formulasearchengine - User contributions [en]

Completeness (order theory)

2014-03-02T05:05:25Z

101.63.202.84: /* Types of completeness properties */

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Current account

2013-12-19T19:32:55Z

101.63.62.44: /* Income */

{{for|the Bernstein polynomial in [[D-module]] theory|Bernstein–Sato polynomial}}
[[Image:Bernstein Approximation.gif|thumb|right|Bernstein polynomials approximating a curve]]
In the [[mathematics|mathematical]] field of [[numerical analysis]], a '''Bernstein polynomial''', named after [[Sergei Natanovich Bernstein]], is a [[polynomial]] in the '''Bernstein form''', that is a [[linear combination]] of '''Bernstein basis polynomials'''.

A [[numerical stability|numerically stable]] way to evaluate polynomials in Bernstein form is [[de Casteljau's algorithm]].

Polynomials in Bernstein form were first used by Bernstein in a constructive proof for the [[Stone–Weierstrass theorem|Stone–Weierstrass approximation theorem]]. With the advent of computer graphics, Bernstein polynomials, restricted to the interval ''x'' ∈ [0, 1], became important in the form of [[Bézier curve]]s.

==Definition==
The ''n'' + 1 '''Bernstein basis polynomials''' of degree ''n'' are defined as

: <math>b_{\nu,n}(x) = {n \choose \nu} x^{\nu} \left( 1 - x \right)^{n - \nu}, \quad \nu = 0, \ldots, n.</math>

where <math>{n \choose \nu}</math> is a [[binomial coefficient]].

The Bernstein basis polynomials of degree ''n'' form a [[basis (linear algebra)|basis]] for the [[vector space]] Π''n'' of polynomials of degree at most ''n''.

A linear combination of Bernstein basis polynomials

:<math>B_n(x) = \sum_{\nu=0}^{n} \beta_{\nu} b_{\nu,n}(x)</math>

is called a '''Bernstein polynomial''' or '''polynomial in Bernstein form''' of degree ''n''. The coefficients <math>\beta_\nu</math> are called '''Bernstein coefficients''' or '''Bézier coefficients'''.

==Example==
The first few Bernstein basis polynomials are:

: <math>
\begin{align}
b_{0,0}(x) & = 1, \\
b_{0,1}(x) & = 1 - x, & b_{1,1}(x) & = x \\
b_{0,2}(x) & = (1 - x)^2, & b_{1,2}(x) & = 2x(1 - x), & b_{2,2}(x) & = x^2 \\
b_{0,3}(x) & = (1 - x)^3, & b_{1,3}(x) & = 3x(1 - x)^2, & b_{2,3}(x) & = 3x^2(1 - x), & b_{3,3}(x) & = x^3 \\
b_{0,4}(x) & = (1 - x)^4, & b_{1,4}(x) & = 4x(1 - x)^3, & b_{2,4}(x) & = 6x^2(1 - x)^2, & b_{3,4}(x) & = 4x^3(1 - x), & b_{4,4}(x) & = x^4
\end{align}
</math>

==Properties==
The Bernstein basis polynomials have the following properties:
* <math>b_{\nu, n}(x) = 0</math>, if <math>\nu < 0</math> or <math>\nu > n</math>.
* <math>b_{\nu, n}(0) = \delta_{\nu, 0}</math> and <math>b_{\nu, n}(1) = \delta_{\nu, n}</math> where <math>\delta</math> is the [[Kronecker delta]] function.
* <math>b_{\nu, n}(x)</math> has a root with multiplicity <math>\nu</math> at point <math>x = 0</math> (note: if <math>\nu = 0</math>, there is no root at 0).
* <math>b_{\nu, n}(x)</math> has a root with multiplicity <math>\left( n - \nu \right)</math> at point <math>x = 1</math> (note: if <math>\nu = n</math>, there is no root at 1).
* <math>b_{\nu, n}(x) \ge 0</math> for <math>x \in [0,\ 1]</math>.
* <math>b_{\nu, n}\left( 1 - x \right) = b_{n - \nu, n}(x)</math>.

