On the final two lines you gotta remember that the notation means which is also equals to , this, codes how varies relatively to vector or vector field, see
Remember also that is an open set of and is injective with jacobian having rank two “along”
With this device you can transport the Euclidean calculus in to calculus in the surface
See how the chain rule is adapted: , from where we can evaluate: … What are here?
As a nontrivial example consider the notion of round functions