Vector coordinate transformation: Help?

[tetris11]

Homework Statement

How does [tex]\delta_{b}C^{d}[/tex] transform?

Also compute [tex]\delta^{'}_{b} C^{'d}[/tex]

The Attempt at a Solution

[tex]\delta_{b} C^{d} = \frac{dC^{d}}{dX^{b}}[/tex]
?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]

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

[tex]\sum_a \frac{\partial C^a}{\partial X^a}[/tex]

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

 

[tetris11]

[tex]C^{'d} = \frac{dX^{'a}}{dX^{b}}C^b[/tex]

not to sure about the other one...

 

[fzero]

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

[tex]\frac{\partial}{\partial X'^a} f(X'^a(X^b)) = ? \frac{\partial}{\partial X^b} f(X^b) [/tex]

 

[tetris11]

Since:
[tex] V'^{a} = \frac{dX'^{a}}{dX^{b}}V^{b} [/tex]

[tex] W'_{b} = \frac{dX^{c}}{dX'^{b}}W_{c} [/tex]
[tex]\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}} = ? [/tex]

I'm still pretty confused.