What does this equation for density mean? $$\rho = \lim_{\Delta V\to\varepsilon^3} \ \frac{\Delta m}{\Delta V}$$


1 Answer 1


In the equation for density:

$$\rho = \lim_{\Delta V\to\varepsilon^3} \ \frac{\Delta m}{\Delta V}$$

$\rho$ is density.

$\Delta V$ represents a small change in volume.

$\Delta m$ represents a small change in mass, corresponding to the change in volume $\Delta V$.

The symbol $\varepsilon^3$ signifies the volume of an infinitesimally small region, and taking the limit as $\Delta V$ approaches $\varepsilon^3$ allows us to consider the density at a single point in space. This concept is often used in calculus and mathematical analysis to define density at a point.

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    $\begingroup$ Thank you so much. This helps but why is it the symbol E^3. And why would the similar equation for pressure be E^2? $\endgroup$
    – sebbbb
    Apr 22 at 11:44
  • $\begingroup$ $\varepsilon^3$ I think represents an incrementally small 1 dimension cubed, i.e. 1d to the power of 3 is volume (3d), or an incrementally small volume. Not zero volume, but near to zero volume. $\endgroup$ Apr 22 at 11:57
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    $\begingroup$ I would have thought $\Delta V \rightarrow 0$ would be a more standard and less confusing notation. $\endgroup$
    – gandalf61
    Apr 23 at 6:34
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    $\begingroup$ The SI units for pressure are kg * m^-1 * s^-2. This is because pressure is a force (kg * m) divided by an area (m^2). To find the pressure at an infinitesimally small point, you'd take the limit as the area the force is divided over approaches 0, ie E^2. $\endgroup$
    – elfeiin
    Apr 23 at 8:02

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