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I am reading this note on the Bernoulli equations with the following derivations:

enter image description here

I am struggling to find a calculus based meaning for the above equations involving the infinitesimal $\delta V$: I tried replacing $\delta V$ with differential form, but $\delta V$ doesn't behave like $dV$, the volume element because $\frac{d}{dt}(\rho dV)\neq 0$; I am trying to replace $\delta V$ with $\int_{\delta V}$, basically summing the quantity $\delta V$ is concatenated with over a small drop of liquid $\delta V$, but this brings the problem of how to show $\frac{d}{dt}(\frac{1}{2}\rho v^2\delta V)=\delta V\rho v\cdot\frac{dv}{dt}$ given $\frac{d}{dt}(\rho\delta V)$.

Could people help me with understanding these equations involving $\delta V$s?

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    $\begingroup$ You can formalize it with big/little-o notation. Simpler example: by $(V+\delta V)^2-V^2=2V\delta V$ we mean these expressions differ by something $o(\delta V)$. $\endgroup$
    – J.G.
    Commented Feb 11, 2023 at 22:05
  • $\begingroup$ Could it be something like the first variation? ( en.wikipedia.org/wiki/First_variation ) $\endgroup$
    – user7427029
    Commented Feb 11, 2023 at 22:38
  • $\begingroup$ Minor comment to the post (v2): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. $\endgroup$
    – Qmechanic ♦
    Commented Feb 12, 2023 at 5:27

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