# Deriving an Euler Equation

In this derivation is it necessary to write the triple integral, as I thought that if we are dealing with one fluid particle it only contains one "point" and hence we do not have to take a sum?

• What @ACuriousMind says. Imgur.com removes un-clicked images after 6 months or so. Linkrot would render this post (v1) unreadable in its current form. – Qmechanic Mar 28 '15 at 21:42

The fluid "particle" in this case is not a mathematical point, it is just a "small" region of the fluid. Indeed, you have explicitly drawn it as an extended body of size $\delta V$ in your accompanying picture.
• 1.I thought that under the continuum hypothesis each particle is represented by a small infinitesimal volume surrounding a point? 2.If $\delta v$ represents a region of fluid then I am unable to use $D/Dt$ as the lagrangian acceleration is only for individual particles? 3.Is the solution correct when it describes the force of gravity as $\rho\delta v\vec{g}$ or is it only correct when it describes the force of gravity in triple integral form? – usainlightning Mar 29 '15 at 0:07
• I understand each particle is a small infinitesimal volume. What I am wondering is whether or not $\int\int\int_{\delta v}\rho\vec{g} dv=\rho\vec{g}\delta v$, as is implied by the notes. To me this would make sense as I thought each fluid particle surrounds only one individual point so the value of $\rho\vec{g}$ is taken at that point and then multiplied by the infinitesimal volume. – usainlightning Mar 30 '15 at 22:09