Notation for differentiation

Jump to: navigation, search

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 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

dy
dx
d 2y
dx2
Main article: Leibniz's notation

The original notation employed by Gottfried Leibniz is used throughout mathematics. It is particularly common when the equation y = f(x) is regarded as a functional relationship between dependent and independent variables y and x. In this case the derivative ( dy/dx ) can be written as:

$\frac{dy}{dx}$

The function whose value at x is the derivative of f at x is therefore written

$\frac{d\bigl(f(x)\bigr)}{dx}\text{ or }\frac{d}{dx}\bigl(f(x)\bigr)$

(although strictly speaking this denotes the variable value of the derivative function rather than the derivative function itself).

Higher derivatives ( d2y/dx2 ··· dny/dxn ) are expressed as

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

for the nth derivative of y = f(x). Historically, this came from the fact that, for example, the third derivative ( d3y/dx3 ) is:

$\frac{d \Bigl(\frac{d \left( \frac{d y} {dx}\right)} {dx}\Bigr)} {dx} = \left(\frac{d}{dx}\right)^3 \bigl(f(x)\bigr)$

which we can loosely write (dropping the brackets in the denominator) as:

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

as above.

With Leibniz's notation, the value of the derivative of y at a point x = a can be written in two different ways:

$\frac{dy}{dx}\left.{\!\!\frac{}{}}\right|_{x=a} = \frac{dy}{dx}(a).$

Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially helpful when considering partial derivatives. It also makes the chain rule easy to remember and recognize:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.$

In the formulation of calculus in terms of limits, the du symbol has been assigned various meanings by various authors.

Some authors do not assign a meaning to du by itself, but only as part of the symbol du/dx.

Others define dx as an independent variable, and use d(x + y) = dx + dy and d(x·y) = dx·y + x·dy as formal axioms for differentiation. See differential algebra.

In non-standard analysis du is defined as an infinitesimal.

It is also interpreted as the exterior derivative du of a function u.

Leibniz's notation for antidifferentiation

y dx  ∫∫ y dx2
For functions of 2 or more variables, see Multiple integral.

Leibniz introduced the integral symbol ( )[1] denoted as the current mathematical notation for the integration ( ∫ y dx )[1] in Analyseos tetragonisticae pars secunda and Methodi tangentium inversae exempla (both from 1675):

$\int y'\,dx = \int f'(x)\,dx = f(x) + C_0 = y + C_0$
$\int y\,dx = \int f(x)\,dx = F(x) + C_1$
$\int \int y\,dx^2 = \int \left ( \int y\,dx \right ) dx = \int_{X\times X} f(x)\,dx = \int F(x)\,dx = g(x) + C_2$
$\underbrace{\int \dots \int}_n y\,\underbrace{dx \dots dx}_n = \int_{\underbrace{X\times\cdots\times X}_n} f(x)\,dx = \int s(x)\,dx = S(x) + C_n$

In order to calculate the nth antiderivative, Cauchy formula for repeated integration may be of help.

Lagrange's notation

f ʹ(x)  f ʺ(x)

One of the most common modern notations for differentiation is due to Joseph Louis Lagrange, defined as φ′(x)[2]

$\phi'(x) \equiv \frac{d\phi (x)}{dx}\,, \quad \text{where}\ \phi(x)\ \text{is an ordinary function of }x\,.$

It uses the prime mark ( f′ f″ f‴ ... f (n) )[3] suffixed to a function: the first three derivatives of f are denoted

$f'\;$ for the first derivative,
$f''\;$ for the second derivative,
$f'''\;$ for the third derivative.

After this, some authors continue by employing Roman numerals such as f IV for the fourth derivative of f, while others put the ordinal of the derivative in parentheses, so that the fourth derivative of f would be denoted f (4). The latter notation extends readily to any number of derivatives, so that the nth derivative of f is denoted f (n).

When there're two independent variables for a function f(x,y), the following convention may be followed:[4]

$f^\prime = \frac{df}{dx} = f_x \,$
$f_\prime = \frac{df}{dy} = f_y \,$
$f^{\prime\prime} = \frac{d^2 f}{dx^2} = f_{xx} \,$
$f_\prime^\prime = \frac{\partial ^2 f}{\partial x \partial y}\ = f_{xy} \,$
$f_{\prime\prime} = \frac{d^2 f}{dy^2} = f_{yy} \,,$

Lagrange's notation for antidifferentiation

f (-1)(x)  f (-2)(x)

When taking the antiderivative, Lagrange followed Leibniz's notation:[2]

$\phi(x) = \int \phi'(x)\,dx = \int y\,dx \,.$

However, Lagrange's notation for higher order operations extends to integration ( f (-1)f (-2)f (-3) ... f (-n) ) as:

$f^{(-1)}\;$ for the first integral (do not confuse with the reciprocal function f −1),
$f^{(-2)}\;$ for the second integral,
$f^{(-3)}\;$ for the third integral.
$f^{(-n)}\;$ for the nth integral.

