This discussion focuses on proving two vector integration problems involving surface and volume integrals. The first problem establishes that the double integral of \( r^{5}n \) over surface \( S \) equals the triple integral of \( 5r^{3}r \) over volume \( V \). The second problem demonstrates that the line integral of \( \phi \) along curve \( c \) is equivalent to the surface integral of \( dS \) multiplied by the gradient \( \nabla\phi \) over surface \( S \). These proofs utilize fundamental concepts of vector calculus and integration techniques.
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zhaene said:(a) Prove that \begin{array}{clcr}\int\int_{S} r^{5}ndS=\int\int\int_{V}5r^{3}rdV
(b) Prove that \begin{array}{clcr}\oint_{c}\phi{d}r=\begin{array}{clcr}\int\int_{S}dS\times\nabla\phi