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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