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An expression of the form
 
An expression of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
u ( x)  = \int\limits _ { D }
 +
h ( | x - y | ) f ( y)  d v ( y) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741202.png" /> is a bounded domain in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741204.png" />, bounded by a closed Lyapunov surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741205.png" /> (a curve for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741206.png" />, cf. [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741207.png" /> is the fundamental solution of the Laplace operator:
+
where $  D $
 +
is a bounded domain in a Euclidean space $  \mathbf R  ^ {N} $,  
 +
$  N \geq  2 $,  
 +
bounded by a closed Lyapunov surface $  S $(
 +
a curve for $  N = 2 $,  
 +
cf. [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]), $  h ( | x - y | ) $
 +
is the fundamental solution of the Laplace operator:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741208.png" /></td> </tr></table>
+
$$
 +
h ( | x - y | )  = \
 +
\left \{
 +
\begin{array}{ll}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p0741209.png" /> is the area of the unit sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412011.png" /> is the distance between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412013.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412014.png" /> is the volume element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412015.png" />.
+
\frac{1}{( N - 2 ) \omega _ {N} | x - y |  ^ {N-} 2 }
 +
, & N \geq  3 ;  \\
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412016.png" />, then the potential is defined for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412018.png" />. In the complementary domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412019.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412020.png" /> then has derivatives of all orders and satisfies the [[Laplace equation|Laplace equation]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412021.png" />, that is, is a [[Harmonic function|harmonic function]]; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412022.png" /> this function is regular at infinity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412023.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412024.png" /> the potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412025.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412026.png" /> and satisfies the [[Poisson equation|Poisson equation]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412027.png" />.
+
\frac{1}{2 \pi }
 +
  \mathop{\rm ln} 
 +
\frac{1}{| x - y | }
 +
, & N = 2 ;  \\
 +
\end{array}
  
These properties can be generalized in various ways. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412028.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412033.png" /> has generalized second derivatives in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412034.png" />, and the Poisson equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412035.png" /> is satisfied almost-everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412036.png" />. Properties of potentials of an arbitrary [[Radon measure|Radon measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412037.png" /> concentrated on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412038.png" />-dimensional domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412039.png" /> have also been studied:
+
\right .$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412040.png" /></td> </tr></table>
+
where  $  \omega _ {N} = 2 \pi  ^ {N/2} / \Gamma ( N / 2 ) $
 +
is the area of the unit sphere in  $  \mathbf R  ^ {N} $,
 +
$  | x - y | $
 +
is the distance between the points  $  x $
 +
and  $  y $,
 +
and  $  d v ( y) $
 +
is the volume element in  $  D $.
  
Here again <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412042.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412044.png" /> almost-everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412046.png" /> is the derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412047.png" /> with respect to Lebesgue measure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412048.png" />. In definition (*) the fundamental solution of the Laplace operator may be replaced by an arbitrary Levi function (see [[#References|[2]]]) for a general second-order elliptic operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412049.png" /> with variable coefficients of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412050.png" />; then the properties listed above still hold with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412051.png" /> replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412052.png" /> (see [[#References|[2]]]–[[#References|[4]]]).
+
If  $  f \in C  ^ {(} 1) ( \overline{D}\; ) $,
 +
then the potential is defined for all  $  x \in \mathbf R  ^ {N} $
 +
and  $  u \in C  ^ {(} 1) ( \mathbf R  ^ {N} ) $.
 +
In the complementary domain  $  \overline{D}\; {}  ^ {c} $,
 +
the function  $  u $
 +
then has derivatives of all orders and satisfies the [[Laplace equation|Laplace equation]]:  $  \Delta u = 0 $,
 +
that is, is a [[Harmonic function|harmonic function]]; for  $  N \geq  3 $
 +
this function is regular at infinity,  $  u ( \infty ) = 0 $.  
 +
In  $  D $
 +
the potential  $  u $
 +
belongs to the class  $  C  ^ {(} 2) ( D) $
 +
and satisfies the [[Poisson equation|Poisson equation]]:  $  \Delta u = - f $.
 +
 
 +
These properties can be generalized in various ways. For example, if  $  f \in L _  \infty  ( D) $,
 +
then  $  u \in C ( \mathbf R  ^ {N} ) $,
 +
$  u \in C  ^  \infty  ( \overline{D}\; {}  ^ {c} ) $,
 +
$  \Delta u = 0 $
 +
in  $  \overline{D}\; {}  ^ {c} $,
 +
$  u $
 +
has generalized second derivatives in  $  D $,
 +
and the Poisson equation  $  \Delta u = - f $
 +
is satisfied almost-everywhere in  $  D $.
 +
Properties of potentials of an arbitrary [[Radon measure|Radon measure]]  $  \mu $
 +
concentrated on an  $  N $-
 +
dimensional domain  $  D $
 +
have also been studied:
 +
 
