Transformation of the Divergence of a Vector Field By a Linear Transformation

The divergence of a vector field  
\[\mathbf{F}= F_1 \mathbf{i} + F_2 \mathbf{j} + F_3 \mathbf{k}\]
  in Cartesian coordinates is  
\[\mathbf{\nabla} \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y}+ \frac{\partial F_3}{\partial z}\]

In a coordinate system  
  the divergence is  
\[\mathbf{\nabla}' \cdot \mathbf{F'} = \frac{\partial F'_1}{\partial x'} + \frac{\partial F'_2}{\partial y'}+ \frac{\partial F'_3}{\partial z'}\]

In general  
\[x', \: y', \: z'\]
  are each functions of  
\[x , \: y, \: z\]
 so we can write  
\[x'=x'(x,y,z), \: y'=y'(x,y,z), \: z'=z'(x,y,z)\]

\[ \begin{equation} \begin{aligned} \mathbf{\nabla} \cdot \mathbf{F} &=\frac{\partial x'}{\partial x}\frac{\partial F'_1}{\partial x'} + \frac{\partial y'}{\partial x}\frac{\partial F'_1}{\partial y'} + \frac{\partial z'}{\partial x}\frac{\partial F'_1}{\partial z'} \\ &+ \frac{\partial y'}{\partial y}\frac{\partial F'_2}{\partial x'} + \frac{\partial y'}{\partial y}\frac{\partial F'_2}{\partial y'} + \frac{\partial z'}{\partial z}\frac{\partial F'_2}{\partial z'} \\ &+ \frac{\partial x'}{\partial z}\frac{\partial F'_3}{\partial x'} + \frac{\partial y'}{\partial z}\frac{\partial F'_3}{\partial y'} + \frac{\partial z'}{\partial z}\frac{\partial F'_3}{\partial z'} \end{aligned} \end{equation}\]

We can write this as  :
\[ \mathbf{\nabla} \cdot \mathbf{F} = \frac{\partial x'_j} {\partial x_i} \frac{\partial F_i}{\partial x'_j}= \frac{\partial x'_j} {\partial x_i} \frac{\partial F'_i}{\partial x'_j} \]

Where repeated indices indicate summation and
\[x=x_1 , \: y=x_2 , \: z=x_3 , x'=x'_1 ,\: y' = x'_2 , \: z=x'_3 \]

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