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The gradient of a function is called a gradient field. A (continuous) gradient field is always a [[conservative vector field]]: its [[line integral]] along any path depends only on the endpoints of the path, and can be evaluated by the [[gradient theorem]] (the fundamental theorem of calculus for line integrals). Conversely, a (continuous) conservative vector field is always the gradient of a function.
==Riemannian manifolds==
For any smooth function f on a [[Riemannian manifold]] (''M'',''g''), the gradient of ''f'' is the [[vector field]] <math>\nabla f</math> such that for any vector field <math>X</math>,
:<math>g(\nabla f, X) = \partial_X f, \qquad \text{i.e.,}\quad g_x((\nabla f)_x, X_x ) = (\partial_X f) (x)</math>
where <math>g_x( \cdot, \cdot )</math> denotes the [[inner product]] of tangent vectors at ''x'' defined by the metric ''g'' and
<math>\partial_X f</math> (sometimes denoted ''X''(''f'')) is the function that takes any point ''x''∈''M'' to the [[directional derivative]] of ''f'' in the direction ''X'', evaluated at ''x''. In other words, in a [[coordinate chart]] <math>\varphi</math> from an open subset of ''M'' to an open subset of '''R'''<sup>''n''</sup>, <math>(\partial_X f)(x)</math> is given by:
:<math>\sum_{j=1}^n X^{j} (\varphi(x)) \frac{\partial}{\partial x_{j}}(f \circ \varphi^{-1}) \Big|_{\varphi(x)},</math>
where ''X''<sup>''j''</sup> denotes the ''j''th component of ''X'' in this coordinate chart.
So, the local form of the gradient takes the form:
:<math> \nabla f= g^{ik}\frac{\partial f}{\partial x^{k}}\frac{\partial}{\partial x^{i}}.</math>
Generalizing the case ''M''='''R'''<sup>''n''</sup>, the gradient of a function is related to its [[exterior derivative]], since <math>(\partial_X f) (x) = df_x(X_x)</math>. More precisely, the gradient <math>\nabla f</math> is the vector field associated to the differential 1-form d''f'' using the [[musical isomorphism]] <math>\sharp=\sharp^g\colon T^*M\to TM</math> (called "sharp") defined by the metric ''g''. The relation between the exterior derivative and the gradient of a function on '''R'''<sup>''n''</sup> is a special case of this in which the metric is the flat metric given by the dot product.
==Non-cartesian coordinate systems==
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