VC5.45

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i).\iint _{S}\phi A\cdot ndS=\iiint _{V}{\mathrm  {div}}(\phi A)dV\,

But {\mathrm  {div}}(\phi A)={\mathrm  {grad}}\phi \cdot A+\phi {\mathrm  {div}}A=\nabla \phi \cdot A+\phi \nabla \cdot A\,

Therefore from the first equation,\iint _{S}\phi A\cdot ndS=\iiint _{V}[\nabla \phi \cdot A+\phi \nabla \cdot A]dV\, or

\iiint _{V}\nabla \phi \cdot AdV=\iint _{S}\phi A\cdot ndS-\iiint _{V}\phi \nabla \cdot AdV\,

ii).Using divergence theorm,we have

\iint _{S}G\times F\cdot dS=\iint _{S}(G\times F)\cdot ndS=\iiint _{V}{\mathrm  {div}}(G\times F)dV\,

But{\mathrm  {div}}(G\times F)=F\cdot {\mathrm  {curl}}G-G\cdot {\mathrm  {curl}}F\,

From the first equation,\iint _{S}G\times F\cdot dS=\iiint _{V}[F\cdot {\mathrm  {curl}}G-G\cdot {\mathrm  {curl}}F]dV\,

or \iiint _{V}F\cdot {\mathrm  {curl}}GdV=\iint _{S}G\times F\cdot dS+\iiint _{V}G\cdot {\mathrm  {curl}}FdV\,

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