In
differential calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve ...
, there is no single uniform notation for differentiation. Instead, various notations for the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of a
function or
variable have been proposed by various mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation (and its opposite operation, the
antidifferentiation or
indefinite integration) are listed below.
Leibniz's notation
The original notation employed by
Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
is used throughout mathematics. It is particularly common when the equation is regarded as a functional relationship between
dependent and independent variables
Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or dema ...
and . Leibniz's notation makes this relationship explicit by writing the derivative as
:
Furthermore, the derivative of at is therefore written
:
Higher derivatives are written as
:
This is a suggestive notational device that comes from formal manipulations of symbols, as in,
:
The value of the derivative of at a point may be expressed in two ways using Leibniz's notation:
:
.
Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially helpful when considering
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s. It also makes the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
easy to remember and recognize:
:
Leibniz's notation for differentiation does not require assigning a meaning to symbols such as or on their own, and some authors do not attempt to assign these symbols meaning. Leibniz treated these symbols as
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
s. Later authors have assigned them other meanings, such as infinitesimals in
non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...
or
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
s.
Some authors and journals set the differential symbol in
roman type
In Latin script typography, roman is one of the three main kinds of historical type, alongside blackletter and italic. Roman type was modelled from a European scribal manuscript style of the 15th century, based on the pairing of inscriptional c ...
instead of
italic: . The
ISO/IEC 80000 scientific style guide recommends this style.
Leibniz's notation for antidifferentiation
Leibniz introduced the
integral symbol in ''Analyseos tetragonisticae pars secunda'' and ''Methodi tangentium inversae exempla'' (both from 1675). It is now the standard symbol for
integration.
:
Lagrange's notation
One of the most common modern notations for differentiation is named after
Joseph Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia[Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...](_blank)
and just popularized by the former. In Lagrange's notation, a
prime mark denotes a derivative. If ''f'' is a function, then its derivative evaluated at ''x'' is written
:
.
It first appeared in print in 1749.
Higher derivatives are indicated using additional prime marks, as in
for the
second derivative
In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...
and
for the
third derivative. The use of repeated prime marks eventually becomes unwieldy. Some authors continue by employing
Roman numeral
Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, ea ...
s, usually in lower case, as in
:
to denote fourth, fifth, sixth, and higher order derivatives. Other authors use Arabic numerals in parentheses, as in
:
This notation also makes it possible to describe the ''n''th derivative, where ''n'' is a variable. This is written
:
Unicode characters related to Lagrange's notation include
*
*
*
*
When there are two independent variables for a function ''f''(''x'', ''y''), the following convention may be followed:
[''The Differential and Integral Calculus'' ( Augustus De Morgan, 1842). pp. 267-268]
:
Lagrange's notation for antidifferentiation
When taking the antiderivative, Lagrange followed Leibniz's notation:
Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia[inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X ...](_blank)
),
: for the second integral,
: for the third integral, and
: for the ''n''th integral.
Euler's notation
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
's notation uses a differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
suggested by Louis François Antoine Arbogast, denoted as (D operator) or (Newton–Leibniz operator).[Weisstein, Eric W. "Differential Operator." From ''MathWorld''--A Wolfram Web Resource. ] When applied to a function , it is defined by
:
Higher derivatives are notated as "powers" of ''D'' (where the superscripts denote iterated composition of ''D''), as in
: for the second derivative,
: for the third derivative, and
: for the ''n''th derivative.
Euler's notation leaves implicit the variable with respect to which differentiation is being done. However, this variable can also be notated explicitly. When ''f'' is a function of a variable ''x'', this is done by writing
: for the first derivative,
: for the second derivative,
: for the third derivative, and
: for the ''n''th derivative.
When ''f'' is a function of several variables, it's common to use " ∂", a stylized cursive lower-case d, rather than "". As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function are:
:
:
:
:
See .
Euler's notation is useful for stating and solving linear differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...
s, as it simplifies presentation of the differential equation, which can make seeing the essential elements of the problem easier.
Euler's notation for antidifferentiation
Euler's notation can be used for antidifferentiation in the same way that Lagrange's notation is as follows
: for a first antiderivative,
: for a second antiderivative, and
: for an ''n''th antiderivative.
Newton's notation
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
's notation for differentiation (also called the dot notation, fluxions, or sometimes, crudely, the flyspeck notation for differentiation) places a dot over the dependent variable. That is, if ''y'' is a function of ''t'', then the derivative of ''y'' with respect to ''t'' is
:
Higher derivatives are represented using multiple dots, as in
:
Newton extended this idea quite far:
:
Unicode characters related to Newton's notation include:
*
*
* ← replaced by "combining diaeresis" + "combining dot above".
* ← replaced by "combining diaeresis" twice.
*
*
*
*
*
Newton's notation is generally used when the independent variable denotes time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
. If location is a function of ''t'', then denotes velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
and denotes acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
. This notation is popular in physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
and mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
. It also appears in areas of mathematics connected with physics such as differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s.
