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At first I want to say that your formula is incorrect. Density is mass per unit volume $\rho = m/V$, and not mass change (differential or delta) per unit volume. Your formula as it is stated shows that environment can be in homogenous, where density can change in some direction. Suppose you measure $dm/dV$ of air going upwards into higher atmosphere altitudes. You will get different values depending on altitude. Thus $\rho = \rho (h) $. However if you divide total mass of atmosphere by total it's volume, you will get an average density of atmosphere. So mass change over elementary volume lets us to evaluate local density or density dynamics, while total values rationratio - average quantities.

As about what "per unit" means. Density is mass per unit volume. It means that if you take a $1~ m^3$ of steel and weight it's mass - what you will get is steel density, aka. "how much unit volume of something weights". Or you can take any volume of steel and divide by it it's total mass and you will get the same density thing. That's why density has meaning of mass per unit volume.

As about when to use differentials/deltas over absolute quantities - not the physicists decides but rather a nature law itself. Differentials are useful when law talks about relationship between changes in quantities. For example body acceleration is defined as it's speed change over time period, thus the formula is $a=\frac {dv} {dt} $ Hope that helps.

At first I want to say that your formula is incorrect. Density is mass per unit volume $\rho = m/V$, and not mass change (differential or delta) per unit volume. Your formula as it is stated shows that environment can be in homogenous, where density can change in some direction. Suppose you measure $dm/dV$ of air going upwards into higher atmosphere altitudes. You will get different values depending on altitude. Thus $\rho = \rho (h) $. However if you divide total mass of atmosphere by total it's volume, you will get an average density of atmosphere. So mass change over elementary volume lets us to evaluate local density or density dynamics, while total values ration - average quantities.

As about what "per unit" means. Density is mass per unit volume. It means that if you take a $1~ m^3$ of steel and weight it's mass - what you will get is steel density, aka. "how much unit volume of something weights". Or you can take any volume of steel and divide by it it's total mass and you will get the same density thing. That's why density has meaning of mass per unit volume.

As about when to use differentials/deltas over absolute quantities - not the physicists decides but rather a nature law itself. Differentials are useful when law talks about relationship between changes in quantities. For example body acceleration is defined as it's speed change over time period, thus the formula is $a=\frac {dv} {dt} $ Hope that helps.

At first I want to say that your formula is incorrect. Density is mass per unit volume $\rho = m/V$, and not mass change (differential or delta) per unit volume. Your formula as it is stated shows that environment can be in homogenous, where density can change in some direction. Suppose you measure $dm/dV$ of air going upwards into higher atmosphere altitudes. You will get different values depending on altitude. Thus $\rho = \rho (h) $. However if you divide total mass of atmosphere by total it's volume, you will get an average density of atmosphere. So mass change over elementary volume lets us to evaluate local density or density dynamics, while total values ratio - average quantities.

As about what "per unit" means. Density is mass per unit volume. It means that if you take a $1~ m^3$ of steel and weight it's mass - what you will get is steel density, aka. "how much unit volume of something weights". Or you can take any volume of steel and divide by it it's total mass and you will get the same density thing. That's why density has meaning of mass per unit volume.

As about when to use differentials/deltas over absolute quantities - not the physicists decides but rather a nature law itself. Differentials are useful when law talks about relationship between changes in quantities. For example body acceleration is defined as it's speed change over time period, thus the formula is $a=\frac {dv} {dt} $ Hope that helps.

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At first I want to say that your formula is incorrect. Density is mass per unit volume $\rho = m/V$, and not mass change (differential or delta) per unit volume. Your formula as it is stated shows that environment can be in homogenous, where density can change in some direction. Suppose you measure $dm/dV$ of air going upwards into higher atmosphere altitudes. You will get different values depending on altitude. Thus $\rho = \rho (h) $. However if you divide total mass of atmosphere by total it's volume, you will get an average density of atmosphere. So mass change over elementary volume lets us to evaluate local density or density dynamics, while total values ration - average quantities.

As about what "per unit" means. Density is mass per unit volume. It means that if you take a $1~ m^3$ of steel and weight it's mass - what you will get is steel density, aka. "how much unit volume of something weights". Or you can take any volume of steel and divide by it it's total mass and you will get the same density thing. That's why density has meaning of mass per unit volume.

As about when to use differentials/deltas over absolute quantities - not the physicists decides but rather a nature law itself. Differentials are useful when law talks about relationship between changes in quantities. For example body acceleration is defined as it's speed change over time period, thus the formula is $a=\frac {dv} {dt} $ Hope that helps.