Editing Differential (mathematics) (section)

==== Differentials as linear maps on R<sup>n</sup> ====

If <math>f</math> is a function from <math>\mathbb{R}^n</math> to <math>\mathbb{R}</math>, then we say that <math>f</math> is ''differentiable''<ref>See, for instance, {{Harvnb|Apostol|1967}}.</ref> at <math>p\in\mathbb{R}^n</math> if there is a linear map <math>df_p</math> from <math>\mathbb{R}^n</math> to <math>\mathbb{R}</math> such that for any <math>\varepsilon>0</math>, there is a [[neighbourhood (mathematics)|neighbourhood]] <math>N</math> of <math>p</math> such that for <math>x\in N</math>,
<math display=block>\left|f(x) - f(p) - df_p(x-p)\right| < \varepsilon \left|x-p\right| .</math>

We can now use the same trick as in the one-dimensional case and think of the expression <math>f(x_1, x_2, \ldots, x_n)</math> as the composite of <math>f</math> with the standard coordinates <math>x_1, x_2, \ldots, x_n</math> on <math>\mathbb{R}^n</math> (so that <math>x_j(p)</math> is the <math>j</math>-th component of <math>p\in\mathbb{R}^n</math>). Then the differentials <math>\left(dx_1\right)_p, \left(dx_2\right)_p, \ldots, \left(dx_n\right)_p</math> at a point <math>p</math> form a [[basis (linear algebra)|basis]] for the [[vector space]] of linear maps from <math>\mathbb{R}^n</math> to <math>\mathbb{R}</math> and therefore, if <math>f</math> is differentiable at <math>p</math>, we can write ''<math>\operatorname{d}f_p</math>'' as a [[linear combination]] of these basis elements:
<math display=block>df_p = \sum_{j=1}^n D_j f(p) \,(dx_j)_p.</math>

The coefficients <math>D_j f(p)</math> are (by definition) the [[partial derivative]]s of <math>f</math> at <math>p</math> with respect to <math>x_1, x_2, \ldots, x_n</math>. Hence, if <math>f</math> is differentiable on all of <math>\mathbb{R}^n</math>, we can write, more concisely:
<math display=block>\operatorname{d}f = \frac{\partial f}{\partial x_1} \,dx_1 + \frac{\partial f}{\partial x_2} \,dx_2 + \cdots +\frac{\partial f}{\partial x_n} \,dx_n.</math>

In the one-dimensional case this becomes
<math display=block>df = \frac{df}{dx}dx</math>
as before.

This idea generalizes straightforwardly to functions from <math>\mathbb{R}^n</math> to <math>\mathbb{R}^m</math>. Furthermore, it has the decisive advantage over other definitions of the derivative that it is [[invariant (mathematics)|invariant]] under changes of coordinates. This means that the same idea can be used to define the [[pushforward (differential)|differential]] of [[smooth map]]s between [[smooth manifold]]s.

Aside: Note that the existence of all the [[partial derivative]]s of <math>f(x)</math> at <math>x</math> is a [[necessary condition]] for the existence of a differential at <math>x</math>. However it is not a [[sufficient condition]]. For counterexamples, see [[Gateaux derivative]].