And it sounds as if they have found a nice balance.
But to visitors from Baghdad the city is a breath of fresh air.
It operates without a curfew unlike the capital and its streets throng with people until late evening, when Baghdad residents fearing criminal attacks have long since gone home...
Mosul is a much smaller and less complex city than Baghdad. It is also outside the heartland of support for Saddam north and west of Baghdad where attacks on U.S. forces are most common.
''People know each other here. It's a smaller city,'' Akil Haydar, an employee at a factory making propane bottles, said in a small store selling shiny black olives in the city centre.
But the 101st Airborne has also gone out of its way to try to ensure anti-American hostility does not build up.
After troops mount nightime raids to arrest someone, they return next day to explain the operation to local people. If a door has to be broken down, soldiers will fix it later.
Often troops surround a house and then knock on the door and ask the occupants to come out. Petraeus says only Saddam's sons Uday and Qusay, killed in a gunbattle in the city on July 22, have put up a fight when faced with that situation.
'Based on what happened to them, I don't think there will be too many more that will do that,'' he said.