Leibniz notation
|
|
In calculus, the Leibniz notation, named in honor of the 17th century German philosopher and mathematician Gottfried Wilhelm Leibniz (pronounced LIBE nits) was originally the use of dx and dy and so forth to represent "infinitely small" increments of quantities x and y, just as Δx and Δy represent finite increments of x and y respectively. According to Leibniz, the derivative of y with respect to x, which mathematicians later came to view as
- <math>\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x}<math>
was the quotient of an infinitely small (i.e., infinitesimal) increment of y by an infinitely small increment of x. Thus if
- <math>y=f(x)<math>
then
- <math>\frac{dy}{dx}=f'(x).<math>
Similarly, where mathematicians may now view an integral as
- <math>\lim_{\Delta x\rightarrow 0}\sum_{i} f(x_i)\,\Delta x,<math>
Leibniz viewed it as the sum of infinitely many infinitely small quantities
- <math>f(x)\,dx.<math>
Well before the end of the 19th century, mathematicians had ceased to take Leibniz's notation for derivatives and integrals literally. It was mainly because the infinitesimal concept contained logical contradictions in the development. A number of 19th century mathematicians (Cauchy, Weierstrass and others) found logically rigorous ways to treat derivatives and integrals without infinitesimals. In the 1950s and 1960s, Abraham Robinson introduced ways of treating infinitesimals both literally and logically rigorously, and so rewriting calculus from that point of view. But Robinson's methods are not used by most mathematicians. (One mathematician, Jerome Keisler, has gone so far as to write a first-year-calculus textbook according to Robinson's point of view.)
Nonetheless, everyone continues to use Leibniz's notation today, and few doubt its utility in certain contexts. Although most people using it do not construe it literally, they find it simpler than alternatives when the technique of separation of variables is used in the solution of differential equations. In physical applications, one may for example regard f(x) as measured in meters per second, and dx in seconds, so that f(x) dx is in meters, and so is the value of its definite integral. In that way the Leibniz notation is in harmony with dimensional analysis. It can clearly be seen by the second derivative in Leibniz's notation:
- <math>\frac{d^2 y}{dx^2}=f''(x)<math>
and has the units of [y]/[x]^2.