Proof That Differentials and Derivatives of Orthogonal Coordinates Form Reciprocal Systems
If
\[u_1 , \: u_2 , \: u_3\]
are general orthogonal coordinates then \[\mathbf{\nabla} u_1 , \: \mathbf{\nabla} u_2 , \: \mathbf{\nabla} u_3 \]
and \[\frac{\partial \mathbf{r}}{\partial u_1 } , \: \frac{\partial \mathbf{r}}{\partial u_2 } , \: \frac{\partial \mathbf{r}}{\partial u_3 }\]
are reciprocal systems of vectors.We have to prove
\[ \frac{\partial \mathbf{r}}{\partial u_i } \cdot \mathbf{\nabla} u_j = \left\{ \begin{array}{cc} 1 & i=j \\ 0 & i \neq j \end{array} \right. \]
.Applying the chain rule we have
\[\begin{equation} \begin{aligned} \frac{\partial \mathbf{r}}{\partial u_i } \cdot \mathbf{\nabla} u_j &= (\frac{\partial x}{\partial u_i } \mathbf{i} + \frac{\partial y}{\partial u_i } \mathbf{j} + \frac{\partial y}{\partial u_i } \mathbf{k} ) \cdot (\frac{\partial u_j}{\partial x} \mathbf{i}+ \frac{\partial u_j}{\partial y} \mathbf{j} + \frac{\partial u_j}{\partial z} \mathbf{k}) \\ &= \frac{\partial x}{\partial u_i } \frac{\partial u_j}{\partial x} + \frac{\partial y}{\partial u_i } \frac{\partial u_j}{\partial y} + \frac{\partial z}{\partial u_i } \frac{\partial u_j}{\partial z} \\ &= \left\{ \begin{array}{cc} 1 & i=j \\ 0 & i \neq j \end{array} \right. \end{aligned} \end{equation}\]
. 
