# 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