#### quasar987

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I want a rigorous description of the density function (in cartesian coordinates) [itex]\rho(x,y,z)[/itex].

I suggest that we define a function M(x,y,z,V'), where V' is a volume of any given shape centered on the point (x,y,z), giving the mass contained in that volume. Then define the density function as

[tex]\rho(x,y,z) = \frac{\partial{M(x,y,z,V')}}{\partial{V'}}\vert_{V'=0}[/tex]

I wrote the volume of the "enclosing volume" as V' to differentiate it from the "volume of integration" V such that

[tex]M_{body} = \int_V \rho dV = \int_{x_1}^{x_2}\int_{y_1}^{y_2}\int_{z_1}^{z_2}\frac{\partial{M(x,y,z,V')}}{\partial{V'}}\vert_{V'=0}dzdydx[/tex]

Any comments?

I suggest that we define a function M(x,y,z,V'), where V' is a volume of any given shape centered on the point (x,y,z), giving the mass contained in that volume. Then define the density function as

[tex]\rho(x,y,z) = \frac{\partial{M(x,y,z,V')}}{\partial{V'}}\vert_{V'=0}[/tex]

I wrote the volume of the "enclosing volume" as V' to differentiate it from the "volume of integration" V such that

[tex]M_{body} = \int_V \rho dV = \int_{x_1}^{x_2}\int_{y_1}^{y_2}\int_{z_1}^{z_2}\frac{\partial{M(x,y,z,V')}}{\partial{V'}}\vert_{V'=0}dzdydx[/tex]

Any comments?

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