For Theorem 1, show that (d)⇒(a), (a)⇒(c), (c)⇒(b), (b)⇒(c), and (c)⇒(a). Reference The...

For Theorem 1, show that (d)⇒(a), (a)⇒(c), (c)⇒(b), (b)⇒(c), and (c)⇒(a).

Reference Theorem 1

Theorem 1

Curl-less (or “irrotational”) fields. The following conditions are equivalent (that is, F satisfies one if and only if it satisfies all the others):

¹⁴In some textbook problems the charge itself extends to infinity (we speak, for instance, of the electric field of an infinite plane, or the magnetic field of an infinite wire). In such cases the normal boundary conditions do not apply, and one must invoke symmetry arguments to determine the fields uniquely.

(a) ∇ × F = 0 everywhere.

(b) is independent of path, for any given end points.

(c) for any closed loop.

(d) F is the gradient of some scalar function: F = −∇V.