Although the gradient, divergence, and curl theorems are the fundamental integral theorems of vector calculus, it is possible to derive a number of corollaries from them. Show that:
(a) \int_{\nu }(∇T ) d\tau =\oint_{S}^{}{}T da .[Hint: Let v = cT, where c is a constant, in the divergence theorem; use the product rules.]
(b) \int_{\nu }(∇ × v) d\tau =-\oint_{S}v\times da . [Hint: Replace v by (v × c) in the divergence theorem.]
(c) \int_{\nu }^{}{} [T∇^2U + (∇T ) · (∇U)] d\tau =\oint_{S}(T∇U) · da. [Hint: Let v = T∇U in the divergence theorem.]
(d) \int_{\nu }(T∇^2U − U∇^2T ) d\tau =\oint_{S}(T∇U − U∇T ) · da. [Comment: This is sometimes called Green’s second identity; it follows from (c), which is known as Green’s identity.]
(e) \int_{S}^{}{}∇T × da = −\oint_{P}^{}{}T dl . [Hint: Let v = cT in Stokes’ theorem.]