He showed that if ρ ( x, t ) is the density of Brownian particles at point x at time t, then ρ satisfies the diffusion equation:

where ρ is the radius of curvature.

where the notation ρ is used to describe the distance of the path from the origin instead of R to emphasize that this distance is not fixed, but varies with time.

By moving the unit vectors so their tails coincide, as seen in the circle at the left of the image above, it is seen that u < sub > ρ </ sub > and u < sub > θ </ sub > form a right-angled pair with tips on the unit circle that trace back and forth on the perimeter of this circle with the same angle θ ( t ) as r ( t ).

To evaluate the velocity, the derivative of the unit vector u < sub > ρ </ sub > is needed.

Because u < sub > ρ </ sub > is a unit vector, its magnitude is fixed, and it can change only in direction, that is, its change du < sub > ρ </ sub > has a component only perpendicular to u < sub > ρ </ sub >.

Therefore the change in u < sub > ρ </ sub > is

As with u < sub > ρ </ sub >, u < sub > θ </ sub > is a unit vector and can only rotate without changing size.

To remain orthogonal to u < sub > ρ </ sub > while the trajectory r ( t ) rotates an amount dθ, u < sub > θ </ sub >, which is orthogonal to r ( t ), also rotates by dθ.

The image above shows the sign to be negative: to maintain orthogonality, if du < sub > ρ </ sub > is positive with dθ, then du < sub > θ </ sub > must decrease.

Substituting the derivatives of u < sub > ρ </ sub > and u < sub > θ </ sub >, the acceleration of the particle is:

The radius of curvature of the path is ρ as found from the rate of rotation of the tangent to the curve with respect to arc length, and is the radius of the osculating circle at position s. The unit circle on the left shows the rotation of the unit vectors with s.

A center of curvature is defined at each position s located a distance ρ ( the radius of curvature ) from the curve on a line along the normal u < sub > n </ sub > ( s ).

The required distance ρ ( s ) at arc length s is defined in terms of the rate of rotation of the tangent to the curve, which in turn is determined by the path itself.

where ρ is the density, m is the mass, and V is the volume.

From the equation for density ( ρ = m / V ), mass density must have units of a unit of mass per unit of volume.

where p is pressure, ρ is density, R < sub > u </ sub > is the gas constant, M is the molar mass and T is temperature.

This equation is Newton's law of universal gravitation, expressed in differential form in terms of the gravitational potential φ ( t, x ) and the mass density ρ ( t, x ).

: ρ < sub > ref </ sub > is the known density ( mass per unit volume ) of the reference liquid ( typically water ).

This weight is equal to the mass of liquid displaced multiplied by g, which in the case of the reference liquid is ρ < sub > ref </ sub > Vg.

The nearest nebula to the Sun where massive stars are being formed is the Orion nebula, away .< ref > However, lower mass star formation is occurring about 400 – 450 light years distant in the ρ Ophiuchi cloud complex.

where r < sub > p </ sub > droplet radius, σ surface tension of droplet, ρ density of liquid, M molar mass, T temperature, and R molar gas constant.

It is equal to the molar mass ( M ) divided by the mass density ( ρ ).

where r is the radius vector of a point within the body, ρ ( r ) is the mass density at point r, and d ( r ) is the distance from point r to the axis of rotation.

* Physical homogeneity ( i. e., uniform mass distribution ) making the density function ρ ( r ) become constant ( elementary calculations, or generally an approximation )

The equivalent field equation in terms of mass density ρ of the attracting mass are:

In terms of the total mass M, the nuclear mass m, the density ρ, and a fudge factor f which takes into account geometrical and other effects, criticality corresponds to

This is applied in implosion-type nuclear weapons where a spherical mass of fissile material that is substantially less than a critical mass is made supercritical by very rapidly increasing ρ ( and thus Σ as well ) ( see below ).

There are two other measures of susceptibility, the mass magnetic susceptibility ( χ < sub > mass </ sub > or χ < sub > g </ sub >, sometimes χ < sub > m </ sub >), measured in m < sup > 3 </ sup >· kg < sup >− 1 </ sup > in SI or in cm < sup > 3 </ sup >· g < sup >− 1 </ sup > in CGS and the molar magnetic susceptibility ( χ < sub > mol </ sub >) measured in m < sup > 3 </ sup >· mol < sup >− 1 </ sup > ( SI ) or cm < sup > 3 </ sup >· mol < sup >− 1 </ sup > ( CGS ) that are defined below, where ρ is the density in kg · m < sup >− 3 </ sup > ( SI ) or g · cm < sup >− 3 </ sup > ( CGS ) and M is molar mass in kg · mol < sup >− 1 </ sup > ( SI ) or g · mol < sup >− 1 </ sup > ( CGS ).

:* ρ < sub > p </ sub > is the mass density of the particles ( kg / m < sup > 3 </ sup >), and

:* ρ < sub > f </ sub > is the mass density of the fluid ( kg / m < sup > 3 </ sup >).

Using F for the drag force, ρ for the density, S for the area of the flat plate, V for the flow velocity, and θ for the inclination angle, his law was expressed as

The symbol most often used for density is ρ ( the lower case Greek letter rho ).

where ρ is the density.

It can be expressed in other specific quantities by h = u + pv, where u is the specific internal energy, p is the pressure, and v is specific volume which is equal to 1 / ρ, where ρ is the density.

Mathematically, incompressibility is expressed by saying that the density ρ of a fluid parcel does not change as it moves in the flow field, i. e.,

* ρ is the density of water in kilograms per cubic metre

where R is the radius of curvature, p is the pressure, ρ is the density, and v is the velocity.

* ρ is air density

Lorentz force ( per unit 3-volume ) f on a continuous charge distribution ( charge density ρ ) in motion.

where f is the force density ( force per unit volume ) and ρ is the charge density ( charge per unit volume ).

The pressure exerted by a column of fluid of height h and density ρ is given by the hydrostatic pressure equation, P = hgρ.

If the fluid being measured is significantly dense, hydrostatic corrections may have to be made for the height between the moving surface of the manometer working fluid and the location where the pressure measurement is desired except when measuring differential pressure of a fluid ( for example across an orifice plate or venturi ), in which case the density ρ should be corrected by subtracting the density of the fluid being measured.

: ρ is density of liquid

where ( ρ ) is the density and ( Q ) is the volume rate of flow of fluid.

The ratios of the respective densities ρ < sub > n </ sub >/ ρ and ρ < sub > s </ sub >/ ρ, with ρ < sub > n </ sub > ( ρ < sub > s </ sub >) the density of the normal ( superfluid ) component, and ρ ( the total density ), depends on temperature and is represented in figure 3.