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 :

\frac.

Furthermore, the derivative of at is therefore written :

\frac(x)\text\frac\text\frac f(x).

Higher derivatives are written as :

\frac, \frac, \frac, \ldots, \frac.

This is a suggestive notational device that comes from formal manipulations of symbols, as in, :

\frac = \left(\frac\right)^2y = \frac.

The value of the derivative of at a point may be expressed in two ways using Leibniz's notation: :

\left.\frac\_ \text \frac(a)

. 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 easy to remember and recognize: :

\frac = \frac \cdot \frac.

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 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. :

\begin
                                                   \int y'\,dx &= \int f'(x)\,dx = f(x) + C_0 = y + C_0 \\
                                                    \int y\,dx &= \int f(x)\,dx = F(x) + C_1 \\
                                             \iint y\,dx^2 &= \int \left ( \int y\,dx \right ) dx = \int_ f(x)\,dx = \int F(x)\,dx = g(x) + C_2 \\
  \underbrace_ y\,\underbrace_n &= \int_ f(x)\,dx = \int s(x)\,dx = S(x) + C_n
\end

Lagrange's notation

One of the most common modern notations for differentiation is named after Joseph Louis Lagrange, even though it was actually invented by

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 ...

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 :

f'(x)

. It first appeared in print in 1749. Higher derivatives are indicated using additional prime marks, as in

f''(x)

for the second derivative and

f(x)

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 :

f^(x), f^(x), f^(x), \ldots,

to denote fourth, fifth, sixth, and higher order derivatives. Other authors use Arabic numerals in parentheses, as in :

f^(x), f^(x), f^(x), \ldots.

This notation also makes it possible to describe the ''n''th derivative, where ''n'' is a variable. This is written :

f^(x).

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 :

\begin
          f^\prime &= \frac = f_x \\
          f_\prime &= \frac = f_y \\
  f^ &= \frac = f_ \\
   f_\prime^\prime &= \frac\ = f_ \\
  f_ &= \frac = f_
\end

Lagrange's notation for antidifferentiation

When taking the antiderivative, Lagrange followed Leibniz's notation:

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiainverse function

f^(x)

), :

f^(x)

for the second integral, :

f^(x)

for the third integral, and :

f^(x)

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 :

(Df)(x) = \frac.

Higher derivatives are notated as "powers" of ''D'' (where the superscripts denote iterated composition of ''D''), as in :

D^2f

for the second derivative, :

D^3f

for the third derivative, and :

D^nf

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 :

D_x f

for the first derivative, :

D^2_x f

for the second derivative, :

D^3_x f

for the third derivative, and :

D^n_x f

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: :

\partial_ f = \frac,

\partial_ f = \frac,

\partial_ f = \frac,

\partial_ f = \frac.

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 :

D^f(x)

for a first antiderivative, :

D^f(x)

for a second antiderivative, and :

D^f(x)

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 :

\dot y

Higher derivatives are represented using multiple dots, as in :

\ddot y, \overset

Newton extended this idea quite far: :

\begin
                 \ddot &\equiv \frac = \frac\left(\frac\right)  = \frac\Bigl(\dot\Bigr) = \frac\Bigl(f'(t)\Bigr) = D_t^2 y = f''(t) = y''_t \\
         \overset &= \dot \equiv \frac = D_t^3 y = f(t) = y_t \\
   \overset &= \overset = \ddot \equiv \frac = D_t^4 y = f^(t) = y^_t \\
   \overset &= \ddot = \dot = \ddot \equiv \frac = D_t^5 y = f^(t) = y^_t \\
   \overset &= \overset \equiv \frac = D_t^6 y = f^(t) = y^_t \\
   \overset &= \dot \equiv \frac = D_t^7 y = f^(t) = y^_t \\
  \overset &= \ddot \equiv \frac = D_t^ y = f^(t) = y^_t \\
   \overset &\equiv \frac = D_t^n y = f^(t) = y^_t
\end

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

\dot y

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

\ddot y

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: :

\frac = \dot:\dot \equiv \frac:\frac = \frac = \frac = \frac\Bigl(f(x)\Bigr) = D y = f'(x) = y'.

Newton developed the following partial differential operators using side-dots on a curved X ( ⵋ ). Definitions given by Whiteside are below: :

\begin
       \mathcal                                                            \ &=\  f(x,y) \,, \\
       \cdot\mathcal                                                       \ &=\  x\frac = xf_x\,, \\
       \mathcal\!\cdot                                                     \ &=\  y\frac = yf_y\,, \\
       \colon\!\mathcal\,\text\,\cdot\!\left(\cdot\mathcal\right) \ &=\  x^2\frac = x^2 f_\,, \\
       \mathcal\colon\,\text\,\left(\mathcal\cdot\right)\!\cdot   \ &=\  y^2\frac = y^2 f_\,, \\
       \cdot\mathcal\!\cdot\                                               \ &=\  xy\frac = xy f_\,,
\end

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 ...

