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enter image description here This is from the book Mathematical Methods (Arfken), Can someone explain how did one del(x) and one dx came here,?

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  • $\begingroup$ what is del(x)? Anyway, its just standard formula $f(x+\Delta,y,z)-f(x,y,z)\approx \frac{\partial f}{\partial x}\Delta $, or are you confused about something else? $\endgroup$
    – Umaxo
    Jun 15 at 5:33
  • $\begingroup$ You mean on the RHS of the highlighted equation? $\endgroup$
    – lineage
    Jun 15 at 5:38
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What the LHS of the highlighted equation represents is the change in $\rho v_x \,dy \,dz$ between two $x$ values. Let the rate of change of this wrt $x$ be $$\frac{\partial (\rho v_x \,dy\, dz)}{\partial x}=\frac{\partial (\rho v_x )}{\partial x} dy\,dz$$

Now $\Delta (\rho v_x)=\frac{\partial \rho v_x}{\partial x} \Delta x$. $\Delta x= x+dx/2-(x-dx/2)=dx$, So the RHS becomes,

$$\Delta (\rho v_x)=\frac{\partial (\rho v_x )}{\partial x}\,dy \,dz \,dx$$

This is a general technique while building differential equations. We calculate the variation of some quantity as a finite difference over a differential element. Then linking that change to the derivative or higher order terms gives a locally valid, differential equation for that quantity.

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