Although the gradient, divergence, and cud theorems are the fundamental integral theorems of vector calculus, it is possible to derive a number of corollaries from them. Show that:

(a) fv (∆T)dτ = fS T da.

(b) fv (∆ x v) dτ = – fS v x da.

(c) fv [T∆2U – U∆2T) ∙ (∆U)] dτ = fS(T∆U) ∙ da.

(d) fv (T∆2U – U∆2T) dτ = fS (T∆U – U∆T) ∙ da. [Comment: This is known as Green's theorem; it follows from (c), which is sometimes called Green's identity.]

(e) fS ∆T x da = – fp T dl.