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| {{Calculus |Differential}}
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| In [[differential calculus]], there is no single uniform '''notation for differentiation'''. Instead, several different notations for the [[derivative]] of a [[function (mathematics)|function]] or [[dependent variable|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 are listed below.
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| == Leibniz's notation ==
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| {{main|Leibniz's notation}}
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| <div style="float:left; margin: 0 10px 10px 0; padding:20px; font-size:400%; line-height: 100%; font-family:Times New Roman, serif; background-color: #ddddff; border: 1px solid #aaaaff;">
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| <div style="display:inline-block; margin: 0 15px"><div style="border-bottom:2px solid black;padding-bottom:6px">''dy''</div><div>''dx''</div></div>
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| <div style="display:inline-block; margin: 0 15px"><div style="border-bottom:2px solid black;padding-bottom:6px">''d''{{resize|30%| }}<sup>2</sup>''y''</div><div>''dx''<sup>2</sup></div></div>
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| </div>
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| {{-}}
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| The original notation employed by [[Gottfried Leibniz]] is used throughout mathematics. It is particularly common when the equation
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| ''y'' = ''f''(''x'') is regarded as a functional relationship between [[dependent and independent variables]] ''y'' and ''x''. In this case the derivative can be written as:
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| : <math>\frac{dy}{dx}</math>
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| The function whose value at ''x'' is the derivative of ''f'' at ''x'' is therefore written
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| : <math>\frac{d\bigl(f(x)\bigr)}{dx}\text{ or }\frac{d}{dx}\bigl(f(x)\bigr)</math>
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| (although strictly speaking this denotes the variable value of the derivative function rather than the derivative function itself).
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| Higher derivatives are expressed as
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| : <math>\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)</math>
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| for the ''n''th derivative of ''y'' = ''f''(''x''). Historically, this came from the fact that, for example, the third derivative is:
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| : <math>\frac{d \Bigl(\frac{d \left( \frac{d y} {dx}\right)} {dx}\Bigr)} {dx} = \left(\frac{d}{dx}\right)^3 \bigl(f(x)\bigr)</math>
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| which we can loosely write (dropping the brackets in the denominator) as:
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| : <math> \frac{d^3}{\left(dx\right)^3} \bigl(f(x)\bigr)=\frac{d^3}{dx^3} \bigl(f(x)\bigr)</math>
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| as above.
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| With Leibniz's notation, the value of the derivative of ''y'' at a point ''x'' = ''a'' can be written in two different ways:
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| : <math>\frac{dy}{dx}\left.{\!\!\frac{}{}}\right|_{x=a} = \frac{dy}{dx}(a).</math>
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| Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially helpful when considering [[partial derivative]]s. It also makes the [[chain rule]] easy to remember and recognize:
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| : <math>\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.</math>
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| In the formulation of calculus in terms of limits, the ''du'' symbol has been assigned various meanings by various authors.
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| Some authors do not assign a meaning to ''du'' by itself, but only as part of the symbol ''du''/''dx''.
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| Others define ''dx'' as an independent variable, and use ''d''(''x'' + ''y'') = ''dx'' + ''dy'' and ''d''(''x''·''y'') = ''dx''·''y'' + ''x''·''dy'' as formal [[axiom]]s for differentiation. See [[differential algebra]].
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| In [[non-standard analysis]] ''du'' is defined as an infinitesimal.
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| It is also interpreted as the [[exterior derivative]] d''u'' of a function ''u''.
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| See [[differential (infinitesimal)]] for further information.
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| == Lagrange's notation ==
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| <div style="float:right; margin: 0 0 10px 10px; padding:40px; font-size:500%; font-family:Times New Roman, serif; background-color: #ddddff; border: 1px solid #aaaaff;">''f ''ʹ(''x'') ''f ''ʺ(''x'')</div>
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| One of the most common modern notations for differentiation is due to [[Joseph Louis Lagrange]] and uses the [[Prime (symbol)|prime mark]]:
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| the first three derivatives of ''f'' are denoted
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| : <math>f'\;</math> for the first derivative,
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| : <math>f''\;</math> for the [[second derivative]],
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| : <math>f'''\;</math> for the third derivative.
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| After this, some authors continue by employing Roman numerals such as ''f''<sup> IV</sup> for the fourth derivative of ''f'', while others put the number of derivatives in brackets, so that the fourth derivative of ''f'' would be denoted ''f''<sup> (4)</sup>. The latter notation extends readily to any number of derivatives, so that the ''n''th derivative of ''f'' is denoted ''f''<sup> (''n'')</sup>.
