NationMaster - Encyclopedia: Leibniz's notation

In calculus, Leibniz's notation, named in honor of the 17th century German philosopher and mathematician Gottfried Wilhelm Leibniz, was originally the use of expressions such as dx and dy and to represent "infinitely small" (or infinitesimal) 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 later came to be viewed as Calculus [from Latin, literally pebble (used in reckoning)] is a major area in mathematics which relates small-scale phenomena with large-scale behavior. ... (16th century - 17th century - 18th century - more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601-1700. ... The philosopher Socrates about to take poison hemlock as ordered by the court. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... It has been suggested that this article be split into multiple articles. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ...

$lim_{Delta xrightarrow 0}frac{Delta y}{Delta x},$

was the quotient of an infinitesimal increment of y by an infinitesimal increment of x. Thus if

$y=f(x) !$

then

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

where the right hand side is Lagrange's notation for the derivative of f at x.

Similarly, although mathematicians usually now view an integral

$int f(x),mathrm dx$

as a limit

$lim_{Delta xrightarrow 0}sum_{i} f(x_i),Delta x$ ,

Leibniz viewed it as the sum (the integral sign denoting summation) of infinitely many infinitesimal quantities f(x) dx.

One advantage of Leibniz's point of view is that it is compatible with dimensional analysis. For, example, in Leibniz's notation, the second derivative (using implicit differentiation) is: Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... In mathematics, to give an implicit function f is to give the graph of a function, as a relation. ...

$frac{mathrm d^2 y}{mathrm dx^2}=f''(x)$

and has the same dimensional units as $frac{y}{x^2}$ .^[1]

1 History
2 Leibniz's notation for differentiation
3 Notes
4 See also

History

The Newton-Leibniz approach to calculus was introduced in the 17th century. In the 19th century, mathematicians ceased to take Leibniz's notation for derivatives and integrals literally. That is, mathematicians saw that the concept of infinitesimals 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 using limits as shown above. 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. (16th century - 17th century - 18th century - more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601-1700. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... Karl Theodor Wilhelm Weierstraß (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. (The letter ß may be transliterated as ss; one often writes Weierstrass. ... In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to re-write an equation so that each of two variables occurs on a different side of the equation. ... Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ...

However, 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.) // Recovering from World War I and its aftermath, the economic miracle emerged in West Germany and Italy. ... The 1960s decade refers to the years from January 1, 1960 to December 31, 1969, inclusive. ... Abraham Robinson Abraham Robinson (October 6, 1918 - April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics. ...

Leibniz's notation for differentiation

In Leibniz's notation for differentiation, the derivative of the function f(x) is written: The term notation can be used in several contexts. ... In some places this article assumes an acquaintance with algebra, analytic geometry, or the limit. ...

$frac{mathrm{d}bigl(f(x)bigr)}{mathrm{d}x}$

If we have a variable representing a function, for example if we set In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ...

y = f (x),

then we can write the derivative as:

$frac{mathrm{d}y}{mathrm{d}x}$

Using Lagrange's notation, we can write:

$frac{mathrm{d}bigl(f(x)bigr)}{mathrm{d}x} = f'(x).$

Using Newton's notation, we can write:

$frac{mathrm{d}x}{mathrm{d}t} = dot{x}.$

For higher derivatives, we express them as follows:

$frac{mathrm{d}^nbigl(f(x)bigr)}{mathrm{d}x^n}$ or $frac{mathrm{d}^ny}{mathrm{d}x^n}$

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

$frac{mathrm{d} Bigl(frac{mathrm{d} left( frac{mathrm{d} left(f(x)right)} {mathrm{d}x}right)} {mathrm{d}x}Bigr)} {mathrm{d}x}$

which we can loosely write as:

$left(frac{mathrm{d}}{mathrm{d}x}right)^3 bigl(f(x)bigr) = frac{mathrm{d}^3}{left(mathrm{d}xright)^3} bigl(f(x)bigr)$

Now drop the brackets and we have:

$frac{mathrm{d}^3}{mathrm{d}x^3}bigl(f(x)bigr) mbox{or} frac{mathrm{d}^3y}{mathrm{d}x^3}$

The chain rule and integration by substitution rules are especially easy to express here, because the "d" terms appear to cancel: In calculus, the chain rule is a formula for the derivative of the composite of two functions. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ...

$frac{mathrm{d}y}{mathrm{d}x} = frac{mathrm{d}y}{mathrm{d}u} cdot frac{mathrm{d}u}{mathrm{d}v} cdot frac{mathrm{d}v}{mathrm{d}w} cdot frac{mathrm{d}w}{mathrm{d}x}$ etc.

and:

$int y , mathrm{d}x = int y frac{mathrm{d}x}{mathrm{d}u} , mathrm{d}u.$

Notes

^ Note that $frac{mathrm d^2 y}{mathrm d x^2}$ is shorthand for $frac{mathrm d{frac{mathrm dy}{mathrm dx}}}{mathrm dx}$ , or in other words the second differential of y over the square of the first differential of x. The denominator is not the differential of x², nor is it the second differential of x.

Contents

Leibniz's notation for differentiation

Notes

See also