VC3.33

From Exampleproblems

Let $V=V_1i+V_2j+V_3k,r=xi+yj+zk\,$

Now, $V\times r=\begin{vmatrix} i & j & k \\ V_1 & V_2 & V_3 \\ x & y & z \end{vmatrix}\,$

= $(V_2z-V_3y)i+(V_3x-V_1z)j+(V_1y-V_2x)k\,$

Therefore, $\nabla\cdot(\nabla\times r)\,$

= $\frac{\partial}{\partial x}(V_2z-V_3y)+\frac{\partial}{\partial y}(V_3x-V_1z)+\frac{\partial}{\partial z}(V_1y-V_2x)\,$ by definition.

= $z\frac{\partial V_2}{\partial x}-y\frac{\partial V_3}{\partial x}+x\frac{\partial V_3}{\partial y}-z\frac{\partial V_1}{\partial z}+y\frac{\partial V_1}{\partial z}-x\frac{\partial V_2}{\partial z}\,$

= $x[\frac{\partial V_3}{\partial y}-\frac{\partial V_2}{\partial z}]+y[\frac{\partial V_1}{\partial z}-\frac{\partial V_3}{\partial x}]+z[\frac{\partial V_2}{\partial x}-y\frac{\partial V_1}{\partial y}]\,$

= $(xi+yj+zk)\cdot[(\frac{\partial V_3}{\partial y}-\frac{\partial V_2}{\partial z})i+(\frac{\partial V_1}{\partial z}-\frac{\partial V_3}{\partial x})j+(\frac{\partial V_2}{\partial x}-y\frac{\partial V_1}{\partial y})k]\,$