## 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.
Let
$(x,y,z)=(x(u_1,u_2,u_3),y(u_1,u_2,u_3),z(u_1,u_2,u_3))$

and
$(x,y,z)=(x(v_1,v_2,v_3),y(v_1,v_2,v_3),z(v_1,v_2,v_3))$

The transformations are onto and one to one, so there exists a transformation from
$U$
to
$V$
.
We can write
$(u_1,u_2,u_3)=(u_1(v_1,v_2,v_3),u_2(v_1,v_2,v_3),u_3(v_1,v_2,v_3))$

The transformation from
$V$
to
$U$
also exists and is one to one.
Differentiating,
$d \mathbf{r} =\frac{\partial \mathbf{r}}{\partial u_1} d u_1 + \frac{\partial \mathbf{r}}{\partial u_2} d u_2 + \frac{\partial \mathbf{r}}{\partial u_3} d u_3 = \frac{\partial \mathbf{r}}{\partial u_i} d u_i$

where the repeated use of the index
$i$
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
$\frac{\mathbf{r}}{\partial v_j} = \frac{\mathbf{r}}{\partial u_j}\frac{\partial u_i}{\partial v_j}$