Any [[vector field]] in [[Euclidean space]] can be broken down into a longitudinal ([[Curl free field | curl-free]]) field and a transverse ([[ divergence-free]]) field. Longitudinal [[field line]]s have sources and sinks, while transverse field lines do not. The vector field can be written<ref> {{cite book |author=Jackson, John D.|title=Classical Electrodynamics (3rd ed.)|publisher=Wiley|year=1998|id=ISBN 047130932X}} p. 242. </ref>

:<math> mathbf J left( mathbf x right) = mathbf J_lleft( mathbf x right) + mathbf J_tleft( mathbf x right) </math>

where the longitudinal field is

:<math> mathbf J_lleft( mathbf x right)
=
-{1 over 4pi} {nabla} left( {nabla} cdot int d^3x^{prime} ;
{ 1 over mid mathbf x - mathbf x^{prime} mid }
mathbf Jleft( mathbf x^{prime} right) right)
= -{1 over 4pi} nabla int d^3x^{prime} ; { nabla^{prime} cdot mathbf Jleft( mathbf x^{prime} right) over mid mathbf x - mathbf x^{prime} mid } </math>

where the last equality holds if the field vanishes at infinity. The transverse field is

:<math>
mathbf J_tleft( mathbf x right)
= {1 over 4pi} {nabla} times {nabla} times int d^3x^{prime} ;
{ 1 over mid mathbf x - mathbf x^{prime} mid }
mathbf Jleft( mathbf x^{prime} right)
. </math>

It is easy to see that the curl of the longitudinal field and the divergence of the transverse field is zero.

==Derivation==

Start with the identity in three dimensions (see [[Common integrals in quantum field theory]])

:<math>
-{1 over 4pi} nabla^2 left( {1 over r} right)
= delta left( mathbf x right)
</math>

where

:<math>
r^2 = mathbf x cdot mathbf x
.</math>

Then

:<math>
mathbf Jleft( mathbf x right)
= int d^3x^{prime} ; delta left( mathbf x^{prime} right)
mathbf Jleft( mathbf x^{prime} right)
</math>

:<math>
=
-{1 over 4pi} int d^3x^{prime} ;
left( {nabla^{prime}}^2 { 1 over mid mathbf x - mathbf x^{prime} mid }right)
mathbf Jleft( mathbf x^{prime} right)
</math>

:<math>
=
-{1 over 4pi} {nabla}^2 int d^3x^{prime} ;
{ 1 over mid mathbf x - mathbf x^{prime} mid }
mathbf Jleft( mathbf x^{prime} right)
</math>

:<math>
=
-{1 over 4pi} {nabla} left( {nabla} cdot int d^3x^{prime} ;
{ 1 over mid mathbf x - mathbf x^{prime} mid }
mathbf Jleft( mathbf x^{prime} right) right)
+
{1 over 4pi} {nabla} times {nabla} times int d^3x^{prime} ;
{ 1 over mid mathbf x - mathbf x^{prime} mid }
mathbf Jleft( mathbf x^{prime} right)
</math>

:<math>
=
{1 over 4pi} {nabla} int d^3x^{prime} ;
left({nabla}^{prime} { 1 over mid mathbf x - mathbf x^{prime} mid }right)
mathbf Jleft( mathbf x^{prime} right)
+
{1 over 4pi} {nabla} times {nabla} times int d^3x^{prime} ;
{ 1 over mid mathbf x - mathbf x^{prime} mid }
mathbf Jleft( mathbf x^{prime} right)
</math>

:<math>
=
-{1 over 4pi} {nabla} int d^3x^{prime} ;
left( { {nabla}^{prime} cdot mathbf Jleft( mathbf x^{prime} right) over mid mathbf x - mathbf x^{prime} mid }right)

+
{1 over 4pi} {nabla} times {nabla} times int d^3x^{prime} ;
{ 1 over mid mathbf x - mathbf x^{prime} mid }
mathbf Jleft( mathbf x^{prime} right)
</math>

where we have used the [[vector identity]]

:<math>
nabla times nabla times mathbf A
=
nabla left( nabla cdot mathbf A right)
- nabla^2 mathbf A
</math>

and integrated by parts neglecting terms at infinity.

==References==
{{reflist}}

[[Category:Vector calculus]]