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 potential

A =

U×r,

(1)

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

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

1 1 ∇ × (-----⃗U ×⃗r ) = ----[(⃗r ⋅ ∇)⃗U − (⃗U ⋅ ∇ )⃗r − (∇ ⋅ ⃗U )⃗r + (∇ ⋅⃗r)⃗U ]. λ+2 λ+2

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 U_x ∂⃗r
∂x + U_y -∂⃗r
∂y + U_z ∂⃗r
∂z , which is U_xi + U_yj + U_zk, 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 for position vector). Thus we get the result

--1--⃗ --1-- ⃗ ⃗ ⃗ ⃗ ⃗ ∇ × (λ+2 U × ⃗r) = λ+2 [λ U − U − 0 + 3U ] = U .

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

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