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| == Euler's notation ==
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| <div style="float:right; margin: 0 0 10px 10px; padding:40px; font-size:500%; font-family:Times New Roman, serif; background-color: #ddddff; border: 1px solid #aaaaff;">''D<sub>x</sub>{{resize|50%| }}y'' ''D''<sup>2</sup>''f''</div>
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| [[Leonhard Euler|Euler]]'s notation uses a [[differential operator]], denoted as ''D'', which is prefixed to the function so that the derivatives of a function ''f'' are denoted by
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| : <math>Df \;</math> for the first derivative,
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| : <math>D^2f \;</math> for the second derivative, and
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| : <math>D^nf \;</math> for the ''n''th derivative, for any positive integer ''n''.
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| 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, leading to the alternative notation
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| : <math>D_x y \;</math> for the first derivative,
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| : <math>D^2_x y\;</math> for the second derivative, and
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| : <math>D^n_x y \;</math> for the ''n''th derivative, for any positive integer ''n''.
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| If there is only one independent variable present, the subscript to the operator is usually dropped, however.
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| Euler's notation is useful for stating and solving [[linear differential equation]]s.
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| == Newton's notation ==
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| <div style="float:right; margin: 0 0 10px 10px; padding:40px; font-size:500%; font-family:Times New Roman, serif; background-color: #ddddff; border: 1px solid #aaaaff;">''ẋ ẍ''</div>
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| Newton's notation for differentiation (also called the dot notation for differentiation) requires placing a dot over the dependent variable and is often used for time derivatives such as [[velocity]]
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| : <math>\dot{y} = \frac{dy}{dt} \,,</math>
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| [[acceleration]]
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| : <math>\ddot{y} = \frac{d^2y}{dt^2} \,,</math>
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| 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''.<ref>Article 567 in Florian Cajori, A History of Mathematical Notations, Dover Publications, Inc. New York. ISBN 0-486-67766-4</ref>
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| Newton's notation is mainly used in [[mechanics]], physics, and the theory of [[ordinary differential equation]]s. It is usually only used for first and second derivatives, and then, only to denote derivatives with respect to time.
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| 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.
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| In [[physics]], [[macroeconomics]] and other fields, Newton's notation is used mostly for [[time derivative]]s, as opposed to [[slope]] or [[position (vector)|position]] [[derivative]]s.
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| Newton did not develop a standard [[mathematical notation]] for [[integral|integration]] but used many different notations.
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| == Partial derivatives ==
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| <div style="float:right; margin: 0 0 10px 10px; padding:40px; font-size:500%; font-family:Times New Roman, serif; background-color: #ddddff; border: 1px solid #aaaaff;">''f<sub>x</sub>'' ''f<sub>xy</sub>''</div>
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| When more specific types of differentiation are necessary, such as in [[multivariate calculus]] or [[tensor analysis]], other notations are common.
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| For a function ''f(x)'', we can express the derivative using subscripts of the independent variable:
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| : <math>f_x = \frac{df}{dx} </math>
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| : <math>f_{x x} = \frac{d^2f}{dx^2}. </math>
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| This is especially useful for taking [[partial derivatives]] of a function of several variables.
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| <div style="float:right; margin: 0 0 10px 10px; padding:20px; font-size:500%; line-height: 100%; font-family:Times New Roman, serif; background-color: #ddddff; border: 1px solid #aaaaff;">
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| <div style="display:inline-block; margin: 0 15px"><div style="border-bottom:2px solid black;padding-bottom:6px">''∂f''</div><div>''∂x''</div></div>
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| </div>
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| 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:
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| : <math>\frac{\partial f}{\partial x} = f_x = \partial_x f = \partial^x f, </math>
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| where the final two notations are equivalent in flat [[Euclidean Space]] but are different in other [[manifolds]]. | |
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| Other generalizations of the derivative can be found in various subfields of mathematics, physics, and engineering.
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| == Notation in vector calculus ==
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| [[Vector calculus]] concerns [[derivative|differentiation]] and [[integral|integration]] of [[vector field|vector]] or [[scalar fields|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 <math>\mathbf{A} = (\mathbf{A}_x, \mathbf{A}_y, \mathbf{A}_z)</math>, and a [[scalar field]] <math>\varphi</math> is <math>\varphi = f(x,y,z)\,</math>.
