# Change of variables in entwined integral

The specific problem I am trying to solve is

$$\int_{-1}^0 dx_1 \int_{-1}^{x_1}dx_2 \cdots \int_{-1}^{x_n}dx_{n+1} f_1(\tilde x_1)\cdots f_{n+1}(\tilde x_{n+1})$$

and doing the change of variables

$$\tilde x_i =x_{i-1}-x_i \quad \forall i=1,\dots, n+1$$

where $x_0 =0$. And what I would like to find out is the boundaries of the $d\tilde x_i$. My guess is that they are still depending on one another.

I did the specific case for 2 dimensions (where I could use basically geometry to find the boundaries) and one gets:

$$\int_{-1}^0 dx_1 \int_{-1}^{x_1}dx_2 = \int_0^1 d\tilde x_1 \int_0^{1-\tilde x_1}d\tilde x_2$$

So I am guessing that the upper bound for $\tilde x_i$ is going to be $1-\sum_{j=1}^{i-1}\tilde x_i$, but I am not sure how to prove this.

• If you use proper MathJax displays rather than your workaround using \quad\quad etc., then you see $\displaystyle\int_{-1}^0 dx_1$, etc., instead of $\int_{-1}^0 dx_1$ and the like. I edited the question accordingly. ${}\qquad{}$ – Michael Hardy Mar 30 '15 at 14:27