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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.

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I think you'll need to provide more context. You're right that the expression takes the same values on the unit sphere whether you use $x$ or $x/|x|$. Whether it makes a difference later on depends on what the meaning of "it" is, how you go on to use this expression. – joriki May 5 '12 at 10:58
You say it should give the same results but you know it doesn't. Can you show an example where using $x$ instead of $x/\lvert x\rvert$ gives different results? – Rahul Jun 13 '12 at 5:11
Your question has been bumped to the front page again. Can you clarify the question as the comments have asked? – Rahul Dec 24 '12 at 1:40
@RahulNarain Sorry, forgot about this question. I redid the calculations and it turns to be to give the same answer so I was mistaken. – soup Dec 24 '12 at 14:58
up vote 1 down vote accepted

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.

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