Editing Partial derivative (section)

==Gradient==
{{Main|Gradient}}
An important example of a function of several variables is the case of a [[scalar-valued function]] <math>f(x_1, \ldots, x_n)</math> on a domain in Euclidean space <math>\R^n</math> (e.g., on <math>\R^2</math> or {{nowrap|<math>\R^3</math>).}} In this case {{mvar|f}} has a partial derivative <math>\partial f/\partial x_j</math> with respect to each variable {{math|''x''<sub>''j''</sub>}}. At the point {{mvar|a}}, these partial derivatives define the vector

<math display="block">\nabla f(a) = \left(\frac{\partial f}{\partial x_1}(a), \ldots, \frac{\partial f}{\partial x_n}(a)\right).</math>

This vector is called the ''[[gradient]]'' of {{mvar|f}} at {{mvar|a}}. If {{mvar|f}} is differentiable at every point in some domain, then the gradient is a vector-valued function {{math|∇''f''}} which takes the point {{mvar|a}} to the vector {{math|∇''f''(''a'')}}. Consequently, the gradient produces a [[vector field]].

A common [[abuse of notation]] is to define the [[del operator]] ({{math|∇}}) as follows in three-dimensional [[Euclidean space]] <math>\R^3</math> with [[unit vectors]] {{nowrap|<math>\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}</math>:}}

<math display="block">\nabla = \left[{\frac{\partial}{\partial x}} \right] \hat{\mathbf{i}} + \left[{\frac{\partial}{\partial y}} \right] \hat{\mathbf{j}} + \left[{\frac{\partial}{\partial z}}\right] \hat{\mathbf{k}}</math>

Or, more generally, for {{mvar|n}}-dimensional Euclidean space <math>\R^n</math> with coordinates <math>x_1, \ldots, x_n</math> and unit vectors {{nowrap|<math>\hat{\mathbf{e}}_1, \ldots, \hat{\mathbf{e}}_n</math>:}}

<math display="block">\nabla = \sum_{j=1}^n \left[\frac{\partial}{\partial x_j} \right] \hat{\mathbf{e}}_j = \left[\frac{\partial}{\partial x_1} \right] \hat{\mathbf{e}}_1 + \left[\frac{\partial}{\partial x_2} \right] \hat{\mathbf{e}}_2 + \dots + \left[\frac{\partial}{\partial x_n} \right] \hat{\mathbf{e}}_n</math>