VC5.26

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Let S be any surface enclosed by a simple closed curve C.Then,applying Stokes'theorm for vector {\mathrm  {curl}}{\mathrm  {grad}}\phi \,,we have

\iint _{S}({\mathrm  {curl}}{\mathrm  {grad}})\cdot ndS=\oint _{C}{\mathrm  {grad}}\phi \cdot dr\, --(1)

Now,{\mathrm  {grad}}\phi \cdot dr=[{\frac  {\partial \phi }{\partial x}}i+{\frac  {\partial \phi }{\partial y}}j+{\frac  {\partial \phi }{\partial z}}k]\cdot (dxi+dyj+dzk)={\frac  {\partial \phi }{\partial x}}dx+{\frac  {\partial \phi }{\partial y}}dy+{\frac  {\partial \phi }{\partial z}}dz=d\phi \,

Therefore,\oint _{c}{\mathrm  {grad}}\phi \cdot dr=\oint _{C}d\phi \,

=[\phi ]_{{A}}^{{A}}=0\, -- (2) A, being any point on curve C.

From (1),(2),\iint _{S}({\mathrm  {curl}}{\mathrm  {grad}}\phi )\cdot ndS=0\, -(3)

Since (3) holds for all values of S,we have {\mathrm  {curl}}{\mathrm  {grad}}\phi =0\,

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