I would like to define a new differential operator that is the tangent gradient for a curve $\Sigma$.

This is defined as $$\nabla_\Sigma=\mathbf{P}\nabla$$ where $\mathbf{P}$ is the projection operator defined as $$\mathbf{P}=\mathbf{I}-\mathbf{n}\otimes\mathbf{n}$$ where $\mathbf{n}$ is the normal at point $x$.

A related question is how to define the Laplace-Beltrami operator for a curve. This is defined as: $$\Delta_\Sigma=\nabla_\Sigma \cdot \nabla_\Sigma$$

For example given a curve defined via the signed distance function $d(x,y)=\sqrt{x^2+y^2}-1$ and a function $u(x,y)$, I want to be able to have a command in Mathematica that gives $\nabla_\Sigma u$ the same way that I can call for example Grad[u(x,y),{x,y}]

  • $\begingroup$ @Jim I just reopened it. How we "usually work" is to do our best to migrate off-topic questions to a place they will get answers. That I'm sure was the intention. As soon as it became clear that this was in fact a Mathematica question the Math.SE mods worked with me to reopen it. $\endgroup$ – Mr.Wizard Mar 8 '13 at 16:50
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    $\begingroup$ Yes, Jim: within just a few minutes you were pointed to an existing answer and your question was presented in another community expected to provide additional information not available here. You are indeed fortunate to be subject to such attention and generosity. $\endgroup$ – whuber Mar 8 '13 at 16:51
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    $\begingroup$ Well, we get several questions/day that should've been asked on Mathematics instead, because the only difference between the two words is a single letter at the end. A lot of new users make this mistake because they do not know about the software. Our policy is migrate those quickly to the right site so that the OP can get their answer sooner rather than later. Your question was no different from them because you didn't have any indication that you were trying to solve this in Mathematica. You needn't be overly explicit — code that you've tried, or error messages, etc. would've given a hint. $\endgroup$ – rm -rf Mar 8 '13 at 16:54
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    $\begingroup$ @rcollyer It means whatever you want. If you can go to a Mma programming site asking for help in TeX, you may well get to a toilette to peruse a sandwich $\endgroup$ – Dr. belisarius Mar 9 '13 at 3:43
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    $\begingroup$ For the record, I asked Jim whether he wanted this site or mathematics in a comment about 1 minute after he asked the question, so we did try to help find the right site... My comment was deleted as soon as the question had been migrated. :) $\endgroup$ – cormullion Mar 9 '13 at 8:40