# Vector Potential

by ManuelF
Tags: potential, vector
 P: 8 Hello! Can you tell me the sufficient condition for the existence of the vector potential? Thank you very much!
P: 707
 Quote by ManuelF Hello! Can you tell me the sufficient condition for the existence of the vector potential? Thank you very much!
when a field is divergence free it has a vector potential.
 P: 8 That is a necessary condition, but I do not think is enough! Are you sure? Thank you.
P: 707

## Vector Potential

 Quote by ManuelF That is a necessary condition, but I do not think is enough! Are you sure? Thank you.
It is necessary and sufficient in Euclidean space.

For instance, in magnetostatics a magnetic field always has a vector potential.

If the field is B = (u,v,w) consider the 2 form udy^dz -vdx^dz + wdx^dy

Since B has zero divergence, the 2 form is closed. By Poincare's Lemma it is therefore the exterior derivative of a 1 form adx + bdy + cdx. The curl of (a,b,c) equals B.
 P: 8 Thank you very much!

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