- #1
center o bass
- 545
- 2
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?
$$Df = \rho$$
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?