The Equations of Motion

The equations of motion for a volume element of fluid, written in Exterior Calculus form, using a system of units which unitizes the distracting arbitrary constants such as the speed of sound, can be written (and translated)  as :


molecular displacement acceleration = opposite of the gradient of pressure change

pressure variation over time is opposite to the divergence of the molecular displacement  velocity

density variation over time is opposite to the divergence of molecular displacement velocity

Now you may well wonder why the mathematical form of these expression seems so opaque when it describes things so easy to say in English, such as the fact that when you crowd air molecules together, the density and pressure goes up, and vice versa.

One answer might be that to those who wish to formulate mathematical descriptions, such descriptions are only acceptable if they are sufficiently detailed to describe and account for all possible situations.  In other words, while “crowding raises the pressure” seems so obvious as to be trite, making this expression into a mathematical equation — involving vector expressions such as gradients and divergences expressed in some form of calculus or differential equation — makes the expression more useful in practical applications where it can be applied to unusual situations and unique geometries.


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