Jacobian and its determinant of $$$\left\{x = 6 u + v, y = 9 u - v\right\}$$$
Your Input
Calculate the Jacobian of $$$\left\{x = 6 u + v, y = 9 u - v\right\}$$$.
Solution
The Jacobian matrix is defined as follows: $$$J{\left(x,y \right)}\left(u, v\right) = \left[\begin{array}{cc}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}\\\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{array}\right].$$$
In our case, $$$J{\left(x,y \right)}\left(u, v\right) = \left[\begin{array}{cc}\frac{\partial}{\partial u} \left(6 u + v\right) & \frac{\partial}{\partial v} \left(6 u + v\right)\\\frac{\partial}{\partial u} \left(9 u - v\right) & \frac{\partial}{\partial v} \left(9 u - v\right)\end{array}\right].$$$
Find the derivatives (for steps, see derivative calculator): $$$J{\left(x,y \right)}\left(u, v\right) = \left[\begin{array}{cc}6 & 1\\9 & -1\end{array}\right]$$$.
The Jacobian determinant is the determinant of the Jacobian matrix: $$$\left|\begin{array}{cc}6 & 1\\9 & -1\end{array}\right| = -15$$$ (for steps, see determinant calculator).
Answer
The Jacobian matrix is $$$\left[\begin{array}{cc}6 & 1\\9 & -1\end{array}\right]$$$A.
The Jacobian determinant is $$$-15$$$A.