Editing Gradient (section)

==Generalizations==

=== Jacobian ===
{{Main|Jacobian matrix and determinant}}

The [[Jacobian matrix]] is the generalization of the gradient for vector-valued functions of several variables and [[differentiable map]]s between [[Euclidean space]]s or, more generally, [[manifold]]s.<ref>{{harvtxt|Beauregard|Fraleigh|1973|pp=87,248}}</ref><ref>{{harvtxt|Kreyszig|1972|pp=333,353,496}}</ref>  A further generalization for a function between [[Banach space]]s is the [[Fréchet derivative]].

Suppose {{math|'''f''' : '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup>}} is a function such that each of its first-order partial derivatives exist on {{math|ℝ<sup>''n''</sup>}}.  Then the Jacobian matrix of {{math|'''f'''}} is defined to be an {{math|''m''×''n''}} matrix, denoted by <math>\mathbf{J}_\mathbb{f}(\mathbb{x})</math> or simply <math>\mathbf{J}</math>. The {{math|(''i'',''j'')}}th entry is <math display="inline">\mathbf J_{ij} = {\partial f_i} / {\partial x_j}</math>. Explicitly
<math display="block">\mathbf J = \begin{bmatrix}
    \dfrac{\partial \mathbf{f}}{\partial x_1} & \cdots & \dfrac{\partial \mathbf{f}}{\partial x_n} \end{bmatrix}
= \begin{bmatrix}
    \nabla^\mathsf{T} f_1 \\  
    \vdots \\
    \nabla^\mathsf{T} f_m   
    \end{bmatrix}
= \begin{bmatrix}
    \dfrac{\partial f_1}{\partial x_1} & \cdots & \dfrac{\partial f_1}{\partial x_n}\\
    \vdots & \ddots & \vdots\\
    \dfrac{\partial f_m}{\partial x_1} & \cdots & \dfrac{\partial f_m}{\partial x_n} \end{bmatrix}.</math>

===Gradient of a vector field===
{{see also|Covariant derivative}}
Since the [[total derivative]] of a vector field is a [[linear mapping]] from vectors to vectors, it is a [[tensor]] quantity.

In rectangular coordinates, the gradient of a vector field {{math|1='''f''' = (&thinsp;''f''{{i sup|1}}, ''f''{{i sup|2}}, ''f''{{i sup|3}})}} is defined by:

<math display="block">\nabla \mathbf{f}=g^{jk}\frac{\partial f^i}{\partial x^j} \mathbf{e}_i \otimes \mathbf{e}_k,</math>

(where the [[Einstein summation notation]] is used and the [[tensor product]] of the vectors {{math|'''e'''<sub>''i''</sub>}} and {{math|'''e'''<sub>''k''</sub>}} is a [[dyadic tensor]] of type (2,0)). Overall, this expression equals the transpose of the Jacobian matrix:

<math display="block">\frac{\partial f^i}{\partial x^j} = \frac{\partial (f^1,f^2,f^3)}{\partial (x^1,x^2,x^3)}.</math>

In curvilinear coordinates, or more generally on a curved [[Riemannian manifold|manifold]], the gradient involves [[Christoffel symbols]]:

<math display="block">\nabla \mathbf{f}=g^{jk}\left(\frac{\partial f^i}{\partial x^j}+{\Gamma^i}_{jl}f^l\right) \mathbf{e}_i \otimes \mathbf{e}_k,</math>

where {{math|''g''{{i sup|''jk''}}}} are the components of the inverse [[metric tensor]] and the {{math|'''e'''<sub>''i''</sub>}} are the coordinate basis vectors.

Expressed more invariantly, the gradient of a vector field {{math|'''f'''}} can be defined by the [[Levi-Civita connection]] and metric tensor:<ref>{{harvnb|Dubrovin|Fomenko|Novikov|1991|pages=348–349}}.</ref>

<math display="block">\nabla^a f^b = g^{ac} \nabla_c f^b ,</math>

where {{math|∇<sub>''c''</sub>}} is the connection.

===Riemannian manifolds===
For any [[smooth function]] {{mvar|f}} on a Riemannian manifold {{math|(''M'', ''g'')}}, the gradient of {{math|''f''}} is the vector field {{math|∇''f''}} such that for any vector field {{math|''X''}},
<math display="block">g(\nabla f, X) = \partial_X f,</math>
that is,
<math display="block">g_x\big((\nabla f)_x, X_x \big) = (\partial_X f) (x),</math>
where {{math|''g''<sub>''x''</sub>( , )}} denotes the [[inner product]] of tangent vectors at {{math|''x''}} defined by the metric {{math|''g''}} and {{math|∂<sub>''X''</sub>&thinsp;''f''}} is the function that takes any point {{math|''x'' ∈ ''M''}} to the directional derivative of {{math|''f''}} in the direction {{math|''X''}}, evaluated at {{math|''x''}}. In other words, in a [[coordinate chart]] {{math|''φ''}} from an open subset of {{math|''M''}} to an open subset of {{math|'''R'''<sup>''n''</sup>}}, {{math|(∂<sub>''X''</sub>&thinsp;''f''&thinsp;)(''x'')}} is given by:
<math display="block">\sum_{j=1}^n X^{j} \big(\varphi(x)\big) \frac{\partial}{\partial x_{j}}(f \circ \varphi^{-1}) \Bigg|_{\varphi(x)},</math>
where {{math|''X''{{isup|''j''}}}} denotes the {{math|''j''}}th component of {{math|''X''}} in this coordinate chart.

So, the local form of the gradient takes the form:

<math display="block">\nabla f = g^{ik} \frac{\partial f}{\partial x^k} {\textbf e}_i .</math>

Generalizing the case {{math|1=''M'' = '''R'''<sup>''n''</sup>}}, the gradient of a function is related to its exterior derivative, since
<math display="block">(\partial_X f) (x) = (df)_x(X_x) .</math>
More precisely, the gradient {{math|∇''f''}} is the vector field associated to the differential 1-form {{math|''df''}} using the [[musical isomorphism]]
<math display="block">\sharp=\sharp^g\colon T^*M\to TM</math>
(called "sharp") defined by the metric {{math|''g''}}. The relation between the exterior derivative and the gradient of a function on {{math|'''R'''<sup>''n''</sup>}} is a special case of this in which the metric is the flat metric given by the dot product.