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