Determining an Exterior Normal

Question

Given a surface , that can be represented by the equation: $F(x,y,z)=0 $ . How can I determine which of the vectors $ -\text{gradF} ,\text {grad} F$ is the exterior normal and which is the interior normal?

In addition, if this surface can be represented as $ z=f(x,y) $ , we know that the vectors $ (f_x,f_y , -1 ) $ are normals. But how can I determine who is the exterior normal and who is the interior one?

Thanks !

Answer 1

Let's assume, as Edgar Matias suggested, that our surface is compact, so we have the interior as the bounded region and the exterior as the unbounded one. I don't think that you can answer this question by considering the gradient locally, since it's not too hard imagine two manifolds with the same gradient at a point, but where in one case the gradient is inward, and in the other case the gradient is outward (imagine a shape that folds over itself).

One possible way of answering this question is to integrate the gradient over the manifold, fixing the outward orientation on the manifold. If the integral is positive, the gradient was the external normal, otherwise it was the internal normal.