Editing Multivariable calculus (section)

== Differentiation ==
{{main article|Partial derivative|Directional derivative}}

=== Directional derivative ===
The derivative of a single-variable function is defined as

{{NumBlk|:|<math>\frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}</math>|{{EquationRef|9}}}}

Using the extension of limits discussed above, one can then extend the definition of the derivative to a scalar-valued function <math>f: \mathbb{R}^n \to \mathbb{R}</math> along some path <math>s(t): \mathbb{R} \to \mathbb{R}^n</math>:

{{NumBlk|:|<math>\left . \frac{df}{dx} \right |_{s(t),t=t_0} = \lim_{h \to 0} \frac{f(s(t_0+h))-f(s(t_0))}{|s(t_0+h)-s(t_0)|}</math>|{{EquationRef|10}}}}

Unlike limits, for which the value depends on the exact form of the path <math>s(t)</math>, it can be shown that the derivative along the path depends only on the tangent vector of the path at <math>s(t_0)</math>, i.e. <math>s'(t_0)</math>, provided that <math>f</math> is [[Lipschitz continuous]] at <math>s(t_0)</math>, and that the limit exits for at least one such path.
<!-- I am not sure in the slightest I got these conditions right. I will look them up at some point, but in the meantime, if you have a better way to put it, please do. This comment will be removed after the reconstruction is finished.-->

{{collapse top|Proof|expand=true}}
For <math>s(t)</math> continuous up to the first derivative (this statement is well defined as <math>s</math> is a function of one variable), we can write the [[Taylor expansion]] of <math>s</math> around <math>t_0</math> using [[Taylor's theorem]] to construct the remainder:

{{NumBlk|:|<math>s(t) = s(t_0) + s'(\tau) (t-t_0) </math>|{{EquationRef|11}}}}

where <math>\tau \in [t_0,t]</math>.

Substituting this into {{EquationNote|10}},

{{NumBlk|:|<math>\left . \frac{df}{dx} \right |_{s(t),t=t_0} = \lim_{h \to 0} \frac{f(s(t_0)+s'(\tau)h)-f(s(t_0))}{|s'(\tau)h|}</math>|{{EquationRef|12}}}}

where <math>\tau(h) \in [t_0,t_0+h]</math>.

Lipschitz continuity gives us <math>|f(x)-f(y)| \leq K|x-y|</math> for some finite <math>K</math>, <math>\forall x,y\in \mathbb{R}^n</math>. It follows that <math>|f(x+O(h))-f(x)| \sim O(h)</math>.

Note also that given the continuity of <math>s'(t)</math>, <math>s'(\tau) = s'(t_0)+O(h)</math> as <math> h \to 0</math>.

Substituting these two conditions into {{EquationNote|12}},

{{NumBlk|:|<math>\left . \frac{df}{dx} \right |_{s(t),t=t_0} = \lim_{h \to 0} \frac{f(s(t_0)+s'(t_0)h)-f(s(t_0))+O(h^2)}{|s'(t_0)h|+O(h^2)}</math>|{{EquationRef|13}}}}

whose limit depends only on <math>s'(t_0)</math> as the dominant term.

{{collapse bottom}}

It is therefore possible to generate the definition of the directional derivative as follows: The directional derivative of a scalar-valued function <math>f:\mathbb{R}^n \to \mathbb{R}</math> along the unit vector <math>\hat{\bold{u}}</math> at some point <math>x_0 \in \mathbb{R}^n</math> is

{{NumBlk|:|<math>\nabla_{\hat{\bold{u}}} f(x_0) = \lim_{t \to 0} \frac{f(x_0+\hat{\bold{u}} t) - f(x_0)}{t}</math>|{{EquationRef|14}}}}
<!-- Do limits need normed spaces too, or is it just derivatives? -->

or, when expressed in terms of ordinary differentiation,

{{NumBlk|:|<math>\nabla_{\hat{\bold{u}}} f(x_0) = \left . \frac{df(x_0+\hat{\bold{u}}t)}{dt} \right |_{t=0}</math>|{{EquationRef|15}}}}

which is a well defined expression because <math>f(x_0+\hat{\bold{u}}t)</math> is a scalar function with one variable in <math>t</math>.

It is not possible to define a unique scalar derivative without a direction; it is clear for example that <math>\nabla_{\hat{\bold{u}}}f(x_0) = - \nabla_{-\hat{\bold{u}}}f(x_0)</math>. It is also possible for directional derivatives to exist for some directions but not for others.

=== Partial derivative ===
{{Main article|Partial derivative}}
The partial derivative generalizes the notion of the derivative to higher dimensions.  A partial derivative of a multivariable function is a [[derivative]] with respect to one variable with all other variables held constant.<ref name="CourantJohn1999"/>{{rp|26ff}}

A partial derivative may be thought of as the directional derivative of the function along a coordinate axis.

Partial derivatives may be combined in interesting ways to create   more complicated expressions of the derivative.  In [[vector calculus]], the [[del]] operator (<math>\nabla</math>) is used to define the concepts of [[gradient]], [[divergence]], and [[Curl (mathematics)|curl]] in terms of partial derivatives.  A matrix of partial derivatives, the '''[[Jacobian matrix and determinant|Jacobian]]''' matrix, may be used to represent the derivative of a function between two spaces of arbitrary dimension.  The derivative can thus be understood as a [[linear transformation]] which directly varies from point to point in the domain of the function.

[[Differential equations]] containing partial derivatives are called [[partial differential equations]] or PDEs.  These equations are generally more difficult to solve than [[ordinary differential equations]], which contain derivatives with respect to only one variable.<ref name="CourantJohn1999"/>{{rp|654ff}}