[permalink] [id link]
In the limit of an infinitesimal volume the density of an inhomogeneous object at a point becomes:, where is an elementary volume at position.
from
Wikipedia
Some Related Sentences
limit and infinitesimal
In the limit where is infinitesimal, the righthand side becomes a space derivative:
* Formally, when working over the reals, as here, this is accomplished by considering the limit as ε → 0 ; but the " infinitesimal " language generalizes directly to Lie groups over general rings.
The Eudoxian definition of proportionality uses the quantifier, " for every ..." to harness the infinite and the infinitesimal, just as do the modern epsilon-delta definitions of limit and continuity.
The approximation becomes exact in the limit as the changes become infinitesimal in size.
* If | ΔP | is infinitesimal ( an infinitely small amount —'— usually expressed in standard analysis as a limit: ), then ΔF ( P ) is known as an infinitesimal difference, with specific denotations of dP and dF ( P ) ( in calculus graphing, the point is almost exclusively identified as " x " and F ( x ) as " y ").
* As the limit of the ordered product of the infinitesimal exponentials:
In the limit of infinitesimal displacements, the ratio of differences becomes a ratio of differentials:
In such context, a marginal change may be an infinitesimal change or a limit.
However, just like a curved surface can be considered flat in the infinitesimal limit of calculus, a curved spacetime can be considered flat at a small scale.
( Newton uses for this derivation – as he does in later proofs in this De Motu, as well as in many parts of the later Principia – a limit argument of infinitesimal calculus in geometric form, in which the area swept out by the radius vector is divided into triangle-sectors.
( Note that we implicitly assumed the standard boundary condition of zero incoming waves from infinity, because otherwise even an infinitesimal loss would eliminate the incoming waves and the limit would not give the lossless solution.
limit and volume
More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,
For a given energy in a given volume, there is an upper limit to the density of information ( the Bekenstein bound ) about the whereabouts of all the particles which compose matter in that volume, suggesting that matter itself cannot be subdivided infinitely many times and there must be an ultimate level of fundamental particles.
Because these changes occur very rapidly they define an upper limit on the volume of a quasar ; quasars are not much larger than the Solar System.
The limit or cap is allocated or sold to firms in the form of emissions permits which, represent the right to emit or discharge a specific volume of the specified pollutant.
One could also limit the theoretical performance of a rather practical " ultimate laptop " with a mass of one kilogram and a volume of one litre.
Here, the size of the scanning volume is determined by the spot size ( close to diffraction limit ) of the optical system because the image of the scanning laser is not an infinitely small point but a three-dimensional diffraction pattern.
In March of that year Congress passed the Cullen-Harrison Act, which legalized " 3. 2 beer " ( i. e., beer 3. 2 % alcohol by weight or 4 % by volume ) and wines of similarly low alcohol content, rather than the 0. 5 % limit defined by the original Volstead Act.
The legality of homebrewing varies from country to country, and some countries limit the volume an individual can legally brew.
The user may optionally apply a BPF-based filter to limit the number of packets seen by tcpdump ; this renders the output more usable on networks with a high volume of traffic.
The seals limit the volume of water entering and leaving the suit.
The reason for the time limit of voluntary apnea is that the rate of breathing and the volume of each breath are tightly regulated to maintain constant values of CO < sub > 2 </ sub > tension and pH of the blood.
Breaths can also be cycled when an alarm condition such as a high pressure limit has been reached, which is a primary strategy in pressure regulated volume control.
It was shown, in the presence of a symmetric Hamiltonian, and in the limit of infinite volume, the system spontaneously adopts a chiral configuration, i. e. breaks mirror plane symmetry.
The seals limit the volume of water entering and leaving the suit.
* Heat transfer: If an object at one temperature is exposed to a medium of another temperature, the temperature difference between the object and the medium follows exponential decay ( in the limit of slow processes ; equivalent to " good " heat conduction inside the object, so that its temperature remains relatively uniform through its volume ).
