gradient of a scalar on a surface

This image represent how to calculate the gradient of a scalar function f:\Sigma\to\mathbb{R}, a measurement on a surface (surface1 , surface2 or surface3)

On the final two lines you gotta remember that the notation Xf means \langle X,{\rm grad}f\rangle which is also equals to \langle {\rm grad}f,X\rangle, this, codes how f varies relatively to X vector or vector field, see

Remember also that \Omega is an open set of \mathbb{R}^2 and  \Phi is injective with jacobian J\Phi having rank two “along” \Omega

With this device you can transport the Euclidean calculus in \mathbb{R}^2 to calculus in the surface \Sigma\subset\mathbb{R}^3

See how the chain rule is adapted: J(f\circ\Phi)=Jf\cdot J\Phi, from where we can evaluate: J(f\circ\Phi)(a)=Jf(\Phi(a))\cdot J\Phi(a)… What are \partial_1,\partial_2 here?

As a nontrivial example consider the notion of round functions

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