In[]:=
f[x]==Integrate[f[y],delta[x,y],{x,-inf,inf},{y,-inf,inf},{z,-inf,inf}]
Out[]=
f[x]
inf
∫
-inf
4
2
inf
delta[x,y]f[y]y
In[]:=
f[r,theta,phi]==Integrate[r'^2Sin[theta']*f[r',theta',phi']*delta[{r,theta,phi},{r'theta',phi'}],{r',0,inf},{theta',0,Pi},{phi',0,2Pi}]
Out[]=
f[r,theta,phi]
inf
∫
0
π
∫
0
2π
∫
0
delta[{r,theta,phi},{
′
r
′
theta
,
′
phi
}]f[
′
r
,
′
theta
,
′
phi
]Sin[
′
theta
]
2
(
′
r
)

′
phi

′
theta

′
r
In[]:=
Integrate[r'^2Sin[theta']*f[r',theta',phi']*delta[r-r']delta[theta-theta']*delta[phi-phi']/(r'2Sin[theta']),{r',0,inf},{theta',0,P},{phi',0,2Pi}]
Out[]=
1
2
inf
∫
0
P
∫
0
2π
∫
0
delta[phi-
′
phi
]delta[r-
′
r
]delta[theta-
′
theta
]f[
′
r
,
′
theta
,
′
phi
]
′
r

′
phi

′
theta

′
r