Euler's notation

Dx y  D2f

Euler's notation uses a differential operator suggested by Arbogast, denoted as D (D operator)[5] or (Newton-Leibniz operator)[6]

$D \equiv \frac{d}{dx} \,,$

which must be prefixed to an expression ( D yD2 y ... Dn y ) so that the derivatives of a function f are denoted by[4]

$Df \;$ for the first derivative,
$D^2f \;$ for the second derivative,
$D^3f \;$ for the third derivative, and
$D^nf \;$ for the nth derivative, for any positive integer n.

When taking the derivative of a dependent variable y = f(x) it is common to add the independent variable x as a subscript to the D notation ( D
x
yD2
x
y ... Dn
x
y
), leading to the alternative notation[4]

$D_x y \;$ for the first derivative,
$D^2_x y\;$ for the second derivative,
$D^3_x y \;$ for the third derivative, and
$D^n_x y \;$ for the nth derivative, for any positive integer n.

If there is only one independent variable present, the subscript to the operator is usually dropped, however. For multiples variables ( D2
xy

yD3
xyz

y ...
)[4]

$D^2_{xy} f \equiv \frac{\partial ^2 f}{\partial x \partial y} = f_{xy} \;,$

Euler's notation is useful for stating and solving linear differential equations, as it simplifies presentation of the differential equation, which can make seeing the essential elements of the problem easier.

Euler's notation for antidifferentiation

D-1
x
y  D-2
f

Euler's notation ( D-1 yD-2 y ... D-n y ) is used for the repeated integral[7] as follows[6]

$D^{-1}f(x) \;$ for the first integral,
$D^{-2}f(x) \;$ for the second integral,

and for the nth integral (for any positive integer n)

$D^{-n}f(x) = \underbrace{\int \dots \int_{0}^{x}}_n f(x)\,\underbrace{dx \dots dx}_n \;.$

Newton's notation

ẋ  ẍ

Newton's notation for differentiation (also called the dot notation for differentiation) requires placing a dot over the dependent variable ( )[8]

$\dot{y} \equiv \frac{dy}{dt} = \frac{d}{dt}\Bigl(f(t)\Bigr) = D_t y = f'(t) = y'_t$

where the independent variable normally denotes time. Higher order derivatives ( ÿÿ̇ÿ̈ÿ̈̇ ...  ...  )[8] are shown below:[9]

$\ddot{y} \equiv \frac{d^2y}{dt^2} = \frac{d}{dt}\left(\frac{dy}{dt}\right) = \frac{d}{dt}\Bigl(\dot{y}\Bigr) = \frac{d}{dt}\Bigl(f'(t)\Bigr) = D_t^2 y = f''(t) = y''_t$
$\overset{...}{y} = \dot{\ddot{y}} \equiv \frac{d^3y}{dt^3} = D_t^3 y = f'''(t) = y'''_t$
$\overset{\,4}{\dot{y}} = \overset{....}{y} = \ddot{\ddot{y}} \equiv \frac{d^4y}{dt^4} = D_t^4 y = f^{IV}(t) = y^{(4)}_t$
$\overset{\,5}{\dot{y}} = \ddot{\overset{...}{y}} = \dot{\ddot{\ddot{y}}} = \ddot{\dot{\ddot{y}}} \equiv \frac{d^5y}{dt^5} = D_t^5 y = f^{V}(t) = y^{(5)}_t$
$\overset{\,6}{\dot{y}} = \overset{...}{\overset{...}{y}} \equiv \frac{d^6y}{dt^6} = D_t^6 y = f^{VI}(t) = y^{(6)}_t$
$\overset{\,7}{\dot{y}} = \dot{\overset{...}{\overset{...}{y}}} \equiv \frac{d^7y}{dt^7} = D_t^7 y = f^{VII}(t) = y^{(7)}_t$
$\overset{\,10}{\dot{y}} = \ddot{\ddot{\ddot{\ddot{\ddot{y}}}}} \equiv \frac{d^{10}y}{dt^{10}} = D_t^{10} y = f^{X}(t) = y^{(10)}_t$
$\overset{\,n}{\dot{y}} \equiv \frac{d^ny}{dt^n} = D_t^n y = f^{(n)}(t) = y^{(n)}_t$

