The industrial revolution . . . as I've already remarked, the industrial age is really the same agricultural society mechanized. When I say industrialism is just agriculturalism I mean the resultant specialization of jobs and class structure.As for that mechanisation . . . it of course started out with the steam engine. The steam engine started out by using the vacuum to pull the piston back down after a fuel bang shot it up.
I don't have a copy of Isaac Asimov's "Chronology of Science and Technology" and it's certainly out of date and only approximate; but, I remember checking it out during some of my school days. In the book, Asimov has a timeline of when technologies occured. I don't remember the precise year, but I know that as far as his timeline is concerned the year when every succeeding year had some science and technological development each year was in the 1790s. This fits the beginnings of the industrial age.
While the technological advances might have hit their stride around 1790, the mathematical advances hit their stride earlier with certainly Newton and Liebniz a hundred years earlier. One could argue earlier with Descartes and Fermat.Most of this advance was in analyses(mathematicians word for the calculus). Newton, in his Principia, tried to tell the immediate future mathematical scientists what the next fields of study must be - electricity and chemistry. The next generation of mathematical scientists did not follow his advice. Those sciences for the most part did not get worked on till the 1800s. The science that was worked on for the next hundred years or so was that of astronomy.
For the hundred or so years after Newton and Liebniz, mathematicians such as Euler, the Bernolies, and Lagrange did analyses. Scientists mostly worked on astronomy. The mathematics that these mathematicians were to work on was differential equations, calculus of variations, complex variables(some), elliptic functions(some), number theory and some initial hints at group theory(Lagrange).Newton and Liebniz had already shown how to derive much of the mathematics the greeks were struggling with. The succeeding mathematics of the Bernollies, Euler, Lagrange(and some others) was of a mass that dwarfed everything that the Greeks had deduced.Analyses of course was to progress in the 1800s; but, algebra and geometry also. But, that's for sometime later.