VC3.12

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We have,\nabla a_{1}=[i{\frac  {\partial }{\partial x}}+j{\frac  {\partial }{\partial y}}+k{\frac  {\partial }{\partial z}}]a_{1}\, by definition.

or \nabla a_{1}=i{\frac  {\partial a_{1}}{\partial x}}+j{\frac  {\partial a_{1}}{\partial y}}+k{\frac  {\partial a_{1}}{\partial z}}\,

Now (\nabla a_{1})\cdot i=[i{\frac  {\partial a_{1}}{\partial x}}+j{\frac  {\partial a_{1}}{\partial y}}+k{\frac  {\partial a_{1}}{\partial z}}]\cdot i={\frac  {\partial a_{1}}{\partial x}}\, [Since i\cdot i=j\cdot j=k\cdot k=0\,]

Similarly, (\nabla a_{2})\cdot j={\frac  {\partial a_{2}}{\partial y}},(\nabla a_{3})\cdot k={\frac  {\partial a_{3}}{\partial z}}\,

Therefore (\nabla a_{1})\cdot i+(\nabla a_{2})\cdot j+(\nabla a_{3})\cdot k={\frac  {\partial a_{1}}{\partial x}}+{\frac  {\partial a_{2}}{\partial y}}+{\frac  {\partial a_{3}}{\partial z}}=\nabla \cdot a\,

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