SORRY I forgot to answer this. Yes the F-D armistice agreement divided France into several parts. All N, N-E and W France including Paris was German-occupied, was called "zone occupée" and comprised all coasts from the Belgian to the Spanish border (Atlantic / Pyrénées). I guess this was about 2/3 of the territory. The rest was the "zone non occupée" or "zone nono" because of the pronunciation (nonoccupée), it was mainly central, south and SE France including the Medit. coast and the harbours of Marseille (biggest French harbour) and Toulon (big naval harbour). As you know the Pétain puppet government was in Vichy, which was chosen because of the numerous hotels, which were used for offices. Vichy-France was not really independent but tried to give the impression it was.
Yes the Germans invaded the rest of France after the Allied landings in French N. Africa, around 20 November 1942 I think. This time they seized all military material, in particular all French aircraft, certainly well over 1,000 AC but by then obsolescent types.
Actually France was not divided in two but in many bits and pieces for along all coasts (thousands of kilometres) there was a land stripe which was strictly "zone interdite", forbidden except to the local population but even the latter was not allowed to move freely. Then there was Alsace and Lorraine, which very quickly were bluntly annexed by the Reich, their male inhabitants forced to become German soldiers, which many hated. Many of them were killed on the Eastern front. There were other special areas but I don't quite remember. Obviously it was "Verboten" around submarine and other naval bases, airfields etc. France was systematically exploited and looted like all other occupied countries, and the industry was forced to produce for Germany as well as the agriculture. Enormous quantities of food were sent to Germany and the French hungered. Well over one million French POWs were kept in Germany as slave labour both in industry and agriculture, later they also forced people living in France to work in factories in Germany.
A book published by a former high-ranking German civil servant (was it Saur? Perhaps, I don't remember) showed me two things :
1. French qualified industry workers were highly appreciated; this man wrote that they were just as good as their German counterparts, which, coming from an ex German nazi, is an almost incredible statement. But indeed this was true, otherwise the French would not have been able 1940 to already LAUNCH the mass production of several superlative fighters equipped with engines with fuel injection, hitherto a German exclusive production (Dewoitine D.523 and 524, D.551 and others), not to mention the remarkable airframes, which needed excellent manpower too (Arsenal fighters, Bloch 174-175, Dewoitine fighters etc.). Fuel injection demanded high-precision machining by highly-qualified manpower. The excellent French tanks - the best in the world - demanded excellent manpower too as well as all the remarkable artillery, AA and anti-tank guns etc..
2. The German top leaders were very keen NOT to have any sort of mass revolt, uprising or revolution in France for this would have brought Germany into an impossible situation. This is precisely what the French communist party, suddenly on the Allied side (for the first time) after their paradise the USSR had been attacked by Germany, after they even had sabotaged French armament production including aircraft 1939-40, systematically tried to trigger through killing German soldiers and hoping for massive German reprisals which would have started a French revolution. The communists didn't care how many French people would die in the process, even millions was no problem as long as it helped the Holy USSR. But the Germans were clever enough not to start any massive retaliation. They were very brutal and inhuman as usual but contended with shooting some "hostages" after every incident.
What I mean is that the German leaders' fear for a French uprising is an interesting element. They simply didn't have the necessary troops to fight an uprising all over France (which has many and very large mountains and forests) and the Red Army at the same time, not to mention the Western Allied air forces and armies. The Germans skilfully managed to exploit France as best they could without getting into real trouble, holding the whole country with but a few divisions.