(Solution) - Although the gradient divergence and cud theorems are the -(2025 Original AI-Free Solution)

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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)d? = fS T da.
(b) fv (? x v) d? = ? fS v x da.
(c) fv [T?2U ? U?2T) ? (?U)] d? = fS(T?U) ? da.
(d) fv (T?2U ? U?2T) d? = 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.