When taking the derivative of a dependent variable ''y'' = ''f''(''x''), an alternative notation exists:
:
Newton developed the following partial differential operators using side-dots on a curved X ( ⵋ ). Definitions given by Whiteside are below:
:
Newton's notation for integration
Newton developed many different notations for integration in his ''Quadratura curvarum'' (1704) and later works: he wrote a small vertical bar or prime above the dependent variable ( ), a prefixing rectangle (), or the inclosure of the term in a rectangle () to denote the '' fluent'' or time integral (absement
In kinematics, absement (or absition) is a measure of sustained displacement of an object from its initial position, i.e. a measure of how far away and for how long. The word ''absement'' is a portmanteau of the words ''absence'' and ''dis ...
).
:
To denote multiple integrals, Newton used two small vertical bars or primes (), or a combination of previous symbols , to denote the second time integral (absity).
:
Higher order time integrals were as follows:
:
This mathematical notation
Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathem ...
did not become widespread because of printing difficulties and the Leibniz–Newton calculus controversy.
Partial derivatives
When more specific types of differentiation are necessary, such as in multivariate calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather th ...
or tensor analysis, other notations are common.
For a function ''f'' of an independent variable ''x'', we can express the derivative using subscripts of the independent variable:
:
This type of notation is especially useful for taking partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s of a function of several variables.
Partial derivatives are generally distinguished from ordinary derivatives by replacing the differential operator ''d'' with a " ∂" symbol. For example, we can indicate the partial derivative of with respect to ''x'', but not to ''y'' or ''z'' in several ways:
:
What makes this distinction important is that a non-partial derivative such as ''may'', depending on the context, be interpreted as a rate of change in relative to when all variables are allowed to vary simultaneously, whereas with a partial derivative such as it is explicit that only one variable should vary.
Other notations can be found in various subfields of mathematics, physics, and engineering; see for example the Maxwell relations
file:Thermodynamic map.svg, 400px, Flow chart showing the paths between the Maxwell relations. P is pressure, T temperature, V volume, S entropy, \alpha coefficient of thermal expansion, \kappa compressibility, C_V heat capacity at constant volu ...
of thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
. The symbol is the derivative of the temperature ''T'' with respect to the volume ''V'' while keeping constant the entropy (subscript) ''S'', while is the derivative of the temperature with respect to the volume while keeping constant the pressure ''P''. This becomes necessary in situations where the number of variables exceeds the degrees of freedom, so that one has to choose which other variables are to be kept fixed.
Higher-order partial derivatives with respect to one variable are expressed as
:
:
and so on. Mixed partial derivatives can be expressed as
:
In this last case the variables are written in inverse order between the two notations, explained as follows:
:
:
So-called multi-index notation is used in situations when the above notation becomes cumbersome or insufficiently expressive. When considering functions on , we define a multi-index to be an ordered list of non-negative integers: . We then define, for , the notation
In this way some results (such as the Leibniz rule Leibniz's rule (named after Gottfried Wilhelm Leibniz) may refer to one of the following:
* Product rule in differential calculus
* General Leibniz rule, a generalization of the product rule
* Leibniz integral rule
* The alternating series test, al ...
) that are tedious to write in other ways can be expressed succinctly -- some examples can be found in the article on multi-indices.
Notation in vector calculus
Vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
concerns differentiation and integration of vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
or scalar fields
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity ( ...
. Several notations specific to the case of three-dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
are common.
Assume that is a given Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, that A is a vector field with components , and that is a scalar field
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantity ...
.
The differential operator introduced by William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
, written ∇ and called del
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes ...
or nabla, is symbolically defined in the form of a vector,
:
where the terminology ''symbolically'' reflects that the operator ∇ will also be treated as an ordinary vector.
* Gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
: The gradient of the scalar field is a vector, which is symbolically expressed by the multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
of ∇ and scalar field '''',
::
* Divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
: The divergence of the vector field A is a scalar, which is symbolically expressed by the dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
of ∇ and the vector A,
::
* Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
: The Laplacian of the scalar field is a scalar, which is symbolically expressed by the scalar multiplication of ∇2 and the scalar field ''φ'',
::
* Rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
: The rotation , or , of the vector field A is a vector, which is symbolically expressed by the cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
of ∇ and the vector A,
::
Many symbolic operations of derivatives can be generalized in a straightforward manner by the gradient operator in Cartesian coordinates. For example, the single-variable product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
has a direct analogue in the multiplication of scalar fields by applying the gradient operator, as in
:
Many other rules from single variable calculus have vector calculus analogues for the gradient, divergence, curl, and Laplacian.
Further notations have been developed for more exotic types of spaces. For calculations in Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
, the d'Alembert operator, also called the d'Alembertian, wave operator, or box operator is represented as , or as when not in conflict with the symbol for the Laplacian.
See also
*
*
*
*
*
*
* Operational calculus
References
External links
Earliest Uses of Symbols of Calculus
maintained by Jeff Miller ().
{{Differential equations topics
Differential calculus
Mathematical notation