept of inward and outward normal makes sense only if the surface is closed.

But even for closed surfaces there is no relation. The level curves $f(x,y,z)=C$ and $-f(x,y,z)=-C$ are the same, but in one case the gradient is an outward normal and in the other an inward normal.