Transformation of the Divergence of a Vector Field By a Linear Transformation
\[\mathbf{F}= F_1 \mathbf{i} + F_2 \mathbf{j} + F_3 \mathbf{k}\]
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
\[(x'.y',z')\]
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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