How is Newton per meter Cubed related to Newton per meter squared (=Pascal)? Is there a way to relate $\frac{N}{m^3}$ to $\frac{N}{m^2}$?
 A: As already pointed out, this is the unit of pressure gradient. But it could also be a “weight density”.
From the standpoint of a physicist, it’s conceptually cleaner to express the weight per unit volume of a substance as a mass density (which for incompressible substances is invariant, and thus a more fundamental characteristic) multiplied by the gravitational acceleration, rather than as a “weight density,” which would have units of newtons per cubic meter.
But ~99.9999% of human activity occurs in a region where gravitational acceleration varies by less than half a percent, so for practical purposes, there is nothing wrong with using a weight density. If you know that your rope is rated for 10 $kN$ and your goop weighs 2000 $N/m^3$, then you can easily calculate that it’s not safe to lift more than 5 cubic meters of the goop with the rope. This calculation would be wrong on the moon, but as you don’t have any goop or rope anywhere but the surface of the earth, that doesn’t matter.
A: Newtons per cubic meter doesn’t work too well as a “force density”, but it may have life as a pressure gradient, .e.g., Pascals per meter, say as you climb K2?
A: I have used force density ($N/m^3$) in calculations of optical forces. That is, when light interacts with a three-dimensional object, how is the optical force distributed in space? A total force ($N$) is applied to the whole object, and then that force is distributed through the object as a density. If there is a direction of interest (say, defined by the normal to a surface), then the optical force density could be integrated along that dimension, resulting in a radiation pressure ($N/m^2$).
So, you relate force density to pressure by integrating the force density over a spatial dimension. Conversely, the force density is the spatial derivative of the pressure. It’s up to you to define exactly how that derivative is calculated based on what is relevant to your problem.
A: You might know that force density f is related to pressure p as
$$f = -∇p$$
This implies,
$$p = \int f·dx$$
Hence is Pressure ($\frac{N}{m²}$) and Force density ($\frac{N}{m³}$) are related.
