Jacobian and its determinant of $$$\left\{u = x, v = y, w = x y\right\}$$$
Your Input
Calculate the Jacobian of $$$\left\{u = x, v = y, w = x y\right\}$$$.
Solution
The Jacobian matrix is defined as follows: $$$J{\left(u,v,w \right)}\left(x, y\right) = \left[\begin{array}{cc}\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y}\\\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\\\frac{\partial w}{\partial x} & \frac{\partial w}{\partial y}\end{array}\right].$$$
In our case, $$$J{\left(u,v,w \right)}\left(x, y\right) = \left[\begin{array}{cc}\frac{\partial}{\partial x} \left(x\right) & \frac{\partial x}{\partial y}\\\frac{\partial y}{\partial x} & \frac{\partial}{\partial y} \left(y\right)\\\frac{\partial}{\partial x} \left(x y\right) & \frac{\partial}{\partial y} \left(x y\right)\end{array}\right].$$$
Find the derivatives (for steps, see derivative calculator): $$$J{\left(u,v,w \right)}\left(x, y\right) = \left[\begin{array}{cc}1 & 0\\0 & 1\\y & x\end{array}\right]$$$.
Since the matrix is not square, the Jacobian determinant doesn't exist.
Answer
The Jacobian matrix is $$$\left[\begin{array}{cc}1 & 0\\0 & 1\\y & x\end{array}\right]$$$A.
The Jacobian determinant doesn't exist.