distribution by In the theory of electromagnetism, the first derivative of the delta function represents a point magnetic dipole situated at the origin. Accordingly, it is referred to as a dipole or the doublet function. The derivative of the delta function satisfies a number of basic properties, including: which can be shown by applying a test function and integrating by parts. The latter of these properties can also be demonstrated by applying distributional derivative definition, Liebnitz's theorem and linearity of inner product: Furthermore, the convolution of with a compactly-supported, smooth function is which follows from the properties of the distributional derivative of a convolution. Higher dimensions More generally, on an open set in the -dimensional Euclidean space , the Dirac delta distribution centered at a point is defined by for all , the space of all smooth functions with compact support on . If is any multi-index with and denotes the associated mixed partial derivative operator, then the -th derivative of is given by That is, the -th derivative of is the distribution whose value on any test function is the -th derivative of at (with the appropriate positive or negative sign). The first partial derivatives of the delta function are thought of as 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 multipoles. Higher derivatives enter into mathematics naturally as the building blocks for the complete structure of distributions with point support. If is any distribution on supported on the set consisting of a single point, then there is an integer and coefficients such that Representations of the delta function The delta function can be viewed as the limit of a sequence of functions where ηε(x) is sometimes called a nascent delta function. This limit is meant in a weak sense: either that for all continuous functions f having compact support, or that this limit holds for all smooth functions f with compact support. The difference between these two slightly different modes of weak convergence is often subtle: the former is convergence in the vague topology of measures, and the latter is convergence in the sense of distributions. Approximations to the identity Typically a nascent delta function ηε can be constructed in the following