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Let $U=\\frac{X}{Y}$ and $V=Y$

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For $0<X<\\var{x}$ and $0 <Y<\\var{y}$

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What is the value of the Jacobian?

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$|J| = \\Bigg|\\begin{vmatrix}
\\frac{\\partial x}{\\partial u} & \\frac{\\partial x}{\\partial v} \\\\
\\frac{\\partial y}{\\partial u} & \\frac{\\partial y}{\\partial v}
\\end{vmatrix}\\Bigg| = \\Bigg|\\begin{vmatrix}
v & u \\\\
0 & 1
\\end{vmatrix}\\Bigg| = |v| = v$ as $v> 0$

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Upper range for x

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Upper range for y

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