Using the limit notation of calculus, the volume may be expressed as:
In the limit of small crystallites, as the volume fraction of grain boundaries approaches 100 %, the material ceases to have any crystalline character, and thus becomes an amorphous solid.
In the low temperature limit, the limitations of the Debye model mentioned above do not apply, and it gives a correct relationship between ( phononic ) heat capacity, temperature, the elastic coefficients, and the volume per atom ( the latter quantities being contained in the Debye temperature ).
A volume may have a quota assigned to it in order to limit the amount of space consumed.
In the limit that the system is 2 dimensional a volume becomes an area and in the limit of 1 dimension it becomes a length.
They also show a maximum loading of ~ 35 wt %, since high aspect ratio particles exhibit a larger per particle excluded volume as well as inter-particle tangling as they attempt to rotate end-over-end, resulting in a limit imposed by high off-state apparent viscosity of the fluids.
limit and density
In this model the two universes do not interact, except via local matter accumulation whose density and pressure would become high enough to connect the two sheets through a bridge without spacetime between them, but with geodesics continuity beyond the radius limit allowing an exchange of matter.
In the classical limit, i. e. at large values of or at small density of states — when wave functions of particles practically do not overlap — both the Bose – Einstein or Fermi – Dirac distribution become the Boltzmann distribution.
It has been inferred that as the white dwarf's mass approaches the Chandrasekhar limit, its central density increases, and, as a result of compressional heating, its temperature also increases.
As the degrees of freedom of a particle are the product of all the degrees of freedom of its sub-particles, were a particle to have infinite subdivisions into lower-level particles, then the degrees of freedom of the original particle must be infinite, violating the maximal limit of entropy density.
It was proved that in the thermodynamic limit, the spectrum of the reduced density matrix of a large block of spins is an exact geometric sequence in this case.
Superparamagnetism sets a limit on the storage density of hard disk drives due to the minimum size of particles that can be used.
A distribution being " smoothed out " by summation, showing original Probability density function | density of distribution and three subsequent summations ; see Illustration of the central limit theorem for further details.
The limit on torque density in universal joints is due to the limited cross sections of the cross and yoke.
These techniques are used for a variety of reasons, including the establishment of secure communications, increasing resistance to natural interference, noise and jamming, to prevent detection, and to limit power flux density ( e. g. in satellite downlinks ).
If its density is similar to Amalthea's, Metis's orbit lies within the fluid Roche limit ; however, since it has not broken up, it must lie outside its rigid Roche limit.
Condensation commonly occurs when a vapor is cooled and / or compressed to its saturation limit when the molecular density in the gas phase reaches its maximal threshold.
While the pin diameter placed a lower limit on the smallest reproducible graphic detail, manufacturers were able to use higher dot density to great effect in improving text quality.
where ρ is the probability density, into the Schrödinger equation and then taking the limit ħ → 0 in the resulting equation, yields the Hamilton – Jacobi equation.
The supernova explosion is caused by a white dwarf or a star core reaching a certain mass / density limit, the Chandrasekhar limit, causing the object to collapse in a fraction of a second.
It was demonstrated that the product of the operating current density and the collector voltage exceeded the theoretical limit of bipolar transistors, 2x10 < sup > 5 </ sup > W / cm < sup > 2 </ sup >, and reached 5x10 < sup > 5 </ sup > W / cm < sup > 2 </ sup >.
The residential system was designed for 4½ people per acre which would limit and control the density of residential development.
The township is largely rural north of Geddes Road, which has been designated by the township government as the northern limit of high density development.
When the mass density of this central body exceeds a particular limit, it triggers a gravitational collapse which, if it occurs with spherical symmetry, produces what is known as a Schwarzschild black hole.
R-7 density, 6-story height limit on new buildings, suggested average apartment size of two bedrooms ( this was a response to the fact that between 1960 and 1970 developers had torn down 2, 300 family-sized units and replaced them with 1, 500 smaller units ).
Forward error correction technology ( as demonstrated in high-data-rate UWB pulse systems such as low density parity check code ) can — perhaps in combination with Reed – Solomon error correction — provide channel performance approaching the Shannon limit.
1.680 seconds.