For Theorem 1, show that (d)⇒(a), (a)⇒(c), (c)⇒(b), (b)⇒(c), and (c)⇒(a).
For Theorem 1, show that (d)⇒(a), (a)⇒(c), (c)⇒(b), (b)⇒(c), and (c)⇒(a).
(d) ⇒ (a): ∇×F = ∇×(−∇U) = 0 (Eq. 1.44 – curl of gradient is always zero).
∇ × (∇T ) = 0. (1.44)
(a)⇒ (c): ∮ F · dl = ∫ (∇×F) · da = 0 (Eq. 1.57–Stokes’ theorem).
\int\limits_{S}^{}{}(∇ × v) · da =\oint\limits_{P}^{}{}v · dl. (1.57)
(c) ⇒ (b): \int_{a I}^{b}F · dl −\int_{a II}^{b}{}F · dl =\int_{a I}^{b}{}F · dl +\int_{b II}^{a}{}F · dl =\oint{}F · dl = 0 , so
\int_{a I}^{b}{} F · dl =\int_{a II}^{b}{} F · dl.
(b) ⇒ (c): same as (c) ⇒ (b), only in reverse; (c) ⇒ (a): same as (a)⇒ (c).