Let $\Gamma$ be the unit 2-sphere, say and let $f:\Gamma \to \mathbb{R}$ be some nice function.
My teacher says when i calculate the surface gradient $$\nabla_\Gamma f = \nabla f – (\nabla f\cdot \nu)\nu$$ where $\nu$ is the unit normal, I need to use $\nu(x) = \frac{x}{|x|}$, and NOT $\nu(x) = x$ as one may expect because we're on the unit sphere (so $|x| = 1$). Why is that? I know that it (whatever it is) doesn't know that we're on a unit sphere but logically it should give the same results should it not (I know it doesn't)?
But when calculating, for example, a second derivative like $D_1(D_1f)$, I can use $D_1f$ with the fact that $|x| = 1$ and not have to worry about it at all. Can someone elaborate on this. Thanks.
Best Answer
I think in this case there should be no difference because the gradient is only a first order derivative. But in general higher order subtitles may be troublesome as you noted. I could not elaborate more without more information available.