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Difference between revisions of "Potential"

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''potential function''
 
''potential function''
  
A characteristic of a [[Vector field|vector field]].
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A characteristic of a [[vector field]].
  
A scalar potential is a scalar function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074090/p0740901.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074090/p0740902.png" /> at every point of the domain of definition of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074090/p0740903.png" /> (sometimes, for example in physics, its negative is called a potential). If such a function exists, the vector field is called a [[Potential field|potential field]].
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A scalar potential is a scalar function $v(M)$ such that $\mathbf{a}(M) = \mathrm{grad}\,v(M)$ at every point of the domain of definition of the field $\mathbf{a}$ (sometimes, for example in physics, its negative is called a potential). If such a function exists, the vector field is called a ''[[potential field]]''.
  
A vector potential is a vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074090/p0740904.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074090/p0740905.png" /> (cf. [[Curl|Curl]]) at every point of the domain of definition of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074090/p0740906.png" />. If such a function exists, the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074090/p0740907.png" /> is called a [[Solenoidal field|solenoidal field]].
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A vector potential is a vector function $\mathbf{A}(M)$ such that $\mathbf{a}(M) = \mathrm{curl}\,\mathbf{A}(M)$ (cf. [[Curl]]) at every point of the domain of definition of the field $\mathbf{a}$. If such a function exists, the vector field $\mathbf{A}(M)$ is called a ''[[solenoidal field]]''.
  
Depending on the distribution of the mass or the charge by which the potential is generated one speaks about a potential of a point-charge, a surface potential (single-layer or double-layer), a volume potential, etc. (see [[Potential theory|Potential theory]]).
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Depending on the distribution of the mass or the charge by which the potential is generated one speaks about a potential of a point-charge, a surface potential (single-layer or double-layer), a volume potential, etc. (see [[Potential theory]]).
  
  
  
 
====Comments====
 
====Comments====
See also [[Double-layer potential|Double-layer potential]]; [[Logarithmic potential|Logarithmic potential]]; [[Multi-field potential|Multi-field potential]]; [[Newton potential|Newton potential]]; [[Non-linear potential|Non-linear potential]]; [[Riesz potential|Riesz potential]].
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See also [[Double-layer potential]]; [[Logarithmic potential]]; [[Multi-field potential]]; [[Newton potential]]; [[Non-linear potential]]; [[Riesz potential]].
  
The use of a vector potential is restricted to three-dimensional vector fields. In this case one can prove the so-called Clebsch lemma, according to which any vector field can be represented as a sum of a potential field and a solenoidal field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074090/p0740908.png" />.
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The use of a vector potential is restricted to three-dimensional vector fields. In this case one can prove the so-called Clebsch lemma, according to which any vector field can be represented as a sum of a potential field and a solenoidal field, $\mathbf{a} = \mathrm{grad}\,v + \mathrm{curl}\,A$.
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Latest revision as of 17:06, 27 December 2017

potential function

A characteristic of a vector field.

A scalar potential is a scalar function $v(M)$ such that $\mathbf{a}(M) = \mathrm{grad}\,v(M)$ at every point of the domain of definition of the field $\mathbf{a}$ (sometimes, for example in physics, its negative is called a potential). If such a function exists, the vector field is called a potential field.

A vector potential is a vector function $\mathbf{A}(M)$ such that $\mathbf{a}(M) = \mathrm{curl}\,\mathbf{A}(M)$ (cf. Curl) at every point of the domain of definition of the field $\mathbf{a}$. If such a function exists, the vector field $\mathbf{A}(M)$ is called a solenoidal field.

Depending on the distribution of the mass or the charge by which the potential is generated one speaks about a potential of a point-charge, a surface potential (single-layer or double-layer), a volume potential, etc. (see Potential theory).


Comments

See also Double-layer potential; Logarithmic potential; Multi-field potential; Newton potential; Non-linear potential; Riesz potential.

The use of a vector potential is restricted to three-dimensional vector fields. In this case one can prove the so-called Clebsch lemma, according to which any vector field can be represented as a sum of a potential field and a solenoidal field, $\mathbf{a} = \mathrm{grad}\,v + \mathrm{curl}\,A$.

How to Cite This Entry:
Potential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Potential&oldid=42618
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article