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 $y$.

$u + v = (3x - 4y) + (x + 4y) = 4x \implies x = \dfrac{u+v}4$

Eliminate $x$.

$u - 3v = (3x - 4y) - 3 (x + 4y) = -16y \implies y = \dfrac{3v-u}{16}$

The Jacobian for this change of coordinates is

$J = \begin{bmatrix} x_u & x_v \\ y_u & y_v \end{bmatrix} = \begin{bmatrix} \dfrac14 & \dfrac14 \\\\ -\dfrac1{16} & \dfrac3{16} \end{bmatrix}$

with determinant

$\det(J) = \dfrac14\cdot\dfrac3{16} - \dfrac14\cdot\left(-\dfrac1{16}\right) = \dfrac1{16}$