 +
$$
 +
u ( x)  =  \int\limits h ( | x - y | )  d \mu ( y) .
 +
$$
 +
 
 +
Here again  $  u \in C  ^  \infty  ( \overline{D}\; {}  ^ {c} ) $
 +
and  $  \Delta u = 0 $
 +
in  $  \overline{D}\; {}  ^ {c} $,
 +
$  \Delta u = - \mu  ^  \prime  $
 +
almost-everywhere in $  D $,  
 +
where $  \mu  ^  \prime  $
 +
is the derivative of $  \mu $
 +
with respect to Lebesgue measure in $  \mathbf R  ^ {n} $.  
 +
In definition (*) the fundamental solution of the Laplace operator may be replaced by an arbitrary Levi function (see [[#References|[2]]]) for a general second-order elliptic operator $  L $
 +
with variable coefficients of class $  C ^ {( 0 , \lambda ) } ( \overline{D}\; ) $;  
 +
then the properties listed above still hold with $  \Delta u $
 +
replaced by $  L u $(
 +
see [[#References|[2]]]–[[#References|[4]]]).
  
 
Potentials of mass distributions are applied in the solution of boundary value problems for elliptic partial differential equations (see [[#References|[2]]]–[[#References|[5]]]).
 
Potentials of mass distributions are applied in the solution of boundary value problems for elliptic partial differential equations (see [[#References|[2]]]–[[#References|[5]]]).
Line 21: Line 99:
 
For the solution of boundary value problems for parabolic partial differential equations the concept of a heat potential of the form
 
For the solution of boundary value problems for parabolic partial differential equations the concept of a heat potential of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412053.png" /></td> </tr></table>
+
$$
 +
v ( x , t )  = \
 +
\int\limits _ { 0 } ^ { t }  d \tau
 +
\int\limits _ { D } G ( x , t ; y, \tau ) f ( y , \tau )  d v ( y)
 +
$$
  
is used, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412054.png" /> is a fundamental solution of the heat equation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412055.png" />:
+
is used, where $  G ( x , t ;  y , \tau ) $
 +
is a fundamental solution of the heat equation in $  \mathbf R  ^ {N} $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412056.png" /></td> </tr></table>
+
$$
 +
G ( x , t ; y , \tau )  = \
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412057.png" /> is the density. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412058.png" /> and its generalizations to the case of an arbitrary second-order parabolic partial differential equation have properties similar to those given above for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074120/p07412059.png" /> (see [[#References|[3]]]–[[#References|[6]]]).
+
\frac{1}{( 2 \sqrt \pi )  ^ {N} ( t - \tau )  ^ {N/2} }
 +
 
 +
\mathop{\rm exp} ^ {- | x - y |  ^ {2} / 4 ( t - \tau ) } ,
 +
$$
 +
 
 +
and  $  f ( y , \tau ) $
 +
is the density. The function $  v ( x , t ) $
 +
and its generalizations to the case of an arbitrary second-order parabolic partial differential equation have properties similar to those given above for $  u $(
 +
see [[#References|[3]]]–[[#References|[6]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.M. Günter,  "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar  (1967)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,  A.A. Samarskii,  "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Friedman,  "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.V. Bitsadze,  "Boundary value problems for second-order elliptic equations" , North-Holland  (1968)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.M. Günter,  "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar  (1967)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,  A.A. Samarskii,  "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Friedman,  "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.V. Bitsadze,  "Boundary value problems for second-order elliptic equations" , North-Holland  (1968)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 14:54, 7 June 2020


An expression of the form

$$ \tag{* } u ( x) = \int\limits _ { D } h ( | x - y | ) f ( y) d v ( y) , $$

where $ D $ is a bounded domain in a Euclidean space $ \mathbf R ^ {N} $, $ N \geq 2 $, bounded by a closed Lyapunov surface $ S $( a curve for $ N = 2 $, cf. Lyapunov surfaces and curves), $ h ( | x - y | ) $ is the fundamental solution of the Laplace operator:

$$ h ( | x - y | ) = \ \left \{ \begin{array}{ll} \frac{1}{( N - 2 ) \omega _ {N} | x - y | ^ {N-} 2 } , & N \geq 3 ; \\ \frac{1}{2 \pi } \mathop{\rm ln} \frac{1}{| x - y | } , & N = 2 ; \\ \end{array} \right .$$

where $ \omega _ {N} = 2 \pi ^ {N/2} / \Gamma ( N / 2 ) $ is the area of the unit sphere in $ \mathbf R ^ {N} $, $ | x - y | $ is the distance between the points $ x $ and $ y $, and $ d v ( y) $ is the volume element in $ D $.

