Editing Dirac delta function (section)

===Higher dimensions===
More generally, on an [[open set]] {{mvar|U}} in the {{mvar|n}}-dimensional [[Euclidean space]] <math>\mathbb{R}^n</math>, the Dirac delta distribution centered at a point {{math|''a'' ∈ ''U''}} is defined by{{sfn|Hörmander|1983|p=56}}
<math display="block">\delta_a[\varphi]=\varphi(a)</math>
for all <math>\varphi \in C_c^\infty(U)</math>, the space of all smooth functions with compact support on {{mvar|U}}.  If <math>\alpha = (\alpha_1, \ldots, \alpha_n)</math> is any [[multi-index]] with <math> |\alpha|=\alpha_1+\cdots+\alpha_n</math> and <math>\partial^\alpha</math> denotes the associated mixed [[partial derivative]] operator, then the {{mvar|α}}-th derivative {{mvar|∂<sup>α</sup>δ<sub>a</sub>}} of {{mvar|δ<sub>a</sub>}} is given by{{sfn|Hörmander|1983|p=56}}

<math display="block">\left\langle \partial^\alpha \delta_{a}, \, \varphi \right\rangle = (-1)^{| \alpha |} \left\langle \delta_{a}, \partial^{\alpha} \varphi \right\rangle = (-1)^{| \alpha |} \partial^\alpha \varphi (x) \Big|_{x = a} \quad \text{ for all } \varphi \in C_c^\infty(U).</math>

That is, the {{mvar|α}}-th derivative of {{mvar|δ<sub>a</sub>}} is the distribution whose value on any test function {{mvar|φ}} is the {{mvar|α}}-th derivative of {{mvar|φ}} at {{mvar|a}} (with the appropriate positive or negative sign).

The first partial derivatives of the delta function are thought of as [[double layer potential|double layers]] along the coordinate planes.  More generally, the [[normal derivative]] of a simple layer supported on a surface is a double layer supported on that surface and represents a laminar magnetic monopole.  Higher derivatives of the delta function are known in physics as [[multipole]]s.

Higher derivatives enter into mathematics naturally as the building blocks for the complete structure of distributions with point support.  If {{mvar|S}} is any distribution on {{mvar|U}} supported on the set {{math|{{brace|''a''}}}} consisting of a single point, then there is an integer {{mvar|m}} and coefficients {{mvar|c<sub>α</sub>}} such that{{sfn|Hörmander|1983|p=56}}{{sfn|Rudin|1991|loc=Theorem 6.25}}
<math display="block">S = \sum_{|\alpha|\le m} c_\alpha \partial^\alpha\delta_a.</math>