It's often used in classical mechanics and mathematics for time derivatives such as velocity ( )[10]

$\dot{x} \equiv \frac{dx}{dt} = x'(t) = x'_t \quad \rightsquigarrow \quad x'\ \text{ or }\ v\,,$
$\ddot{x} \equiv \frac{d^2x}{dt^2} = \frac{d}{dt}\Bigl(\dot{x}\Bigr) = x''(t) = x''_t \quad \rightsquigarrow \quad x''\ \text{ or }\ a\,,$

and so on. It can also be used as a direct substitute for the prime in Lagrange's notation. Again this is common for functions f(t) of time. Newton referred to this as a fluxion.[12]

Newton's notation is mainly used in mechanics, physics, and the theory of ordinary differential equations. It is usually only used for first and second derivatives, and then, only to denote derivatives with respect to time.

Dot notation is not very useful for higher-order derivatives, but in mechanics and other engineering fields, the use of higher than second-order derivatives is limited.

In physics, economics and other fields, Newton's notation is used mostly for time derivatives, as opposed to slope or position derivatives.

When taking the derivative of a dependent variable y = f(x), an alternative notation exists:[13]

$\frac{\dot{y}}{\dot{x}} = \dot{y}:\dot{x} \equiv \frac{dy}{dt}:\frac{dx}{dt} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{dy}{dx} = \frac{d}{dt}\Bigl(f(x)\Bigr) = D y = f'(x) = y'$

Newton developed the following partial differential operators using side-dots on a curved X ( ⵋ ). Definitions given by Whiteside are below:[14][15]

$\mathcal{X} \ = \ f(x,y) \,,$
$\cdot\mathcal{X} \ = \ x\frac{\partial f}{\partial x} = xf_x\,,$
$\mathcal{X}\cdot \ = \ y\frac{\partial f}{\partial y} = yf_y\,,$
$\colon\mathcal{X}\,\text{ or }\,\cdot\colon\mathcal{X} \ = \ x^2\frac{\partial^2 f}{\partial x^2} = x^2 f_{xx}\,,$
$\mathcal{X}\colon\,\text{ or }\,\cdot\mathcal{X}\cdot \ = \ y^2\frac{\partial^2 f}{\partial y^2} = y^2 f_{yy}\,,$
$\cdot\mathcal{X}\cdot\,\text{ or }\,\mathcal{X}\colon\cdot \ = \ xy\frac{\partial^2 f}{\partial x\partial y} = xy f_{xy}\,,$

Newton's notation for antidifferentiation

x̍   x̎

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 ( ),[8] a prefixing rectangle ( y )[8] or the inclosure of the term in a rectangle ( y )[8] to denote the fluent or [time] integral (absement)

$y = \Box \dot{y} \equiv \int \dot{y} \,dt = \int f'(t) \,dt = D_t^{-1} (D_t y) = f(t) + C_0 = y_t + C_0$
$\overset{\,\prime}{y} = \Box y \equiv \int y \,dt = \int f(t) \,dt = D_t^{-1} y = F(t) + C_1$

two small vertical bars or primes ( ),[8] or a combination of previous symbols ( ),[8] to denote the second time integral (absity)

$\overset{\,\prime\prime}{y} = \Box \overset{\,\prime}{y} \equiv \int \overset{\,\prime}{y} \,dt = \int F(t) \,dt = D_t^{-2} y = g(t) + C_2$

and higher order time integrals ( y̎̍y̎̎ ...  )[8] as follows:[16]

$\overset{\,\prime\prime\prime}{y} = \Box \overset{\,\prime\prime}{y} \equiv \int \overset{\,\prime\prime}{y} \,dt = \int g(t) \,dt = D_t^{-3} y = G(t) + C_3$
$\overset{\,\prime\prime\prime\prime}{y} = \Box \overset{\,\prime\prime\prime}{y} \equiv \int \overset{\,\prime\prime\prime}{y} \,dt = \int G(t) \,dt = D_t^{-4} y = h(t) + C_4$
$\overset{\;n}\overset{\,\prime}{y} = \Box \overset{\;n-1}\overset{\,\prime}y \equiv \int \overset{\;n-1}\overset{\,\prime}y \,dt = \int s(t) \,dt = D_t^{-n} y = S(t) + C_n$

This mathematical notation didn't became widespread because of printing difficulties and other controversies.

