## Relations Between Contravariant Components of a Vector in Two Coordinate Systems

Supp pose a vector field
$\mathbf{F}$
is defined in two coordinate systems
$U(u_1,u_2,u_3), \: V(v_1,v_2,v_3)$
.
Let
$\alpha_1, \: \alpha_2, \: \alpha_3$
be the contravariant components of
$\mathbf{F}$
in the
$U(u_1,u_2,u_3)$
coordinate system, so that
$\mathbf{F} = \alpha_1 \frac{\partial \mathbf{r}}{\partial u_1 } + + \alpha_2 \frac{\partial \mathbf{r}}{\partial u_2 } + \alpha_3 \frac{\partial \mathbf{r}}{\partial u_3 } = \alpha_i \frac{\partial \mathbf{r}}{\partial u_i }$

with the repeated index indicating summation.
In the
$V(v_1,v_2,v_3)$
system
$\mathbf{F} = \beta_1 \frac{\partial \mathbf{r}}{\partial v_1 } + + \beta_2 \frac{\partial \mathbf{r}}{\partial v_2 } + \beta_3 \frac{\partial \mathbf{r}}{\partial v_3 } = \beta_j \frac{\partial \mathbf{r}}{\partial u_j }$

Hence
$\alpha_i \frac{\partial \mathbf{r}}{\partial u_i } = \beta_j \frac{\partial \mathbf{r}}{\partial u_j }$

From the Chain Rule
$\frac{\partial \mathbf{r}}{\partial v_j} = \frac{\partial \mathbf{r}}{\partial u_i} \frac{\partial u_i}{\partial v_j}$

Hence
$\alpha_i \frac{\partial \mathbf{r}}{\partial u_i } = \beta_j \frac{\partial \mathbf{r}}{\partial u_i} {\frac{\partial u_i}{\partial v_j} }$

Equating coefficients gives
$\alpha_i = \beta_j {\frac{\partial u_i}{\partial v_j} }$