how to extend a locally defined function to a smooth function on the whole manifold ,by using a bump function?
For instance, say F is defined on a coordinate chart U diffeomorphic to R^n. Let D(1) be the closed subset of U corresponding to the closed unit disk under the coordinate map, and let B(2) be the open subset of U corresponding to the open disk of radius 2 under the coordinate map. Then you know there is a smooth bump function b that is equal to 1 on all of D(1), and has support in B(2), right? So define F', an "extension" of F, by settingHi,
i quote from the text:If F is a smooth function on a neighbourhood of x,we can multiply it by a bump function to extend it to M ".here M is a differential manifold so there are local coordinates at each point.F is a real function.
i don't see how such an extention is done.