Question

Solve the system u=3x-4y, v=x+4y for x and y in terms of u and v. then find the value of the jacobian

Answer 1

Eliminate
`y`.

`u + v = (3x - 4y) + (x + 4y) = 4x \implies x = \frac{u+v}4`

Eliminate
`x`.

`u - 3v = (3x - 4y) - 3 (x + 4y) = -16y \implies y = (3v-u)/(16)`

The Jacobian for this change of coordinates is

`J = \begin{bmatrix} x_u & x_v \\ y_u & y_v \end{bmatrix} = \begin{bmatrix} \frac14 & \frac14 \\\\ -\frac1{16} & \frac3{16} \end{bmatrix}`

with determinant

`\det(J) = \frac14\cdot\frac3{16} - \frac14\cdot\left(-\frac1{16}\right) = \frac1{16}`