For Theorem 2, show that (d)⇒(a), (a)⇒(c), (c)⇒(b), (b)⇒(c), and (c)⇒(a). Reference Th...

For Theorem 2, show that (d)⇒(a), (a)⇒(c), (c)⇒(b), (b)⇒(c), and (c)⇒(a).

Reference Theorem 2,

Divergence-less (or “solenoidal”) fields. The following conditions are equivalent:

(a) ∇ · F = 0 everywhere.

(b) ? F · da is independent of surface, for any given boundary line.

(c) for any closed surface.

(d) F is the curl of some vector function: F = ∇ × A.