I need to integrate the following expression involving a $\delta$-function
$$\int_0^1 \mathrm{d}x \int_0^1 \mathrm{d}y \int_0^1 \mathrm{d}z \, \delta(x+y+z-1)$$ The textbook I'm using suggests this can be simplified by taking the $z$ integral first, then what remains is
$$\int_0^1 \mathrm{d}x \int_0^{1-x} \mathrm{d}y $$
This seems fine to me, although I'm not sure if my understanding is correct: after $z$-integration, the $\delta$-function imposes the constraint $x+y=1$, which is just a straight line $y=-x+1$. Graphically, we have the following 2D integration region, which shows that for every allowed $x$, $\, 0 \le y \le 1-x$, which is reflected by the bounds of integration.
I also have to deal with longer versions of the above integral. If we add another variable and integrate over $x_4$, we now have this 3D integration region and so my guess is that:
$$
\small \int_0^1 \mathrm{d}x_1 \int_0^1 \mathrm{d}x_2 \int_0^1 \mathrm{d}x_3 \int_0^1 \mathrm{d}x_4 \, \delta(x_1+x_2+x_3+x_4-1) =
\int_0^1 \mathrm{d}x_1 \int_0^{1-x_1} \mathrm{d}x_2 \int_0^{1-x_1-x_2} \mathrm{d}x_3
$$
In general, I would like to evaluate this for an arbitrary numer of variables,
$$\int_0^1 \prod_{i=1}^n\mathrm{d}x_i \; \delta \left(\sum_{i=1}^n x_i -1 \right)$$
where for $n>4$ such drawings are not possible any more as the integration region becomes 4D.
First of all, I would like to clarify if the two results above for $n=3$ and $n=4$ are correct. Moreover, is there another way of deriving these results, without having to resort to graphical interpretations of the constraint imposed by the $\delta$-function?
Best Answer
You are correct.
$ \int_0^1 \delta(x+y+z-1)\; dz = 1$ if $0$ is in the interval $(x+y-1, x+y)$, i.e. $0 < x+y < 1$, and $0$ if $x+y > 1$. Since $x+y>0$ is guaranteed when $0 < x < 1$ and $0 < y < 1$, the constraint is $x + y < 1$.
More generally,
$$ \int_0^1 \delta(x_1 + \ldots + x_n - 1)\; d x_n = 1\ \text{if}\ 0 < x_1 + \ldots + x_{n-1} < 1$$ so your constraint is $x_1 + \ldots + x_{n-1} < 1$, and the integral can be written as
$$\int_0^1 dx_1 \int_0^{1-x_1} dx_2 \ldots \int_0^{1-x_1 - \ldots - x_{n-2}} dx_{n-1} \; 1$$