Are you looking for a heuristic reason why you should believe $G \in C^2$, or a full rigorous proof of the fact?
– TomNov 13 '13 at 01:07
Why G \in C^2? And I wonder whether Rudin's proof is ture. I cheek the rudin's proof about Poincaré Lemma.I found it seem have a mistake. Through Google,I found another guy else has ever noted such an erro. More detail,http://mathforum.org/kb/thread.jspa?forumID=13&threadID=2141549&messageID=7208112
– gilliattNov 13 '13 at 09:21