Why would the Jacobian not be zero in this case?

Question

Find the jacobian of the transformation x = u, y = 3uv in the uv plane.

Why would $U_y$ not be zero in this case, if the equation U = x contains no mentions of y?

Answer 1

The Jacobian is $$ \frac{\partial(x, y)}{\partial(u, v)} = \left[ \begin{matrix} \displaystyle \frac{\partial x}{\partial u} & \displaystyle\frac{\partial x}{\partial v} \\[6pt] \displaystyle\frac{\partial y}{\partial u} & \displaystyle\frac{\partial y}{\partial v} \end{matrix} \right] = \left[ \begin{matrix} 1 & 0 \\3v & 3u \end{matrix} \right] $$

You were trying to find $\displaystyle\frac{\partial u}{\partial y}$, which would be part of the reverse Jacobian (from $uv$ to $xy$)

Answer 2

What is $U_y$? The jacobian matrix is

$J = \begin{pmatrix} 1 & 0 \\ 3v & 3u\end{pmatrix}$

It's determinant is $3u$

Answer 3

Because y is a function of $x: y=3ux$(And viceversa), $u=x$, then $y=3xv $ and $x=y/3v$, so you get $u=y/3v$, so that $u$ is a non-constant function of $y$, and so $U_y \neq 0$.