## Relations Between Covariant 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 coravariant components of
$\mathbf{F}$
in the
$U(u_1,u_2,u_3)$
coordinate system, so that
$\mathbf{F} = \alpha_1 \mathbf{\nabla} u_1 + \alpha_2 \mathbf{\nabla} u_2 + \alpha_3 \mathbf{\nabla} u_3 = \alpha_i \mathbf{\nabla} u_i$

with the repeated index indicating summation.
In the
$V(v_1,v_2,v_3)$
system
$\mathbf{F} = \beta_1 \mathbf{\nabla} v_1 + \beta_2 \mathbf{\nabla} v_2 + \beta_3 \mathbf{\nabla} v_3 = \beta_i \mathbf{\nabla} v_i$

Hence
$\alpha_i \mathbf{\nabla} u_i = \beta_j \mathbf{\nabla} v_j$

Since
$u_i =u_i(v_1,v_2,v_3)$
and
$v_j =v_j(u_1,u_2,u_3)$
,
$\frac{\partial v_j}{\partial x} = \frac{\partial v_j}{\partial u_1} \frac{\partial u_1}{\partial x}+ \frac{\partial v_j}{\partial u_2} \frac{\partial u_2}+ \frac{\partial v_j}{\partial u_3} \frac{\partial u_3}= \frac{\partial v_j}{\partial u_k} \frac{\partial u_k}{\partial x}$

Let
$x=x_1, \: y=x_2 \: z=x_3$
then the last result becomes
$\frac{\partial v_j}{\partial x_1} = \frac{\partial v_j}{\partial u_k} \frac{\partial u_k}{\partial x_1}$

and in general
$\frac{\partial v_j}{\partial x_k} = \frac{\partial v_j}{\partial u_n} \frac{\partial u_n}{\partial x_k}$
(1)
By cycling the
$j$
;s and
$k$
's we obtain nine equations.
We have
$\alpha_i \mathbf{\nabla} u_i = \alpha_j \frac{\partial u_j}{\partial x_1} \mathbf{i} + \alpha_j \frac{\partial u_j}{\partial x_2} \mathbf{j} + \alpha_j \frac{\partial u_j}{\partial x_3} \mathbf{k}$

and
$\beta_i \mathbf{\nabla} v_i = \beta_j \frac{\partial u_j}{\partial x_1} \mathbf{i}+ \beta_j \frac{\partial u_j}{\partial x_2} \mathbf{j} + \beta_j \frac{\partial u_j}{\partial x_3} \mathbf{k}$

Equating components of
$\mathbf{i} , \:\mathbf{j} , \: \mathbf{k}$
gives
$\alpha_i \frac{\partial u_i}{\partial x_k} = \beta_j \frac{\partial v_j}{\partial x_k}$

Use (1) in this result to get
$\alpha_i \frac{\partial u_i}{\partial x_k} = \beta_j \frac{\partial v_j}{\partial u_i} \frac{\partial u_i}{\partial x_k}$

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

Now multiply by
$\frac{\partial u_j}{\partial v_k}$
to get
$\alpha_i \frac{\partial u_j}{\partial v_k}= \beta_i \frac{\partial v_i}{\partial u_j} \frac{\partial u_j}{\partial v_k}$

Remeber
$\frac{\partial v_i}{\partial u_j} \frac{\partial u_j}{\partial v_k}= \delta_{ik}$

Then
$\alpha_j \frac{\partial u_j}{\partial v_k} =\delta{ik}$

Then
$\alpha_j = \beta_i \frac{\partial v_i}{\partial u_j}$
and
$\beta_j = \alpha_i \frac{\partial u_i}{\partial u_j}$