Editing Gradient (section)

==Definition==
[[File:3d-gradient-cos.svg|thumb|350px|The gradient of the function {{math|''f''(''x'',''y'') {{=}} −(cos<sup>2</sup>''x'' + cos<sup>2</sup>''y'')<sup>2</sup>}} depicted as a projected [[vector field]] on the bottom plane.]]

The gradient (or gradient vector field) of a scalar function {{math|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, …, ''x<sub>n</sub>'')}} is denoted {{math|∇''f''}} or {{math|{{vec|∇}}''f''}}  where {{math|∇}} ([[nabla symbol|nabla]]) denotes the vector [[differential operator]], [[del]]. The notation {{math|grad ''f''}}  is also commonly used to represent the gradient. The gradient of {{math|''f''}} is defined as the unique vector field whose dot product with any [[Euclidean vector|vector]] {{math|'''v'''}} at each point {{math|''x''}} is the directional derivative of {{math|''f''}} along {{math|'''v'''}}. That is,

<math display="block">\big(\nabla f(x)\big)\cdot \mathbf{v} = D_{\mathbf v}f(x)</math>

where the right-hand side is the [[directional derivative]] and there are many ways to represent it. Formally, the derivative is ''dual'' to the gradient; see [[#Derivative|relationship with derivative]].

When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only (see [[Spatial gradient]]).

The magnitude and direction of the gradient vector are [[Invariant (mathematics)|independent]] of the particular [[Coordinate system|coordinate representation]].<ref>{{harvtxt|Kreyszig|1972|pp=308–309}}</ref><ref>{{harvtxt|Stoker|1969|p=292}}</ref>

===Cartesian coordinates===
In the three-dimensional [[Cartesian coordinate system]] with a [[Euclidean metric]], the gradient, if it exists, is given by

<math display="block">\nabla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k},</math>

where {{math|'''i'''}}, {{math|'''j'''}}, {{math|'''k'''}} are the [[standard basis|standard]] unit vectors in the directions of the {{math|''x''}}, {{math|''y''}} and {{math|''z''}} coordinates, respectively. For example, the gradient of the function
<math display="block">f(x,y,z)= 2x+3y^2-\sin(z)</math>
is
<math display="block">\nabla f(x, y, z) = 2\mathbf{i}+ 6y\mathbf{j} -\cos(z)\mathbf{k}.</math>
or 
<math display="block">\nabla f(x, y, z) = 
\begin{bmatrix}
 2 \\
 6y \\
 -\cos z
\end{bmatrix}.
</math>

In some applications it is customary to represent the gradient as a [[row vector]] or [[column vector]] of its components in a rectangular coordinate system; this article follows the convention of the gradient being a column vector, while the derivative is a row vector.

===Cylindrical and spherical coordinates===
{{main|Del in cylindrical and spherical coordinates}}

In [[cylindrical coordinate system#Definition|cylindrical coordinates]], the gradient is given by:<ref name="Schey-1992" />

<math display="block">\nabla f(\rho, \varphi, z) = \frac{\partial f}{\partial \rho}\mathbf{e}_\rho + \frac{1}{\rho}\frac{\partial f}{\partial \varphi}\mathbf{e}_\varphi + \frac{\partial f}{\partial z}\mathbf{e}_z,</math>

where {{math|''ρ''}} is the axial distance, {{math|''φ''}} is the azimuthal or azimuth angle, {{math|''z''}} is the axial coordinate, and {{math|'''e'''<sub>''ρ''</sub>}}, {{math|'''e'''<sub>''φ''</sub>}} and {{math|'''e'''<sub>''z''</sub>}} are unit vectors pointing along the coordinate directions.

In [[spherical coordinate system#Definition|spherical coordinates]] with a Euclidean metric, the gradient is given by:<ref name="Schey-1992">{{harvnb|Schey|1992|pp=139–142}}.</ref>

<math display="block">\nabla f(r, \theta, \varphi) = \frac{\partial f}{\partial r}\mathbf{e}_r + \frac{1}{r}\frac{\partial f}{\partial \theta}\mathbf{e}_\theta + \frac{1}{r \sin\theta}\frac{\partial f}{\partial \varphi}\mathbf{e}_\varphi,</math>

where {{math|''r''}} is the radial distance, {{math|''φ''}} is the azimuthal angle and {{math|''θ''}} is the polar angle, and {{math|'''e'''<sub>''r''</sub>}}, {{math|'''e'''<sub>''θ''</sub>}} and {{math|'''e'''<sub>''φ''</sub>}} are again local unit vectors pointing in the coordinate directions (that is, the normalized [[Curvilinear coordinates#Covariant and contravariant bases|covariant basis]]).

For the gradient in other [[orthogonal coordinate system]]s, see [[Orthogonal coordinates#Differential operators in three dimensions|Orthogonal coordinates (Differential operators in three dimensions)]].

===General coordinates===
We consider [[Curvilinear coordinates|general coordinates]], which we write as {{math|''x''<sup>1</sup>, …, ''x''<sup>''i''</sup>, …, ''x''<sup>''n''</sup>}}, where {{mvar|n}} is the number of dimensions of the domain. Here, the upper index refers to the position in the list of the coordinate or component, so {{math|''x''<sup>2</sup>}} refers to the second component—not the quantity {{math|''x''}} squared. The index variable {{math|''i''}} refers to an arbitrary element {{math|''x''<sup>''i''</sup>}}. Using [[Einstein notation]], the gradient can then be written as:

<math display="block">\nabla f = \frac{\partial f}{\partial x^{i}}g^{ij} \mathbf{e}_j</math> (Note that its [[Dual space|dual]] is <math display="inline">\mathrm{d}f = \frac{\partial f}{\partial x^{i}}\mathbf{e}^i</math>),

where <math>\mathbf{e}^i = \mathrm{d}x^i</math> and <math>\mathbf{e}_i = \partial \mathbf{x}/\partial x^i</math> refer to the unnormalized local [[Curvilinear coordinates#Covariant and contravariant bases|covariant and contravariant bases]] respectively, <math>g^{ij}</math> is the [[Metric tensor#Inverse metric|inverse metric tensor]], and the Einstein summation convention implies summation over ''i''  and ''j''.

If the coordinates are orthogonal we can easily express the gradient (and the [[Differential form|differential]]) in terms of the normalized bases, which we refer to as  <math>\hat{\mathbf{e}}_i</math> and  <math>\hat{\mathbf{e}}^i</math>, using the scale factors (also known as [[Lamé coefficients]])  <math>h_i= \lVert \mathbf{e}_i \rVert = \sqrt{g_{i i}} = 1\, / \lVert \mathbf{e}^i \rVert</math> :

<math display="block">\nabla f = \frac{\partial f}{\partial x^{i}}g^{ij} \hat{\mathbf{e}}_{j}\sqrt{g_{jj}} = \sum_{i=1}^n \, \frac{\partial f}{\partial x^{i}} \frac{1}{h_i} \mathbf{\hat{e}}_i</math> (and <math display="inline">\mathrm{d}f = \sum_{i=1}^n \, \frac{\partial f}{\partial x^{i}} \frac{1}{h_i} \mathbf{\hat{e}}^i</math>),

where we cannot use Einstein notation, since it is impossible to avoid the repetition of more than two indices. Despite the use of upper and lower indices, <math>\mathbf{\hat{e}}_i</math>, <math>\mathbf{\hat{e}}^i</math>, and <math>h_i</math> are neither contravariant nor covariant.

The latter expression evaluates to the expressions given above for cylindrical and spherical coordinates.