In one dimension, the expansion of 
around 
using a taylor series
is
This equation can be converted into an integral interpolant form, so that
where h is the halfwidth of the interpolation function and is called
the smoothing length. Here we have used 
as
the interpolating function. We could just as easily have used the
more conventional W as the interpolating function. The advantage of
the equation above is that the integral on the right reduces to -1
(under the assumption that 
at the surface). If we had
used W as the interpolating function we would have to calculate a
normalizing integral.
The more general equation in one dimension is calculated by expanding
around 
and 
around 
to get
It should be noted that the term 
is well
behaved as 
.
Converting the integral into a summation, produces the SPH form of the second derivative,
It is easily seen by substituting equation (8) into the energy equation (1), and summing the entropy over all the particles, that the time derivative of the total entropy will always be positive.
From equation (8) we can see that a hotter particle
will loose heat to a cooler particle, this means that as
, or the particles become dispersed
in space they cannot cool. This can also be demonstrated by rewriting
equation (1) in terms of the total thermal energy and summing
over all the particles so that
The change in the total thermal energy is zero, so the domain is insulated and a surface radiative boundary condition will have to be supplied.
One advantage of using the B-Spline kernel 
[8] is
that the SPH form of the hydrodynamic equations reduce to recognizable
finite difference equations. With the assumption of constant mass
particles separated by a distance h
equation (8) reduces to
The methodology used above can be extended to construct the SPH form in higher dimensions, which is
As in the one dimensional case the last term on the right of
equation (11) remains well behaved even if
