Question

In Exercises \(1-8,\) find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. \(x=a u+b v, y=c u+d v\)

Short answer

The Jacobian for the indicated change of variables is \(ad - bc\).

Step-by-step solution

Derive Partial Derivatives

To start off, we derive functions as follows: \(\frac{\partial x}{\partial u} = a\), \(\frac{\partial x}{\partial v} = b\), \(\frac{\partial y}{\partial u} = c\), \(\frac{\partial y}{\partial v} = d\).

Construct the Jacobian Matrix

Next, we form the Jacobian matrix: \[ J = \begin{{bmatrix}} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{{bmatrix}} = \begin{{bmatrix}} a & b \ c & d \end{{bmatrix}}. \]

Calculate the Determinant of the Jacobian Matrix

Finally, we will calculate the determinant of this matrix: \( D = ad - bc\).