I'm integrating a function of the variables $x_i,y_i,z_i$ for $i=1,2,3$ and I define new variables given by $$u_1=x_1+y_3 z_3,$$ $$u_2=x_2+y_1 z_1,$$ $$u_3=x_3+y_2 z_2.$$ NOTE: I don't actually want to evaluate the integral, I just want to it to end up in a particular form.
Let's say my integral has the following form $$\int_0^{\infty}\prod_i dx_i dy_i dz_i f(x,y,z),$$ where $$f(x,y,z)=(y_2(x_1+x_2)+x_2x_3+x_1(x_2+x_3))(y_1(x_1+x_3)+x_2x_3+x_1(x_2+x_3))(y_3(x_2+x_3)+x_2x_3+x_1(x_2+x_3))z_1z_2z_3.$$
To introduce the new variable I used a dirac delta function (I'm a physics student so have mercy) that should evaluate to 1, $$\int_0^{\infty}\prod_i dx_i dy_i dz_i f(x,y,z)\int_0^{\infty}\prod_i du_i\delta(u_1-x_1-y_3 z_3)\delta(u_2-x_2-y_1 z_1)\delta(u_3-x_3-y_2 z_2),$$
Now, I want to isolate these new coordinates so that an overall factor multiplies my integral. This happens for the function if I rescale
$$x_i\rightarrow u_i q$$ $$y_1\rightarrow -u_2 q$$ $$y_2\rightarrow -u_3 q$$ $$y_3\rightarrow -u_1 q,$$ where I hold $u_i$ constant. I don't think this transformation really makes sense though as I should have 3 new variables rather than one, but defining 3 variables results in the function $f$ having an undesirable form. I am fairly certain it's possible to get it into the form I want, so maybe there is a better transformation I can make.
Can someone help me make sense of this?