example of vector potential

If the solenoidal vector  U=U(x,y,z)  is a homogeneous function of degree λ (-2),  then it has the vector potentialMathworldPlanetmath

A=1λ+2U×r, (1)

where  r=xi+yj+zk  is the position vector.

Proof.  Using the entry nabla acting on products, we first may write


In the brackets the first product is, according to Euler’s theorem on homogeneous functions, equal to λU.  The second product can be written as  Uxrx+Uyry+Uzrz, which is Uxi+Uyj+Uzk, i.e. U.  The third product is, due to the sodenoidalness, equal to  0r=0.  The last product equals to 3U (see the first formula (http://planetmath.org/PositionVector) for position vector).  Thus we get the result


This means that U has the vector potential (1).

Title example of vector potential
Canonical name ExampleOfVectorPotential
Date of creation 2013-03-22 15:42:56
Last modified on 2013-03-22 15:42:56
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Example
Classification msc 26B12