I have been given the following problem and a unsure if I have established the correct answer or not? See attachment
Thank You in advance for any help
I think the problem is asking you to choose a particular alpha and beta, and show that the expression on the left equals the Kronecker delta. Recall,
$\displaystyle {\delta^{\alpha}}_{\beta} = \left\{ \begin{array}{cc} 1, & \mbox{ if } \alpha = \beta \\ 0, & \mbox{ if } \alpha \neq \beta \end{array} \right. $
Let's do the example $\displaystyle \alpha = \beta = r $, where $\displaystyle r = \sqrt{x^2 + y^2 + z^2} $
$\displaystyle \frac{\partial z^{\alpha}}{\partial x^{\mu}} \frac{\partial x^{\mu}}{\partial z^{\beta}} = \frac{\partial r}{\partial x} \frac{\partial x}{\partial r} + \frac{\partial r}{\partial y} \frac{\partial y}{\partial r} + \frac{\partial r}{\partial z} \frac{\partial z}{\partial r} $
$\displaystyle = \frac{x \sin{\theta} \cos{\phi}}{\sqrt{x^2 + y^2 + z^2}} + \frac{y \sin{\theta} \sin{\phi}}{\sqrt{x^2 + y^2 + z^2}} + \frac{z \cos{\theta}}{\sqrt{x^2 + y^2 + z^2}} $
$\displaystyle = \frac{x^2 + y^2 + z^2}{r^2} $
$\displaystyle = 1 $
See if you can do it for other choices of alpha and beta.