In which of these regions did the industrial revolution begin

I dont think it can be narrowed down to a specific region of the US since all reigons of the US after the Civil War had a hand in starting the Industrial Revolution. During the Reconstructionist Era the advent of steam technology created steam powered engines and technology which in turn drove other industries such as metalworking, textiles and resource refinement. Labor for these industrial enterprises was fueled by Irish immigrants as well as newly freed slaves. But if there were one place that would have to be named as the origin on the revolution, I'd say it was in Massachusets on the Merrimak river where steam and water power were first harnessed along the river. The creation of the Lowell system was a very significant technological milestone in America's technological movement.