Joint pdf of standard normal and chi sq.

$\displaystyle X_{1},...,X_{N}$ is a random sample from a $\displaystyle N(\mu,\sigma^2)$ distribution, where $\displaystyle \mu$ and $\displaystyle \sigma$ are unknown.

I'm given:

$\displaystyle Z = \frac{\sqrt{n}(\overline{X} - \mu)}{\sigma}$

$\displaystyle U = \frac{(n - 1)S^2}{\sigma^2}$

where

$\displaystyle \overline{X} = \sum\frac{X_{i}}{n}$

$\displaystyle S^2 = \frac{\sum(X_{i} - \overline{X})^2}{n-1}$

Now, I think that Z is the standard normal dist, and U is dist. Chi Sq with n-1 degrees of freedom.

I'm now asked to find the joint pdf of Z and U; how would I do this?

I'm also given:

$\displaystyle T = \frac{\sqrt{n}(\overline{X} - \mu)}{S}$

and asked to deduce the joint pdf of T and U; again, I think T has the t dist. with n-1 degrees of freedom, but I'm not sure how to go about doing this question.