In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent "infinitely small" (or infinitesimal) increments of x and y, just as Δx and Δy represent finite increments of x and y. For y as a function of x, or
the derivative of y with respect to x, which later came to be viewed as
was, according to Leibniz, the quotient of an infinitesimal increment of y by an infinitesimal increment of x, or
where the right hand side is Lagrange's notation for the derivative of f at x. From the point of view of modern infinitesimal theory, is an infinitesimal x-increment, is the corresponding y-increment, and the derivative is the standard part of the infinitesimal ratio:
Then one sets, so that by definition, is the ratio of dy by dx.
Similarly, although mathematicians sometimes now view an integral
as a limit
where Δx is an interval containing xi, Leibniz viewed it as the sum (the integral sign denoting summation) of infinitely many infinitesimal quantities f(x) dx. From the modern viewpoint, it is more correct to view the integral as the standard part of an infinite sum of such quantities.
Famous quotes containing the word leibniz:
“Navarette, a Chinese missionary, agrees with Leibniz and says that It is the special providence of God that the Chinese did not know what was done in Christendom; for if they did, there would be never a man among them, but would spit in our faces.”
—Matthew Tindal (16531733)