Relationships Between Derivatives of Coordinate Systems

Given two coordinate systems  
\[U(u_1 ,u_2 ,u_3) , \: V(v_1 ,v_2 ,v_3)\]
, we can express the derivatives of one in terms of the derivatives of the other.


The transformations are onto and one to one, so there exists a transformation from  
We can write  

The transformation from  
  also exists and is one to one.
\[d \mathbf{r} =\frac{\partial \mathbf{r}}{\partial u_1} d u_1 + \frac{\partial \mathbf{r}}{\partial u_2} d u_2 + \frac{\partial \mathbf{r}}{\partial u_3} d u_3 = \frac{\partial \mathbf{r}}{\partial u_i} d u_i\]

where the repeated use of the index  
  indicates summation. Similarly  
\[d \mathbf{r} = \frac{\partial \mathbf{r}}{\partial v_i}dv_i\]

We can equate these two expressions, so  
\[\frac{\partial \mathbf{r}}{\partial u_i}du_i = \frac{\partial \mathbf{r}}{\partial v_i}dv_i\]

\[u_i =u_i (v_1,v_2,v_3) \rightarrow du_i =\frac{\partial u_i}{\partial v_j} dv_j\]

Combining the last two equations gives  
\[\frac{\mathbf{r}}{\partial v_j} dv_j = \frac{\mathbf{r}}{\partial u_j}\frac{\partial u_i}{\partial v_j} dv_j \]

Cancelling then gives  
\[\frac{\mathbf{r}}{\partial v_j} = \frac{\mathbf{r}}{\partial u_j}\frac{\partial u_i}{\partial v_j} \]