\[\mathbf{F}\]
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
\[\alpha_i = \beta_j {\frac{\partial u_i}{\partial v_j} } \]