Sure - this is a great way to think about calculus intuitively, but imprecisely; rigorous, mathematical proofs based on the hyperreals, on the other hand, may b

Sure - this is a great way to think about calculus intuitively, but imprecisely; rigorous, mathematical proofs based on the hyperreals, on the other hand, may become unwieldy, and, as you rightly noted, they do not extend to the modern treatments of, say, differential geometry which, too, plays a huge role, as a framework, in the modern theoretical physics.