## 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
$(x'.y',z')$
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{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}

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

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