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| First, a differential operator, or a [[Hamilton operator]] [[nabla symbol|∇]] which is called [[del]] is symbolically defined in the form of a vector,
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| :<math>\nabla = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right)</math>,
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| where the terminology ''symbolically'' reflects that the operator ∇ will also be treated as an ordinary vector. | |
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| <div style="float:right; margin: 0 0 10px 10px; padding:30px; font-size:500%; font-family:Times New Roman, serif; background-color: #ddddff; border: 1px solid #aaaaff;">∇''φ''</div>
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| * '''[[Gradient]]''': The gradient <math>\mathrm{grad\,} \varphi\,</math> of the scalar field <math>\varphi</math> is a vector, which is symbolically expressed by the [[multiplication]] of ∇ and scalar field ''<math>\varphi</math>'',
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| : <math> \mathrm{grad\,}\,\varphi = \left( \frac{\partial \varphi}{\partial x}, \frac{\partial \varphi}{\partial y}, \frac{\partial \varphi}{\partial z} \right) </math> ,
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| ::: <math>= \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \varphi </math> ,
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| ::: <math>= \nabla \varphi</math> .
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| <div style="float:right; margin: 0 0 10px 10px; padding:30px; font-size:500%; font-family: Serif; background-color: #ddddff; border: 1px solid #aaaaff;">∇∙'''A'''</div>
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| * '''[[Divergence]]''': The divergence <math>\mathrm{div}\,\mathbf{A}\,</math> of the vector '''A''' is a scalar, which is symbolically expressed by the [[dot product]] of ∇ and the vector '''A''',
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| : <math> \mathrm{div\,} \mathbf{A} = {\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}</math> ,
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| ::: <math>= \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \cdot \mathbf{A}</math>,
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| ::: <math> = \nabla \cdot \mathbf{A}</math> .
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| <div style="float:right; margin: 0 0 0px 0px; padding:10px 30px 30px 30px; font-size:500%; font-family: Times New Roman, serif; background-color: #ddddff; border: 1px solid #aaaaff;">∇<sup>2</sup>''φ''</div>
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| * '''[[Laplacian]]''': The Laplacian <math>\mathrm{div} \, \mathrm{grad} \, \varphi\,</math> of the scalar field <math>\varphi</math> is a scalar, which is symbolically expressed by the scalar multiplication of ∇<sup>2</sup> and the scalar field ''φ'',
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| : <math>\mathrm{div} \, \mathrm{grad} \, \varphi\, = \nabla \cdot (\nabla \varphi)</math>
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| :::: <math> = (\nabla \cdot \nabla) \varphi = \nabla^2 \varphi = \Delta \varphi </math> ,
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| :where, <math>\Delta = \nabla^2</math> is called a [[Laplacian operator]].
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| <div style="float:right; margin: 0 0 10px 10px; padding:30px; font-size:500%; font-family: Serif; background-color: #ddddff; border: 1px solid #aaaaff;">∇×'''A'''</div>
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| * '''[[Curl (mathematics)|Rotation]]''': The rotation <math>\mathrm{curl}\,\mathbf{A}\,</math>, or <math>\mathrm{rot}\,\mathbf{A}\,</math>, of the vector '''A''' is a vector, which is symbolically expressed by the [[cross product]] of ∇ and the vector '''A''',
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| :<math> \mathrm{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) </math>,
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| :::<math>= \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}</math>,
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| :::<math>=
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| \begin{vmatrix}
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| \mathbf{i} & \mathbf{j} & \mathbf{k} \\[5pt]
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| \cfrac{\partial}{\partial x} & \cfrac{\partial}{\partial y} & \cfrac{\partial}{\partial z} \\[12pt]
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| A_x & A_y & A_z
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| \end{vmatrix}
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| </math> ,
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| :::<math>= \nabla \times \mathbf{A}</math> .
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| 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
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| :<math>(f g)' = f' g+f g' ~~~ \Longrightarrow ~~~ \nabla(\phi \psi) = (\nabla \phi) \psi + \phi (\nabla \psi).</math>
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| 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 <math>\Box</math>, or as <math>\Delta</math> when not in conflict with the symbol for the Laplacian.
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| ==See also==
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| * [[Derivative]]
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| * [[Jacobian matrix]]
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| * [[Hessian matrix]]
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| ==External links==
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| *[http://jeff560.tripod.com/calculus.html Earliest Uses of Symbols of Calculus], maintained by Jeff Miller.
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| {{reflist}}
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| [[Category:Differential calculus]]
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| [[Category:Mathematical notation]]
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| [[de:Differential (Mathematik)#Notationen der Ableitung]]
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