Partial derivatives

fx  fxy

When more specific types of differentiation are necessary, such as in multivariate calculus or tensor analysis, other notations are common.

For a function f(x), we can express the derivative using subscripts of the independent variable:

$f_x = \frac{df}{dx}$
$f_{x x} = \frac{d^2f}{dx^2}.$

This is especially useful for taking partial derivatives of a function of several variables.

∂f
∂x

Partial derivatives will generally be distinguished from ordinary derivatives by replacing the differential operator d with a "" symbol. For example, we can indicate the partial derivative of f(x,y,z) with respect to x, but not to y or z in several ways:

$\frac{\partial f}{\partial x} = f_x = \partial_x f = \partial^x f,$

where the final two notations are equivalent in flat Euclidean Space but are different in other manifolds.

Other notations can be found in various subfields of mathematics, physics, and engineering, see for example the Maxwell relations of thermodynamics. The symbol $\left(\frac{\partial T}{\partial V}\right)_S$ is the derivative of the temperature T with respect to the volume V while keeping constant the entropy S, while $\left(\frac{\partial T}{\partial V}\right)_P$ is the derivative of the temperature with respect to the volume while keeping constant the pressure P.

Notation in vector calculus

Vector calculus concerns differentiation and integration of vector or scalar fields particularly in a three-dimensional Euclidean space, and uses specific notations of differentiation. In a Cartesian coordinate o-xyz, assuming a vector field A is $\mathbf{A} = (\mathbf{A}_x, \mathbf{A}_y, \mathbf{A}_z)$, and a scalar field $\varphi$ is $\varphi = f(x,y,z)\,$.

First, a differential operator, or a Hamilton operator which is called del is symbolically defined in the form of a vector,

$\nabla = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right),$

where the terminology symbolically reflects that the operator ∇ will also be treated as an ordinary vector.

φ
• Gradient: The gradient $\mathrm{grad\,} \varphi\,$ of the scalar field $\varphi$ is a vector, which is symbolically expressed by the multiplication of ∇ and scalar field $\varphi$,
\begin{align} \operatorname{grad} \varphi &= \left( \frac{\partial \varphi}{\partial x}, \frac{\partial \varphi}{\partial y}, \frac{\partial \varphi}{\partial z} \right) \\ &= \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \varphi \\ &= \nabla \varphi \end{align}
∇∙A
• Divergence: The divergence $\mathrm{div}\,\mathbf{A}\,$ of the vector field A is a scalar, which is symbolically expressed by the dot product of ∇ and the vector A,
\begin{align} \operatorname{div} \mathbf{A} &= {\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z} \\ &= \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \cdot \mathbf{A} \\ &= \nabla \cdot \mathbf{A} \end{align}
2φ
• Laplacian: The Laplacian $\operatorname{div} \operatorname{grad} \varphi$ of the scalar field $\varphi$ is a scalar, which is symbolically expressed by the scalar multiplication of ∇2 and the scalar field φ,
\begin{align} \operatorname{div} \operatorname{grad} \varphi &= \nabla \cdot (\nabla \varphi) \\ &= (\nabla \cdot \nabla) \varphi \\ &= \nabla^2 \varphi \\ &= \Delta \varphi \\ \end{align}
where, $\Delta = \nabla^2$ is called a Laplacian operator.
∇×A
• Rotation: The rotation $\mathrm{curl}\,\mathbf{A}\,$, or $\mathrm{rot}\,\mathbf{A}\,$, of the vector field A is a vector, which is symbolically expressed by the cross product of ∇ and the vector A,
\begin{align} \operatorname{curl} \mathbf{A} &= \left( {\partial A_z \over {\partial y} } - {\partial A_y \over {\partial z} }, {\partial A_x \over {\partial z} } - {\partial A_z \over {\partial x} }, {\partial A_y \over {\partial x} } - {\partial A_x \over {\partial y} } \right) \\ &= \left( {\partial A_z \over {\partial y} } - {\partial A_y \over {\partial z} } \right) \mathbf{i} + \left( {\partial A_x \over {\partial z} } - {\partial A_z \over {\partial x} } \right) \mathbf{j} + \left( {\partial A_y \over {\partial x} } - {\partial A_x \over {\partial y} } \right) \mathbf{k} \\ &= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\[5pt] \cfrac{\partial}{\partial x} & \cfrac{\partial}{\partial y} & \cfrac{\partial}{\partial z} \\[12pt] A_x & A_y & A_z \end{vmatrix} \\ &= \nabla \times \mathbf{A} \end{align}

