Simply cancelling the #dx# here is not a proof! In real analysis (pure mathematics) it must be proved otherwise. infinitesimal generators of Markov processes satisfy the positive maximum. (3) An arbitrary infinitesimal is either a positive infinitesimal, a. If we used non-standard values, I believe by closure properties, the integral itself would be non-standard -valued. c (Rd) can be written as a pseudo-differential operator with negative definite. (2) A negative infinitesimal is a negative hyperreal greater than every negative real. Yes, by the Archimedean principle, a Real number cannot be indefinitely small without being 0 but in doing Riemann Integration, AFAIK, we are just requiring that the partition width dx <=||P|| goes to 0 but I don't see how we're requiring that dx be less than _every_(rather than any) Real, forcing the AP to kick in or else allowing dx to take non-standard Real values. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the ' infinity - th ' item in a sequence. In any case, #dx# in the integral is not a real number.Įdit:Well, not the dx itself but the values dx assumes. In mathematics, an infinitesimal or infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. Perhaps you didn't phrase this precisely, but a real number is either zero or it's not. How is that possible, given that the integrand is always positive over the domain of integration, and an integral adds up all these infinitesimal positive.
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