This paper presents a mathematically complete derivation of the minimum-ene
rgy divergence-free vector fields of fixed helicity, defined on and tangent
to the boundary of solid balls and spherical shells. These fields satisfy
the equation del xV=lambda V, where lambda is the eigenvalue of curl having
smallest nonzero absolute value among such fields. It is shown that on the
ball the energy minimizers are the axially symmetric spheromak fields foun
d by Woltjer and Chandrasekhar-Kendall, and on spherical shells they are sp
heromak-like fields. The geometry and topology of these minimum-energy fiel
ds, as well as of some higher-energy eigenfields, are illustrated. (C) 2000
American Institute of Physics. [S1070-664X(00)04005-2].