Vector coordinate transformation: Help?

[tetris11]

1. The problem statement, all variables and given/known data

How does \delta_{b}C^{d} transform?

Also compute \delta^{'}_{b} C^{'d}


3. The attempt at a solution
\delta_{b} C^{d} = \frac{dC^{d}}{dX^{b}}
?


I think Im supposed to prove that its a scalar, but I really have no starting point.
Any extensive help would be really great.

[fzero]

\frac{\partial C^d}{\partial X^b} is not a scalar, but

\sum_a \frac{\partial C^a}{\partial X^a}

is. Do you know how C^d and \partial/\partial X^b transform on their own?

[tetris11]

C^{'d} = \frac{dX^{'a}}{dX^{b}}C^b

not to sure about the other one....

[fzero]

For the other one, use the chain rule, thinking of X'^a as a function of X^b. In other words, compute

\frac{\partial}{\partial X'^a} f(X'^a(X^b)) = ???? \frac{\partial}{\partial X^b} f(X^b)

[tetris11]

Since:
V'^{a} = \frac{dX'^{a}}{dX^{b}}V^{b}

W'_{b} = \frac{dX^{c}}{dX'^{b}}W_{c}



\frac{dC^{d}}{dX^{b}} *\delta_{'b}C^{'d} = \frac{dC^{d}}{dX^{b}}* \frac{dC^{'d}}{dX^{'b}} = \frac{dC^{d}}{dX^{'b}}* \frac{dC^{'d}}{dX^{b}} = \frac{W'_{b}}{W_{b}}*\frac{V'^{d}}{V^{b}} = ?

I'm still pretty confused.