). :

\begin
                      y &= \Box \dot \equiv \int \dot \,dt = \int f'(t) \,dt = D_t^ (D_t y) = f(t) + C_0 = y_t + C_0 \\
  \overset &= \Box y \equiv \int y \,dt = \int f(t) \,dt = D_t^ y = F(t) + C_1
\end

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). :

\overset = \Box \overset \equiv \int \overset \,dt = \int F(t) \,dt = D_t^ y = g(t) + C_2

Higher order time integrals were as follows: :

\begin
        \overset &= \Box \overset \equiv \int \overset \,dt = \int g(t) \,dt = D_t^ y = G(t) + C_3 \\
  \overset &= \Box \overset \equiv \int \overset \,dt = \int G(t) \,dt = D_t^ y = h(t) + C_4 \\
       \overset\overset &= \Box \overset\oversety \equiv \int \overset\oversety \,dt = \int s(t) \,dt = D_t^ y = S(t) + C_n
\end

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: :

\begin
      f_x &= \frac \\
  f_ &= \frac.
\end

This type of notation is especially useful for taking

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: :

\frac = f_x = \partial_x f.

What makes this distinction important is that a non-partial derivative such as

\textstyle \frac

''may'', depending on the context, be interpreted as a rate of change in

f

relative to

x

when all variables are allowed to vary simultaneously, whereas with a partial derivative such as

\textstyle \frac

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 ...

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

\left(\frac\right)_

is the derivative of the temperature ''T'' with respect to the volume ''V'' while keeping constant the entropy (subscript) ''S'', while

\left(\frac\right)_

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 :

\frac = f_,

\frac = f_,

and so on. Mixed partial derivatives can be expressed as :

\frac = f_.

In this last case the variables are written in inverse order between the two notations, explained as follows: :

(f_)_ = f_,

\frac\!\left(\frac\right) = \frac.

So-called multi-index notation is used in situations when the above notation becomes cumbersome or insufficiently expressive. When considering functions on

\R^n

, we define a multi-index to be an ordered list of

n

non-negative integers:

\alpha = (\alpha_1,..,\alpha_n), \ \alpha_i \in \Z_

. We then define, for

f:\R^n \to X

, the notation

\partial^\alpha f = \frac \cdots \frac f

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 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

\mathbf = (\mathbf_x, \mathbf_y, \mathbf_z)

, and that

\varphi = \varphi(x,y,z)

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 or nabla, is symbolically defined in the form of a vector, :

\nabla = \left( \frac, \frac, \frac \right)\!,

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

\mathrm \varphi

of the scalar field

\varphi

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 ''

\varphi

'', ::

\begin
  \operatorname \varphi
    &= \left( \frac, \frac, \frac \right) \\
    &= \left( \frac, \frac, \frac \right) \varphi \\
    &= \nabla \varphi
\end

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

\mathrm\,\mathbf

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, ::

\begin
  \operatorname \mathbf
    &=  +  +  \\
    &= \left( \frac, \frac, \frac \right)  \cdot \mathbf \\
    &= \nabla \cdot \mathbf
\end

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

\operatorname \operatorname \varphi

of the scalar field

\varphi

is a scalar, which is symbolically expressed by the scalar multiplication of ∇² and the scalar field ''φ'', ::

\begin
  \operatorname \operatorname \varphi
    &= \nabla \cdot (\nabla \varphi) \\
    &= (\nabla \cdot \nabla) \varphi \\
    &= \nabla^2 \varphi \\
    &= \Delta \varphi \\
\end

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

\mathrm\,\mathbf

, or

\mathrm\,\mathbf

, 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, ::

\begin
  \operatorname \mathbf
    &= \left(
          - ,
          - ,
          - 
       \right) \\
    &= \left(  -  \right) \mathbf +
       \left(  -  \right) \mathbf +
       \left(  -  \right) \mathbf \\
    &= \begin
         \mathbf & \mathbf & \mathbf \\
         \cfrac & \cfrac & \cfrac \\
         A_x & A_y & A_z
       \end \\
    &= \nabla \times \mathbf
\end

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 has a direct analogue in the multiplication of scalar fields by applying the gradient operator, as in :

(f g)' = f' g+f g' ~~~ \Longrightarrow ~~~ \nabla(\phi \psi) = (\nabla \phi) \psi + \phi (\nabla \psi).

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, the d'Alembert operator, also called the d'Alembertian, wave operator, or box operator is represented as

\Box

, or as

\Delta

when not in conflict with the symbol for the Laplacian.

References

External links

Earliest Uses of Symbols of Calculus
maintained by Jeff Miller (). {{Differential equations topics Differential calculus Mathematical notation