If $ f \in C ^ {(} 1) ( \overline{D}\; ) $, then the potential is defined for all $ x \in \mathbf R ^ {N} $ and $ u \in C ^ {(} 1) ( \mathbf R ^ {N} ) $. In the complementary domain $ \overline{D}\; {} ^ {c} $, the function $ u $ then has derivatives of all orders and satisfies the Laplace equation: $ \Delta u = 0 $, that is, is a harmonic function; for $ N \geq 3 $ this function is regular at infinity, $ u ( \infty ) = 0 $. In $ D $ the potential $ u $ belongs to the class $ C ^ {(} 2) ( D) $ and satisfies the Poisson equation: $ \Delta u = - f $.

These properties can be generalized in various ways. For example, if $ f \in L _ \infty ( D) $, then $ u \in C ( \mathbf R ^ {N} ) $, $ u \in C ^ \infty ( \overline{D}\; {} ^ {c} ) $, $ \Delta u = 0 $ in $ \overline{D}\; {} ^ {c} $, $ u $ has generalized second derivatives in $ D $, and the Poisson equation $ \Delta u = - f $ is satisfied almost-everywhere in $ D $. Properties of potentials of an arbitrary Radon measure $ \mu $ concentrated on an $ N $- dimensional domain $ D $ have also been studied:

$$ u ( x) = \int\limits h ( | x - y | ) d \mu ( y) . $$

Here again $ u \in C ^ \infty ( \overline{D}\; {} ^ {c} ) $ and $ \Delta u = 0 $ in $ \overline{D}\; {} ^ {c} $, $ \Delta u = - \mu ^ \prime $ almost-everywhere in $ D $, where $ \mu ^ \prime $ is the derivative of $ \mu $ with respect to Lebesgue measure in $ \mathbf R ^ {n} $. In definition (*) the fundamental solution of the Laplace operator may be replaced by an arbitrary Levi function (see [2]) for a general second-order elliptic operator $ L $ with variable coefficients of class $ C ^ {( 0 , \lambda ) } ( \overline{D}\; ) $; then the properties listed above still hold with $ \Delta u $ replaced by $ L u $( see [2][4]).

Potentials of mass distributions are applied in the solution of boundary value problems for elliptic partial differential equations (see [2][5]).

For the solution of boundary value problems for parabolic partial differential equations the concept of a heat potential of the form

$$ v ( x , t ) = \ \int\limits _ { 0 } ^ { t } d \tau \int\limits _ { D } G ( x , t ; y, \tau ) f ( y , \tau ) d v ( y) $$

is used, where $ G ( x , t ; y , \tau ) $ is a fundamental solution of the heat equation in $ \mathbf R ^ {N} $:

$$ G ( x , t ; y , \tau ) = \ \frac{1}{( 2 \sqrt \pi ) ^ {N} ( t - \tau ) ^ {N/2} } \mathop{\rm exp} ^ {- | x - y | ^ {2} / 4 ( t - \tau ) } , $$

and $ f ( y , \tau ) $ is the density. The function $ v ( x , t ) $ and its generalizations to the case of an arbitrary second-order parabolic partial differential equation have properties similar to those given above for $ u $( see [3][6]).

References

[1] N.M. Günter, "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar (1967) (Translated from French)
[2] C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)
[3] A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[4] V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)
[5] A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)
[6] A.V. Bitsadze, "Boundary value problems for second-order elliptic equations" , North-Holland (1968) (Translated from Russian)

Comments

A Levi function of a linear partial differential equation is also called a fundamental solution of this equation, or a parametrix of this equation. This function is named after E.E. Levi, who anticipated [a1], [a2] what is known today as the parametrix method.

See also Potential theory; Logarithmic potential; Newton potential; Non-linear potential; Riesz potential; Bessel potential.

References

[a1] E.E. Levi, "Sulle equazioni lineari alle derivate parziali totalmente ellittiche" Rend. R. Acc. Lincei, Classe Sci. (V) , 16 (1907) pp. 932–938
[a2] E.E. Levi, "Sulle equazioni lineari totalmente ellittiche alle derivate parziali" Rend. Circ. Mat. Palermo , 24 (1907) pp. 275–317
[a3] O.D. Kellogg, "Foundations of potential theory" , F. Ungar (1929) (Re-issue: Springer, 1967)
How to Cite This Entry:
Potential of a mass distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Potential_of_a_mass_distribution&oldid=49370
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article