# A Jacobian Transformation

#### Char. Limit

##### PF SAS Commando
Gold Member
1. The problem statement, all variables and given/known data

Find the Jacobian of the transformation:

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

2. Relevant equations

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

3. 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?

Last edited:

#### Char. Limit

##### PF SAS Commando
Gold Member
Fixed a bit of LaTeX.

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