Leibniz notation
Encyclopedia
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.
was introduced in the 17th century. While Newton did not have a standard notation for integration, Leibniz began using the
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, but he had been using it in private manuscripts at least since 1675.
In the 19th century, mathematicians ceased to take Leibniz's notation for derivatives and integrals literally. That is, mathematicians felt that the concept of infinitesimal
s contained logical contradictions in its 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
.
In the 1960s, building upon earlier work by Edwin Hewitt
and Jerzy Łoś, Abraham Robinson
developed rigorous mathematical explanations for Leibniz' intuitive notion of the "infinitesimal," 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 to Robinson's approach.
for differentiation
, the derivative of the function f(x) is written:
If we have a variable
representing a function, for example if we set
then we can write the derivative as:
Using Lagrange's notation
, we can write:
Using Newton's notation, we can write:
For higher derivatives, we express them as follows:
denotes the nth derivative of ƒ(x) or y respectively. Historically, this came from the fact that, for example, the third derivative is:
which we can loosely write as:
Now drop the parentheses and we have:
The chain rule
and integration by substitution
rules are especially easy to express here, because the "d" terms appear to cancel:
etc., and:
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
, Leibniz's notation, named in honor of the 17th-century German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
philosopher
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
and mathematician
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
Gottfried Wilhelm Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
, uses the symbols dx and dy to represent "infinitely small" (or infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
) increments of x and y, just as Δx and Δy represent finite increments of x and y. For y as a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
of x, or
the derivative of y with respect to x, which later came to be viewed as
was, according to Leibniz, the quotient
Quotient
In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...
of an infinitesimal increment of y by an infinitesimal increment of x, or
where the right hand side is Lagrange's notation
Notation for differentiation
In differential calculus, there is no single uniform notation for differentiation. Instead, several different notations for the derivative of a function or variable have been proposed by different mathematicians. The usefulness of each notation varies with the context, and it is sometimes...
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.
History
The Newton-Leibniz approach to infinitesimal calculusInfinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...
was introduced in the 17th century. While Newton did not have a standard notation for integration, Leibniz began using the
character. He based the character on the Latin word summa ("sum"), which he wrote ſumma with the elongated sLong s
The long, medial or descending s is a form of the minuscule letter s formerly used where s occurred in the middle or at the beginning of a word, for example "ſinfulneſs" . The modern letterform was called the terminal, round, or short s.-History:The long s is derived from the old Roman cursive...
commonly used in Germany at the time. This use first appeared publicly in his paper De Geometria, published in Acta Eruditorum
Acta Eruditorum
Acta Eruditorum was the first scientific journal of the German lands, published from 1682 to 1782....
of June 1686, but he had been using it in private manuscripts at least since 1675.
In the 19th century, mathematicians ceased to take Leibniz's notation for derivatives and integrals literally. That is, mathematicians felt that the concept of infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
s contained logical contradictions in its 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
Separation of variables
In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....
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
Dimensional analysis
In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...
.
In the 1960s, building upon earlier work by Edwin Hewitt
Edwin Hewitt
Edwin Hewitt was an American mathematician known for his work in abstract harmonic analysis and for his discovery, in collaboration with Leonard Jimmie Savage, of the Hewitt–Savage zero-one law.He received his Ph.D...
and Jerzy Łoś, Abraham Robinson
Abraham Robinson
Abraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics....
developed rigorous mathematical explanations for Leibniz' intuitive notion of the "infinitesimal," and developed non-standard analysis
Non-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
based on these ideas. Robinson's methods are used by only a minority of mathematicians. Jerome Keisler wrote a first-year-calculus textbook based to Robinson's approach.
Leibniz's notation for differentiation
In Leibniz's notationMathematical notation
Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...
for differentiation
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
, the derivative of the function f(x) is written:
If we have a variable
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
representing a function, for example if we set
then we can write the derivative as:
Using Lagrange's notation
Notation for differentiation
In differential calculus, there is no single uniform notation for differentiation. Instead, several different notations for the derivative of a function or variable have been proposed by different mathematicians. The usefulness of each notation varies with the context, and it is sometimes...
, we can write:
Using Newton's notation, we can write:
For higher derivatives, we express them as follows:
denotes the nth derivative of ƒ(x) or y respectively. Historically, this came from the fact that, for example, the third derivative is:
which we can loosely write as:
Now drop the parentheses and we have:
The chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
and integration by substitution
Integration by substitution
In calculus, integration by substitution is a method for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool for mathematicians...
rules are especially easy to express here, because the "d" terms appear to cancel:
etc., and:
See also
- Notation for differentiationNotation for differentiationIn differential calculus, there is no single uniform notation for differentiation. Instead, several different notations for the derivative of a function or variable have been proposed by different mathematicians. The usefulness of each notation varies with the context, and it is sometimes...
- Newton's notation
- Leibniz and Newton calculus controversy