How to separate a partial differential equation where R is a function of three variables? Using the method of separation of variables, how can I separate each X,Y,Z if the differential equation has a function of R(x,y,z)?
Example: 
$ R_{xx} + R_{yy} + R_{zz} = 0 $
I understand how to apply the method if R is only a funtion of X and Y, but when it comes to three variables, I am completely lost.
 A: The argument parallels the two variable case.  Setting
$R(x, y, z) = X(x)Y(y)Z(z), \tag{1}$
we have
$X_{xx}(x)Y(y)Z(z) + X(x)Y_{yy}(y)Z(z) + X(x)Y(y)Z_{zz}(z) = 0, \tag{2}$
and dividing through by $X(x)Y(y)Z(z)$ we obtain
$X_{xx} / X + Y_{yy} / Y + Z_{zz} / Z = 0, \tag{3}$
which we write as
$X_{xx} / X = -Y_{yy} / Y - Z_{zz} / Z. \tag{4}$
Now we note that, since the two sides depend upon different independent variables, there must be a constant, call it $-k_x^2$, to which they are each equal, thus:
$X_{xx} / X = -k_x^2, \tag{5}$
or
$X_{xx} + k_x^2X = 0, \tag{5A}$
and
$Y_{yy} / Y + Z_{zz} / Z = k_x^2. \tag{6}$
Having separated out the $x$ dependence, we write (6) as
$Y_{yy} / Y = k_x^2 - Z_{zz} / Z, \tag{7}$
and once again observe that the two sides depend on different independent variables, so again each must equal some constant value, call it $-k_y^2$ this time:
$Y_{yy} / Y = -k_y^2  = k_x^2 - Z_{zz} / Z, \tag{8}$
which leads to
$Y_{yy} + k_y^2Y = 0 \tag{9}$
and
$Z_{zz} + k_z^2Z = 0, \tag{10}$
where we have set
$k_z^2 = -(k_x^2 + k_y^2). \tag{11}$
It should be noted that
$k_x^2 + k_y^2 + k_z^2 = 0, \tag{12}$
so that at least one of the three numbers $k_x, k_y, k_z$ must be complex.  In the typical case occurring in practical applications, the $k_x, k_y, k_z$ are either real or pure imaginary, leading to solutions of (5A), (9), (10) which are respectively periodic or exponential, again analogous to the two-dimensional case.  
Finally, it is worth noting that the techniques outlined above easily extend to the $n$-dimensional case of the equation
$\sum_1^n R_{x_jx_j} = 0; \tag{13}$
if we set
$R = \prod_1^nX_j(x_j), \tag{13A}$
we obtain $n$ equations of the form
$d^2X_j / dx_j^2 + k_j^2X_j = 0 \tag{14}$
with
$\sum_1^nk_j^2 = 0; \tag{15}$
the details are easy to execute and left to the reader.  As is well-known, the solutions
$X_j(x_j$) are of the form
$X_j(x_j) = a_+e^{ik_jx_j} + a_-e^{-ik_jx_j} \tag{16}$
for suitably chosen $a_\pm$.
Hope this helps.  Cheerio,
and as always,
Fiat Lux!!!
