TWO
DEFINITIONS
OF
a(?)
103
Figure
5.4
A
Single
Point
Mass
at
the
Origin
Integrating the mass density over a volume
V
gives the total mass enclosed:
d7
p,(F)
=
total mass inside
V.
(5.5)
Thus, if there is a single point of mass
m,
located at the origin, as shown in Figure
5.4,
any volume integral which includes the origin must give the total mass as
m,.
Integrals
which exclude the origin must give zero. In mathematical terms:
(5.6)
origin included in
V
origin excluded from
V
.
ld7pm(F)
=
mo
0
{
Using Dirac &functions, this mass density function becomes
(5.7)
Equation
5.6
can easily be checked by expanding the integral as
Application of Equation
5.3
on the three independent integrals gives
m,
only when
V
includes the origin. If, instead of the origin, the point mass is located at the point
(x,,
yo,
z,),
shifted arguments are used in each
of
the &functions:
p,(F)
=
m,Wx
-
x&%y
-
Y,)~(z
-
GI.
(5.9)
Dirac 6-functions can
also
be
used, in a
similar
way, to represent point charges
in
electromagnetism.
5.2
TWO
DEFINITIONS
OF
S(t)
There are two common ways to define the Dirac &function. The more rigorous
approach, from the theory of generalized functions, defines it by its behavior inside
integral operations. In fact, the &function is actually never supposed to exist outside
an integral. In general, scientists and engineers are a bit more lax and use a second
definition. They often define the &function as the limit of
an
infinite sequence
of
