# Green's function in n-dim, but with one independent variable.

1. Jul 21, 2014

### center o bass

Suppose we have some partial differential equation for a scalar $f$
$$Df = \rho$$
taking values in $\mathbb{R}^n$, and further suppose that the differential equation is completely independent of the variable $y:=x^n$ so that the differential operator $D$ only contains derivatives with respect to $x^1,\ldots, x^{n-1}$, and $f$ as well as $\rho$ is also independent of $x^n$. Would it then be correct to use a Green's function

$$D G(\vec r- \vec r') = \delta^{(n-1)}(\vec r - \vec r')$$
for $\vec r, \vec r' \in \mathbb{R}^{n-1}$ and with
$$f(\vec r) = \int_{\mathbb{R}^{n-1}} d^{n-1} \vec r' G(\vec r - \vec r') \rho(\vec r')$$?

In other words, would it be correct to just use the method of greens functions in $n-1$ dimensions, and pretend that the $n$'th dimension is not there?

2. Aug 5, 2014