Sejam dois sistemas de coordenadas $(x,y,z)$ e $(\epsilon,\eta, \zeta)$ conectados por meio das seguintes equações de transformação:
\begin{matrix} x=x(\xi,\eta, \zeta) \\ y=y(\xi,\eta, \zeta) \\ z=z(\xi,\eta, \zeta) \end{matrix}
verifiquemos como o vetor posição
muda infinitesimalmente de acordo com a representação de cada sistema de coordenadas.
Seja,
\begin{equation} d\vec{r}=dx(\xi,\eta, \zeta)\hat{x}+dy(\xi,\eta, \zeta)\hat{y}+dz(\xi,\eta, \zeta)\hat{z} \end{equation}
\begin{equation} =\left(\frac{\partial x}{\partial \xi}d\xi+\dfrac{\partial x}{\partial \xi}d\eta+\frac{\partial x}{\partial \xi}d\zeta\right)\hat{x}+\left(\frac{\partial x}{\partial \eta}d\xi+\dfrac{\partial x}{\partial \eta}d\eta+\frac{\partial x}{\partial \eta}d\zeta\right)\hat{y}+\left(\frac{\partial x}{\partial \zeta}\xi+\dfrac{\partial x}{\partial \zeta}\eta+\frac{\partial x}{\partial \zeta}\zeta\right)\hat{z} \end{equation}
\begin{equation} =\left(\frac{\partial x}{\partial \xi}\hat{x}+\dfrac{\partial y}{\partial \xi}\hat{y}+\frac{\partial z}{\partial \xi}\hat{z}\right)d\xi+\left(\frac{\partial x}{\partial \eta}\hat{x}+\dfrac{\partial x}{\partial \eta}\hat{y}+\frac{\partial x}{\partial \eta}\hat{z}\right)d\eta+\left(\frac{\partial x}{\partial \zeta}\hat{x}+\dfrac{\partial x}{\partial \zeta}\hat{y}+\frac{\partial x}{\partial \zeta}\hat{z}\right)d\zeta \end{equation}
\begin{equation}=\left ( \frac{\partial \vec{r}}{\partial \xi} \right )_{\eta,\zeta}d\xi+\left ( \frac{\partial \vec{r}}{\partial \eta} \right )_{\xi,\zeta}d\eta+\left ( \frac{\partial \vec{r}}{\partial \zeta} \right )_{\xi,\eta}d\zeta \end{equation}
com
\begin{matrix} \left ( \frac{\partial \vec{r}}{\partial \xi} \right )_{\eta,\zeta}=\frac{\partial x}{\partial \xi}\hat{x}+\frac{\partial y}{\partial \xi}\hat{y}+\frac{\partial z}{\partial \xi}\hat{z} \\ \left ( \frac{\partial \vec{r}}{\partial \xi} \right )_{\xi,\zeta}=\frac{\partial x}{\partial \eta}\hat{x}+\frac{\partial y}{\partial \eta}\hat{y}+\frac{\partial z}{\partial \eta}\hat{z} \\ \left ( \frac{\partial \vec{r}}{\partial \xi} \right )_{\xi,\eta}=\frac{\partial x}{\partial \zeta}\hat{x}+\frac{\partial y}{\partial \zeta}\hat{y}+\frac{\partial z}{\partial \zeta}\hat{z} \end{matrix}