= Locality of weak differentiability
A locally integrable function has <weak derivatives> on an open set exactly when it has them near every point. Local derivatives agree almost everywhere on overlaps because they define the same <distributional derivative>. A countable local cover therefore gives globally defined, locally integrable derivative functions. A smooth <partition of unity> subordinate to the cover splits each compactly supported <test function>; summing the local integration identities yields the global weak-derivative identity.
Back to article page
