In this paper, the scaling-law vector calculus, which is connected
between the vector calculus and the scaling law in fractal geometry, is
addressed based on the Leibniz derivative and Stieltjes integral for the
first time. The scaling-law Gauss-Ostrogradsky-like, Stokes-like, and
Green-like theorems, and Green-like identities are considered in sense
of the scaling-law vector calculus. The scaling-law Navier-Stokes-like
equations are obtained in detail. The obtained result is as a
potentially mathematical tool proposed to develop an important way of
approaching this challenge for the scaling-law flows.