Here is a problem which I cannot fully understand how it is solved: Suppose $Z,Y$ are random variables that are independent and $Z\sim\,N(0,1)$ and $Y\sim\chi^2(m)$. Find the distribution of $T=\frac{Z}{\sqrt{\frac{Y}{m}}}$ and the limit distribution as $m\to\infty$. My first issue is how it had calculated the Jacobian. It has written:
$$ t=\frac{z}{\sqrt{\frac{y}{m}}}\quad, z=t\sqrt{\frac{y}{m}}\quad,y=u\quad,u=y $$ and then concluded that: $$ |J|=\begin{vmatrix} \sqrt{\frac{u}{r}} & \frac{\partial z}{\partial u}\\ 0 & 1 \end{vmatrix} $$ How such a conclusion was derived? In particular how were $\sqrt{\frac{u}{r}}$ and $0,1$ were deduced?
That distribution is just the definition of t-student with $m$ degrees of freedom.