Editing Distribution (mathematics) (section)

====Differentiation of distributions====

Let <math>A : \mathcal{D}(U) \to \mathcal{D}(U)</math> be the partial derivative operator <math>\tfrac{\partial}{\partial x_k}.</math> To extend <math>A</math> we compute its transpose:
<math display=block>\begin{align}
\langle {}^{t}A(D_\psi), \phi \rangle
&= \int_U \psi (A\phi) \,dx && \text{(See above.)} \\
&= \int_U \psi \frac{\partial\phi}{\partial x_k} \, dx \\[4pt]
&= -\int_U \phi \frac{\partial\psi}{\partial x_k}\, dx && \text{(integration by parts)} \\[4pt]
&= -\left\langle \frac{\partial\psi}{\partial x_k}, \phi \right\rangle \\[4pt]
&= -\langle A \psi, \phi \rangle = \langle - A \psi, \phi \rangle
\end{align}</math>

Therefore <math>{}^{t}A = -A.</math> Thus, the partial derivative of <math>T</math> with respect to the coordinate <math>x_k</math> is defined by the formula
<math display=block>\left\langle \frac{\partial T}{\partial x_k}, \phi \right\rangle = - \left\langle T, \frac{\partial \phi}{\partial x_k} \right\rangle \qquad \text{ for all } \phi \in \mathcal{D}(U).</math>

With this definition, every distribution is infinitely differentiable, and the derivative in the direction <math>x_k</math> is a [[linear operator]] on <math>\mathcal{D}'(U).</math>

More generally, if <math>\alpha</math> is an arbitrary [[multi-index]], then the partial derivative <math>\partial^\alpha T</math> of the distribution <math>T \in \mathcal{D}'(U)</math> is defined by
<math display=block>\langle \partial^\alpha T, \phi \rangle = (-1)^{|\alpha|} \langle T, \partial^\alpha \phi \rangle \qquad \text{ for all } \phi \in \mathcal{D}(U).</math>

Differentiation of distributions is a continuous operator on <math>\mathcal{D}'(U);</math> this is an important and desirable property that is not shared by most other notions of differentiation.

If <math>T</math> is a distribution in <math>\R</math> then
<math display=block>\lim_{x \to 0} \frac{T - \tau_x T}{x} = T'\in \mathcal{D}'(\R),</math>
where <math>T'</math> is the derivative of <math>T</math> and <math>\tau_x</math> is a translation by <math>x;</math> thus the derivative of <math>T</math> may be viewed as a limit of quotients.{{sfn|Rudin|1991|p=180}}