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I don't understand why in some texts they put that infinitesimal volume $dV = dx dy dz$. If $ V = V(x,y, z)$ infinitesimal volume should be $$dV = \frac{\partial V}{\partial x} dx +\frac{\partial V}{\partial y} dy + \frac{\partial V}{\partial z} dz$$

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Would Mathematics be a better home for this question? –  Qmechanic Oct 13 '13 at 18:34
    
May be, but I posted it here because modern mathematicians almost wiped out the concept of infinitesimal from calculus, although still it is stillwidely used in sciences, like physics. (See math.oregonstate.edu/bridge/papers/differentials.pdf) So, I thought physicists could better explain me that. –  Girolamo Oct 13 '13 at 18:54
    
Actually, it's been proved that infinitesimals don't lead to any problems after all. For a discussion at an elementary level, see math.wisc.edu/~keisler/calc.html . –  Ben Crowell Oct 13 '13 at 20:17
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If $V$ is a function, writing $\mathrm dV$ for the volume element is just a bad thing to do. I'd write $\mathrm d^3x$ or $\omega_V$, depending on who's gonna read it. –  NiftyKitty95 Oct 13 '13 at 22:30
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2 Answers

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The volume of a finite box is $\Delta x \Delta y \Delta z$. The infinitesimals are simply thought of as very small deltas -- so small that they're smaller than any real number. (This can be formalized in terms of non-standard analysis.)

What may have misled you was the notation $dV$. This simply means an infinitesimally small volume. It doesn't mean that $V$ is a function being differentiated.

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There are two independent concepts here.

  1. One is that of an infinitesimal volume, also referred to as a differential volume element. $$dV = dxdydz$$ This represents a small volume that you would use to derive conservation equations for example. See also infinitesimal for a more general definition.

2 .The second is a concept of a differential, $$dV = \frac{\partial V }{\partial x } dx+ \frac{\partial V }{\partial y }dy+ \frac{\partial V }{\partial z }dz$$

This differential represents a change in the linearization of a function.

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