44
4 Three-Dimensional Harmonic Function
Let us consider the integral
Q =
(V )
ϕdV ,
(4.27)
where (V )is the area inside the sphere with a radius of R, and ϕ is the function
harmonic in (V ). We will calculate the integral (4.27) by integrating inside a sphere
of some radius r, (r < R) and then along the normal line to such sphere. Taking
into account formula (4.26), we obtain
Q =
4
3
πR
3 ϕ(ξ, η, ζ ),
or
ϕ(ξ, η, ζ ) =
1
V
(V )
ϕdV ,
(4.28)
where V is the volume of such sphere.
In this manner, the value of the harmonic function in the center of the sphere
(ξ, η, ζ ) equals the average value of this function upon the sphere volume.
References
1. B. Budak, S. Fomin, Kratnye integraly i ryady [Multiple integrals and series] (Nauka Publ.,
Moscow, 1965)
2. A. Tikhonov, A. Samarskii, Uravneniya matematicheskoi fiziki: uchebnoe posobie. 6-e izd., ispr.
i dop. [Mathematical physics equations: tutorial. 6th ed., correct and additional] (Izd-vo MGU
Publ., Moscow, 1999)
4 Three-Dimensional Harmonic Function
Let us consider the integral
Q =
(V )
ϕdV ,
(4.27)
where (V )is the area inside the sphere with a radius of R, and ϕ is the function
harmonic in (V ). We will calculate the integral (4.27) by integrating inside a sphere
of some radius r, (r < R) and then along the normal line to such sphere. Taking
into account formula (4.26), we obtain
Q =
4
3
πR
3 ϕ(ξ, η, ζ ),
or
ϕ(ξ, η, ζ ) =
1
V
(V )
ϕdV ,
(4.28)
where V is the volume of such sphere.
In this manner, the value of the harmonic function in the center of the sphere
(ξ, η, ζ ) equals the average value of this function upon the sphere volume.
References
1. B. Budak, S. Fomin, Kratnye integraly i ryady [Multiple integrals and series] (Nauka Publ.,
Moscow, 1965)
2. A. Tikhonov, A. Samarskii, Uravneniya matematicheskoi fiziki: uchebnoe posobie. 6-e izd., ispr.
i dop. [Mathematical physics equations: tutorial. 6th ed., correct and additional] (Izd-vo MGU
Publ., Moscow, 1999)