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

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

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

1. ^ a b For Unicode characters related to Leibniz's notation, see Integral symbol
2. ^ a b Lagrange, Nouvelle méthode pour résoudre les équations littérales par le moyen des séries (1770), p. 25-26. http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN308900308%7CLOG_0017&physid=PHYS_0031
Lagrange used his notation in earlier unpublished works.
3. ^ Unicode characters related to Lagrange's notation:
• U+2032 ◌′ prime (derivative)
• U+2033 ◌″ double prime (double derivative)
• U+2034 ◌‴ triple prime (third derivative)
• U+2057 ◌⁗ quadruple prime (fourth derivative) * ← f IV or f (4) are preferred.
4. ^ a b c d The Differential and Integral Calculus (Augustus De Morgan, 1842). pp. 267-268
5. ^ http://www.codecogs.com/library/maths/calculus/differential/the-d-operator.php
6. ^ a b Weisstein, Eric W. "Differential Operator." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DifferentialOperator.html
7. ^ Weisstein, Eric W. "Repeated Integral." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RepeatedIntegral.html
8. Unicode characters related to Newton's notation:
• U+0307 ◌̇ combining dot above (derivative)
• U+0308 ◌̈ combining diaeresis (double derivative)
• U+20DB ◌⃛ combining three dots above (third derivative) * ← replaced by "combining diaeresis" + "combining dot above".
• U+20DC ◌⃜ combining four dots above (fourth derivative) * ← replaced by "combining diaeresis" twice.
• U+030D ◌̍ combining vertical line above (integral)
• U+030E ◌̎ combining double vertical line above (second integral)
• U+25AD white rectangle (integral)
• U+20DE ◌⃞ combining enclosing square (integral) *
• U+1DE0 ◌ᷠ combining latin small letter n (nth derivative) *
* Because of rendering issues, markup language has been used.
9. ^ Newton's notation reproduced from:
• 1st to 5th derivatives: Quadratura curvarum (Newton, 1704), p. 7 (p. 5r in original MS: http://cudl.lib.cam.ac.uk/view/MS-ADD-03962/9).
• 1st to 7th, nth and (n+1)th derivatives: Method of Fluxions (Newton, 1736), pp. 313-318 and p. 265 (p. 163 in original MS: http://cudl.lib.cam.ac.uk/view/MS-ADD-03960/257)
• 1st to 5th derivatives : A Treatise of Fluxions (Colin MacLaurin, 1742), p. 613
• 1st to 4th and nth derivatives: Articles "Differential" and "Fluxion", Dictionary of Pure and Mixed Mathematics (Peter Barlow, 1814)
• 1st to 4th, 10th and nth derivatives: Articles 622, 580 and 579 in A History of Mathematical Notations (F .Cajori, 1929)
• 1st to 6th and nth derivatives: The Mathematical Papers of Isaac Newton Vol. 7 1691-1695 (D. T. Whiteside, 1976), pp.88 and 17
• 1st to 3rd and nth derivatives: A History of Analysis (Hans Niels Jahnke, 2000), pp. 84-85
The dot for nth derivative may be omitted ( $\overset{\,n}{y}$ )
10. ^ Weisstein, Eric W. "Overdot." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Overdot.html
11. ^ Weisstein, Eric W. "Double Dot." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DoubleDot.html
12. ^ Article 567 in Florian Cajori, A History of Mathematical Notations (1929), Dover Publications, Inc. New York. ISBN 0-486-67766-4
13. ^ Article 580 in Florian Cajori, A History of Mathematical Notations (1929), Dover Publications, Inc. New York. ISBN 0-486-67766-4
14. ^ "Patterns of Mathematical Thought in the Later Seventeenth Century", Archive for History of Exact Sciences Vol. 1, No. 3 (D. T. Whiteside, 1961), pp. 179-388
15. ^ S.B. Engelsman has given more strict definitions in Families of Curves and the Origins of Partial Differentiation (2000), pp. 223-226
16. ^ Newton's notation for integration reproduced from:
The nth integral notation is deducted from the nth derivative. It could be used in Methodus Incrementorum Directa & Inversa (Brook Taylor, 1715)