Call Us 07766496223
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  
\[x'=x'(x,y,z), \: y'=y'(x,y,z), \: z'=z'(x,y,z)\]

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

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