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Question 1.51: For Theorem 1, show that (d)⇒(a), (a)⇒(c), (c)⇒(b), (b)⇒(c),...

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

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(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).

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