What is the Jacobian of a Transformation with x=u/(u+v) and y=v/(u-v)?

[Char. Limit]

Homework Statement

Find the Jacobian of the transformation:

$$x=\frac{u}{u+v}, y=\frac{v}{u-v}$$

Homework Equations

Jacobian = $$\left|\stackrel{\frac{\partial x}{\partial u}}{\frac{\partial x}{\partial v}} \stackrel{\frac{\partial y}{\partial u}}{\frac{\partial y}{\partial v}}\right| =\left(\frac{\partial x}{\partial u}\right) \left(\frac{\partial y}{\partial v}\right) - \left(\frac{\partial x}{\partial v}\right) \left(\frac{\partial y}{\partial u}\right)$$

The Attempt at a Solution

Now, I got for my four partial derivatives...

$$\frac{\partial x}{\partial u} = \frac{v}{\left(u+v\right)^2}$$

$$\frac{\partial x}{\partial v} = - \frac{u}{\left(u+v\right)^2}$$

$$\frac{\partial y}{\partial u} = - \frac{v}{\left(u-v\right)^2}$$

$$\frac{\partial y}{\partial v} = \frac{u}{\left(u-v\right)^2}$$

So, multiplying these together gave me...

$$Jacobian = \frac{vu}{(u+v)^2 (u-v)^2} - \frac{uv}{(u+v)^2 - (u-v)^2} = 0$$

Am I supposed to get a Jacobian of 0?

 

[Char. Limit]

Fixed a bit of LaTeX.