Infinitesimal volume using differentials

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Discussion Overview

The discussion revolves around the concept of infinitesimal volume in the context of calculus, specifically addressing the expression for infinitesimal volume dV in relation to the volume function V = xyz. Participants explore the differences in interpretations and formulations of dV, including the use of differentials and partial differentiation.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants question the expression dV = dx dy dz, arguing that it should be derived from partial differentiation, leading to dV = y z dx + x z dy + x y dz.
  • Others assert that dV = dx * dy * dz represents the volume of a box with dimensions dx, dy, and dz, suggesting a different perspective on the interpretation of infinitesimal volume.
  • A participant mentions that the reasoning for dA = dx * dy is analogous to the reasoning for dV, implying a connection between area and volume calculations.
  • Another participant describes dx, dy, and dz as independent small changes in the axis variables, suggesting they span a small rectangular solid.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct formulation of infinitesimal volume, with multiple competing views presented regarding the use of differentials and partial differentiation.

Contextual Notes

There are unresolved assumptions regarding the definitions of infinitesimals and the context in which these expressions are applied, which may influence the interpretations of dV.

girolamo
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Hi, I don't understand why in some texts they put that infinitesimal volume dV = dx dy dz. If V = x y z, infinitesimal volume should be dV = y z dx + x z dy + x y dz, by partial differentiation. Thanks
 
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girolamo said:
Hi, I don't understand why in some texts they put that infinitesimal volume dV = dx dy dz.
Because the dimensions of the box are dx, dy, and dz. The volume of this box is dV, which is dx * dy * dz.
girolamo said:
If V = x y z, infinitesimal volume should be dV = y z dx + x z dy + x y dz, by partial differentiation. Thanks

This would be the change in volume of a box whose dimensions are x, y, and z, and whose dimensions change by dx, dy, and dz.
 
girolamo said:
Hi, I don't understand why in some texts they put that infinitesimal volume dV = dx dy dz. If V = x y z, infinitesimal volume should be dV = y z dx + x z dy + x y dz, by partial differentiation. Thanks

It's for the same reason that dA = dx*dy.
 
dx dy dx dz are three independent small changes in the axis variables. You can think of the as spanning a small rectangular solid.
 

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