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Or, more generally, for ''n''-dimensional Euclidean space '''R'''<sup>''n''</sup> with coordinates (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>,...,x<sub>''n''</sub>) and unit vectors (<math>\mathbf{\hat{e}_1}, \mathbf{\hat{e}_2}, \mathbf{\hat{e}_3}, \dots , \mathbf{\hat{e}_n}</math>):
:<math>\nabla = \sum_{j=1}^n \bigg[{\frac{\partial}{\partial x_j}}\bigg] \mathbf{\hat{e}_j} = \bigg[{\frac{\partial}{\partial x_1}}\bigg] \mathbf{\hat{e}_1} + \bigg[{\frac{\partial}{\partial x_2}}\bigg] \mathbf{\hat{e}_2} + \bigg[{\frac{\partial}{\partial x_3}}\bigg] \mathbf{\hat{e}_3} + \dots + \bigg[{\frac{\partial}{\partial x_n}}\bigg] \mathbf{\hat{e}_n}</math>
===Formal definition===
Like ordinary derivatives, the partial derivative is defined as a [[limit of a function|limit]]. Let ''U'' be an [[open set|open subset]] of '''R'''<sup>''n''</sup> and ''f'' : ''U'' → '''R''' a function. The partial derivative of ''f'' at the point '''''a''''' = (''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>) ∈ ''U'' with respect to the ''i''-th variable ''a''<sub>''i''</sub> is defined as
:<math>\frac{ \partial }{\partial a_i }f(\mathbf{a}) =
\lim_{h \rightarrow 0}{
f(a_1, \dots , a_{i-1}, a_i+h, a_{i+1}, \dots ,a_n) -
f(a_1, \dots, a_i, \dots ,a_n) \over h }
</math>
Even if all partial derivatives ∂''f''/∂''a''<sub>''i''</sub>(''a'') exist at a given point ''a'', the function need not be [[continuous function|continuous]] there. However, if all partial derivatives exist in a [[neighborhood (topology)|neighborhood]] of ''a'' and are continuous there, then ''f'' is [[total derivative|totally differentiable]] in that neighborhood and the total derivative is continuous. In this case, it is said that ''f'' is a C<sup>1</sup> function. This can be used to generalize for vector valued functions (''f'' : ''U'' → ''R'''<sup>''m''</sup>) by carefully using a componentwise argument.
The partial derivative <math>\frac{\partial f}{\partial x}</math> can be seen as another function defined on ''U'' and can again be partially differentiated. If all mixed second order partial derivatives are continuous at a point (or on a set), ''f'' is termed a C<sup>2</sup> function at that point (or on that set); in this case, the partial derivatives can be exchanged by [[Symmetry of second derivatives#Clairaut.27s theorem|Clairaut's theorem]]:
:<math>\frac{\partial^2f}{\partial x_i\, \partial x_j} = \frac{\partial^2f} {\partial x_j\, \partial x_i}.</math>
==Examples==
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