VC3.37

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Since v=v_{1}i+v_{2}j+v_{3}k\,,we have

\nabla \times v=[{\frac  {\partial v_{3}}{\partial y}}-{\frac  {\partial v_{2}}{\partial z}}]i+[{\frac  {\partial v_{1}}{\partial z}}-{\frac  {\partial v_{3}}{\partial x}}]j+[{\frac  {\partial v_{2}}{\partial x}}-{\frac  {\partial v_{1}}{\partial y}}]k\, --(1)

Now,\nabla v_{1}\times i=[i{\frac  {\partial v_{1}}{\partial x}}+j{\frac  {\partial v_{2}}{\partial y}}+k{\frac  {\partial v_{3}}{\partial z}}]\times i={\frac  {\partial v_{1}}{\partial y}}(j\times i)+{\frac  {\partial v_{1}}{\partial z}}(k\times i)=-{\frac  {\partial v_{1}}{\partial y}}k+{\frac  {\partial v_{1}}{\partial z}}j\, --(2)

\nabla v_{2}\times j=[i{\frac  {\partial v_{2}}{\partial x}}+j{\frac  {\partial v_{2}}{\partial y}}+k{\frac  {\partial v_{2}}{\partial z}}]\times j={\frac  {\partial v_{2}}{\partial x}}(i\times j)+{\frac  {\partial v_{2}}{\partial z}}(k\times i)={\frac  {\partial v_{2}}{\partial x}}k-{\frac  {\partial v_{2}}{\partial z}}i\, --(3)

and \nabla v_{3}\times k=[i{\frac  {\partial v_{3}}{\partial x}}+j{\frac  {\partial v_{3}}{\partial y}}+k{\frac  {\partial v_{3}}{\partial z}}]\times k={\frac  {\partial v_{3}}{\partial x}}(i\times k)+{\frac  {\partial v_{3}}{\partial y}}(j\times k)=-{\frac  {\partial v_{3}}{\partial x}}j+{\frac  {\partial v_{3}}{\partial y}}i\, --(4)

Adding (2),(3),(4), we have

\nabla v_{1}\times i+\nabla v_{2}\times j+\nabla v_{3}\times k=[{\frac  {\partial v_{3}}{\partial y}}-{\frac  {\partial v_{2}}{\partial z}}]i+[{\frac  {\partial v_{1}}{\partial z}}-{\frac  {\partial v_{3}}{\partial x}}]j+[{\frac  {\partial v_{2}}{\partial x}}-{\frac  {\partial v_{3}}{\partial y}}]k\, implies

\nabla v_{1}\times i+\nabla v_{2}\times j+\nabla v_{3}\times k=\nabla \times v\, by (1)

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