Vector coordinate transformation: Help?

[tetris11]

Homework Statement

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

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

The Attempt at a Solution

$$\delta_{b} C^{d} = \frac{dC^{d}}{dX^{b}}$$
?I think I am 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.