## Transforming the Gradient of a Function by a Linear Transformation

Suppose coordinate systems
$S(x,y,z)$
and
$S'(x',y',z')$
are related by the linear transformation
$T$
so that
$\begin{pmatrix}x\\y\\z\end{pmatrix}=T \begin{pmatrix}x'\\y'\\z'\end{pmatrix}$

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

\mathbf{\nabla} \begin{aligned} f &=(\frac{\partial x'}{\partial x}\frac{\partial f}{\partial x'} + \frac{\partial y'}{\partial x}\frac{\partial f}{\partial y'} + \frac{\partial z'}{\partial x}\frac{\partial f}{\partial z'}) \mathbf{i} \\ &+ (\frac{\partial y'}{\partial y}\frac{\partial f}{\partial x'} + \frac{\partial y'}{\partial y}\frac{\partial f}{\partial y'} + \frac{\partial z'}{\partial z}\frac{\partial f}{\partial z'}) \mathbf{j} \\ &+ (\frac{\partial x'}{\partial z}\frac{\partial f}{\partial x'} + \frac{\partial y'}{\partial z}\frac{\partial f}{\partial y'} + \frac{\partial z'}{\partial z}\frac{\partial f}{\partial z'}) \mathbf{k}\end{aligned}

Hence
$\begin{pmatrix}{\partial f}/{\partial x}\\ {\partial f}/{\partial y}\\ {\partial f}/{\partial y}\end{pmatrix} = \left( \begin{array}{ccc} \frac{\partial x}{\partial x'} & \frac{\partial x}{\partial y'} & \frac{\partial x}{\partial z'} \\ \frac{\partial y}{\partial x'} & \frac{\partial y}{\partial y'} & \frac{\partial y}{\partial z'} \\ \frac{\partial z}{\partial x'} & \frac{\partial z}{\partial y'} & \frac{\partial z}{\partial z'} \end{array} \right) \begin{pmatrix}{\partial f'}/{\partial x'}\\ {\partial f'}/{\partial y'}\\ {\partial f'}/{\partial y'}\end{pmatrix}$

But
$\left( \begin{array}{ccc} \frac{\partial x}{\partial x'} & \frac{\partial x}{\partial y'} & \frac{\partial x}{\partial z'} \\ \frac{\partial y}{\partial x'} & \frac{\partial y}{\partial y'} & \frac{\partial y}{\partial z'} \\ \frac{\partial z}{\partial x'} & \frac{\partial z}{\partial y'} & \frac{\partial z}{\partial z'} \end{array} \right)$
is just the matrix
$T$
so

$\begin{pmatrix}{\partial f}/{\partial x}\\ {\partial f}/{\partial y}\\ {\partial f}/{\partial y}\end{pmatrix} = T \begin{pmatrix}{\partial f'}/{\partial x'}\\ {\partial f'}/{\partial y'}\\ {\partial f'}/{\partial y'}\end{pmatrix}$