Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/62

in general, a scalar quantity as well as a vector. The scalar part, however, which we have called the convergence of the vector-function, vanishes when the vector-function satisfies the solenoidal condition

${\frac {dF}{d\xi }}+{\frac {dG}{d\eta }}+{\frac {dH}{d\zeta }}=0.$

(27)

By differentiating the expressions for F, G, H in equations (22), we find that this equation is satisfied by these quantities.

We may therefore write the relation between the magnetic induction and its vector-potential

${\mathfrak {B}}=\nabla {\mathfrak {A}},$

which may be expressed in words by saying that the magnetic induction is the curl of its vector-potential. See Art. 25.