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Gradient
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===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|β''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|β<sub>''X''</sub> ''f''}} is the function that takes any point {{math|''x'' β ''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|''Ο''}} from an open subset of {{math|''M''}} to an open subset of {{math|'''R'''<sup>''n''</sup>}}, {{math|(β<sub>''X''</sub> ''f'' )(''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|β''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.
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