Hey everybody! Physicists have no problem differentiating a function of many variables - in flat space R^n. But I dont like how many books dont give examples of how this done in a manifold- even if it may be easy when one finally understands it. For example, how do I differentiate a function f on a circle if that function only lives on the circle and not on an ambient space? For example, if given a prescription of a differentiation (contained in some vector V at a point p), how does V differentiate f explicitly? Can someone explain + give a non-trivial example? Also, I would very much like to see an example on a sphere too - where many directions may be chosen in which to differentiate a function.