Leibniz's notation

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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.[1] For y as a function of x, or

y=f(x) \,,

the derivative of y with respect to x, which later came to be viewed as

\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x} = \lim_{\Delta x\rightarrow 0}\frac{f(x + \Delta x)-f(x)}{(x + \Delta x)-x},

was, according to Leibniz, the quotient of an infinitesimal increment of y by an infinitesimal increment of x, or

\frac{dy}{dx}=f'(x),

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, \Delta x is an infinitesimal x-increment, \Delta y is the corresponding y-increment, and the derivative is the standard part of the infinitesimal ratio:

f'(x)={\rm st}\Bigg( \frac{\Delta y}{\Delta x} \Bigg).

Then one sets dx=\Delta x, dy = f'(x) dx\,, so that by definition, f'(x)\, is the ratio of dy by dx.

Similarly, although mathematicians sometimes now view an integral

\int f(x)\,dx

as a limit

\lim_{\Delta x\rightarrow 0}\sum_{i} f(x_i)\,\Delta x,

where Δx is an interval containing xi, Leibniz viewed it as the sum (the integral sign denoting summation) of infinitely many infinitesimal quantities f(xdx. From the modern viewpoint, it is more correct to view the integral as the standard part of such an infinite sum.

History[edit]

The Newton-Leibniz approach to infinitesimal calculus was introduced in the 17th century. While Newton did not have a standard notation for integration, Leibniz began using the \int character. He based the character on the Latin word summa ("sum"), which he wrote ſumma with the elongated s commonly used in Germany at the time. This use first appeared publicly in his paper De Geometria, published in Acta Eruditorum of June 1686,[2] but he had been using it in private manuscripts at least since 1675.[3]

At the end of the 19th century, Weierstrass's followers ceased to take Leibniz's notation for derivatives and integrals literally. That is, mathematicians felt that the concept of infinitesimals contained logical contradictions in its development. A number of 19th century mathematicians (Weierstrass and others) found logically rigorous ways to treat derivatives and integrals without infinitesimals using limits as shown above, while Cauchy exploited both infinitesimals and limits (see Cours d'Analyse). Nonetheless, Leibniz's notation is still in general use. Although the notation need not be taken literally, it is usually 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.

In the 1960s, building upon earlier work by Edwin Hewitt and Jerzy Łoś, Abraham Robinson developed mathematical explanations for Leibniz's infinitesimals that were acceptable by contemporary standards of rigor, and developed non-standard analysis based on these ideas. Robinson's methods are used by only a minority of mathematicians. Jerome Keisler wrote a first-year-calculus textbook based on Robinson's approach.

Leibniz's notation for differentiation[edit]

In Leibniz's notation for differentiation, the derivative of the function f(x) is written:

\frac{d\bigl(f(x)\bigr)}{dx}\,.

If we have a variable representing a function, for example if we set

y=f(x) \,,

then we can write the derivative as:

\frac{dy}{dx}\,.

Using Lagrange's notation, we can write:

\frac{d\bigl(f(x)\bigr)}{dx} = f'(x)\,.

Using Newton's notation, we can write:

\frac{dx}{dt} = \dot{x}\,.

For higher derivatives, we express them as follows:

\frac{d^n\bigl(f(x)\bigr)}{dx^n}\text{ or }\frac{d^ny}{dx^n}

denotes the nth derivative of ƒ(x) or y respectively. Historically, this came from the fact that, for example, the third derivative is:

\frac{d \left(\frac{d \left( \frac{d \left(f(x)\right)} {dx}\right)} {dx}\right)} {dx}\,,

which we can loosely write as:

\left(\frac{d}{dx}\right)^3 \bigl(f(x)\bigr) =
\frac{d^3}{\left(dx\right)^3} \bigl(f(x)\bigr)\,.

Now drop the parentheses and we have:

\frac{d^3}{dx^3}\bigl(f(x)\bigr)\ \mbox{or}\ \frac{d^3y}{dx^3}\,.

The chain rule and integration by substitution rules are especially easy to express here, because the "d" terms appear to cancel:

\frac{dy}{dx} = \frac{dy}{du_1} \cdot \frac{du_1}{du_2} \cdot \frac{du_2}{du_3}\cdots \frac{du_n}{dx}\,,

etc., and:

\int y \, dx = \int y \frac{dx}{du} \, du.

See also[edit]

Notes[edit]

  1. ^ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 0-495-01166-5. 
  2. ^ Mathematics and its History, John Stillwell, Springer 1989, p. 110
  3. ^ Early Mathematical Manuscripts of Leibniz, J. M. Child, Open Court Publishing Co., 1920, pp. 73–74, 80.