# How to define the tangent gradient operator? [duplicate]

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}]