Classical Electrodynamics · Problem 1.2

Boundary-Value Problems in Electrostatics: I — John David Jackson

Physics — E&M · dirac-delta-function-orthogonal-coordinates

Problem

The Dirac delta function in three dimensions can be taken as the improper limit as

\alpha \to 0

of the Gaussian function

D(\alpha; x, y, z) = (2\pi)^{-3/2}\alpha^{-3} \exp\left[-\frac{1}{2\alpha^2}(x^2 + y^2 + z^2)\right]

Consider a general orthogonal coordinate system specified by the surfaces

u =

constant,

v =

constant,

w =

constant, with length elements

du/U

dv/V

dw/W

in the three perpendicular directions. Show that

\delta(\mathbf{x} - \mathbf{x}') = \delta(u - u')\, \delta(v - v')\, \delta(w - w') \cdot UVW

by considering the limit of the Gaussian above. Note that as

\alpha \to 0

only the infinitesimal length element need be used for the distance between the points in the exponent.