smooth compatibility of charts

Suppose two observers map the same patch of a manifold with two different coordinate systems. Where their patches overlap, each point gets two sets of coordinates, and there is a dictionary translating one set into the other. Smooth compatibility is the demand that this dictionary be infinitely differentiable in both directions. It is the local condition out of which the global notion of a smooth structure is built.

Precisely: let (U, phi) and (V, psi) be two charts on M, with phi: U -> R^n and psi: V -> R^n homeomorphisms onto open sets. They are smoothly compatible if either U and V are disjoint, or else the transition map psi composed with phi^{-1}, defined on phi(U intersect V) inside R^n, is a C^infinity map, and so is its inverse phi composed with psi^{-1}. Note both maps go between open subsets of R^n, so 'C^infinity' is just ordinary multivariable calculus — there is nothing exotic to check, only the standard partial derivatives of all orders.

The reason this is the right gluing condition is the chain rule. If transitions are smooth, then a function that looks smooth in one chart looks smooth in every overlapping chart, so 'smooth function on M' becomes a well-defined, chart-independent notion. An atlas in which all pairs of charts are smoothly compatible is itself called smooth; every smooth atlas extends to a unique maximal one, which is the smooth structure. A common slip is to check smoothness of the transition but forget its inverse — you need both, i.e. the transition is a diffeomorphism between open sets, not merely a smooth bijection.