Editing Partial derivative (section)

==Definition==
Like ordinary derivatives, the partial derivative is defined as a [[limit of a function|limit]]. Let {{mvar|U}} be an [[open set|open subset]] of <math>\R^n</math> and <math>f:U\to\R</math> a function. The partial derivative of {{mvar|f}} at the point <math>\mathbf{a}=(a_1, \ldots, a_n) \in U</math> with respect to the {{mvar|i}}-th variable {{math|''x''<sub>''i''</sub>}} is defined as

<math display="block">\begin{align}
\frac{\partial }{\partial x_i }f(\mathbf{a}) & = \lim_{h \to 0} \frac{f(a_1, \ldots , a_{i-1}, a_i+h, a_{i+1}\, \ldots ,a_{n})\ -
f(a_1, \ldots, a_i, \dots ,a_n)}{h} \\ 
& = \lim_{h \to 0} \frac{f(\mathbf{a}+h\mathbf{e_i}) - f(\mathbf{a})}{h}\,.
\end{align}</math>

Where <math>\mathbf{e_i}</math> is the [[unit vector]] of {{mvar|i}}-th variable {{math|''x''<sub>''i''</sub>}}. Even if all partial derivatives <math>\partial f / \partial x_i(a)</math> exist at a given point {{mvar|a}}, the function need not be [[continuous function|continuous]] there. However, if all partial derivatives exist in a [[neighborhood (topology)|neighborhood]] of {{mvar|a}} and are continuous there, then {{mvar|f}} is [[total derivative|totally differentiable]] in that neighborhood and the total derivative is continuous. In this case, it is said that {{mvar|f}} is a {{math|''C''<sup>1</sup>}} function. This can be used to generalize for vector valued functions, {{nowrap|<math>f:U \to \R^m</math>,}} by carefully using a componentwise argument.

The partial derivative <math display="inline">\frac{\partial f}{\partial x}</math> can be seen as another function defined on {{mvar|U}} and can again be partially differentiated. If the direction of derivative is {{em|not}} repeated, it is called a '''''mixed partial derivative'''''. If all mixed second order partial derivatives are continuous at a point (or on a set), {{mvar|f}} is termed a {{math|''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#Schwarz's theorem|Clairaut's theorem]]:

<math display="block">\frac{\partial^2f}{\partial x_i \partial x_j} = \frac{\partial^2f} {\partial x_j \partial x_i}.</math>