* The [[derivative]] can be written as a combination of two polynomials of lower degree:
*: <math>b'_{\nu, n}(x) = n \left( b_{\nu - 1, n - 1}(x) - b_{\nu, n - 1}(x) \right).</math>

* The [[integral]] is constant for a given <math>n</math>
*: <math>\int_{0}^{1}b_{\nu, n}(x)dx = \frac{1}{n+1} \forall \nu = 0,1 \dots n</math>

* If <math>n \ne 0</math>, then <math>b_{\nu, n}(x)</math> has a unique local maximum on the interval <math>[0,\ 1]</math> at <math>x = \frac{\nu}{n}</math>. This maximum takes the value:
*: <math>\nu^\nu n^{-n} \left( n - \nu \right)^{n - \nu} {n \choose \nu}.</math>

* The Bernstein basis polynomials of degree <math>n</math> form a [[partition of unity]]:
*: <math>\sum_{\nu = 0}^n b_{\nu, n}(x) = \sum_{\nu = 0}^n {n \choose \nu} x^\nu \left( 1 - x \right)^{n - \nu} = \left(x + \left( 1 - x \right) \right)^n = 1.</math>

* By taking the first derivative of <math>(x+y)^n</math> where <math>y = 1-x</math>, it can be shown that
*: <math>\sum_{\nu=0}^{n}\nu b_{\nu, n}(x) = nx</math>

* The second derivative of <math>(x+y)^n</math> where <math>y = 1-x</math> can be used to show
*: <math>\sum_{\nu=1}^{n}\nu(\nu-1) b_{\nu, n}(x) = n(n-1)x^2</math>

* A Bernstein polynomial can always be written as a linear combination of polynomials of higher degree:
*: <math>b_{\nu, n - 1}(x) = \frac{n - \nu}{n} b_{\nu, n}(x) + \frac{\nu + 1}{n} b_{\nu + 1, n}(x).</math>

==Approximating continuous functions==
Let ''&fnof;'' be a [[continuous function]] on the interval [0, 1]. Consider the Bernstein polynomial

: <math>B_n(f)(x) = \sum_{\nu = 0}^n f\left( \frac{\nu}{n} \right) b_{\nu,n}(x).</math>

It can be shown that

: <math>\lim_{n \to \infty}{ B_n(f)(x) } = f(x) \,</math>

[[uniform convergence|uniformly]] on the interval [0, 1]. This is a stronger statement than the proposition that the limit holds for each value of ''x'' separately; that would be [[pointwise convergence]] rather than [[uniform convergence]]. Specifically, the word ''uniformly'' signifies that

: <math>\lim_{n \to \infty} \sup \left\{\, \left| f(x) - B_n(f)(x) \right| \,:\, 0 \leq x \leq 1 \,\right\} = 0.</math>

Bernstein polynomials thus afford one way to prove the [[Stone–Weierstrass theorem#Weierstrass_approximation_theorem|Weierstrass approximation theorem]] that every real-valued continuous function on a real interval [''a'', ''b''] can be uniformly approximated by polynomial functions over '''R'''.

A more general statement for a function with continuous ''k''th derivative is

: <math>{\left\| B_n(f)^{(k)} \right\|}_\infty \le \frac{ (n)_k }{ n^k } \left\| f^{(k)} \right\|_\infty \text{ and } \left\| f^{(k)}- B_n(f)^{(k)} \right\|_\infty \to 0</math>

where additionally

: <math>\frac{ (n)_k }{ n^k } = \left( 1 - \frac{0}{n} \right) \left( 1 - \frac{1}{n} \right) \cdots \left( 1 - \frac{k - 1}{n} \right)</math>

is an [[eigenvalue]] of ''B''''n''; the corresponding eigenfunction is a polynomial of degree ''k''.

===Proof===

Suppose ''K'' is a [[random variable]] distributed as the number of successes in ''n'' independent [[Bernoulli trial]]s with probability ''x'' of success on each trial; in other words, ''K'' has a [[binomial distribution]] with parameters ''n'' and ''x''. Then we have the [[expected value]] E(''K''/''n'') = ''x''.

By the [[law of large numbers|weak law of large numbers]] of [[probability theory]],
: <math>\lim_{n \to \infty}{ P\left( \left| \frac{K}{n} - x \right|>\delta \right) } = 0</math>
for every ''δ'' > 0. Moreover, this relation holds uniformly in ''x'', which can be seen from its proof via [[Chebyshev's inequality]], taking into account that the variance of ''K''/''n'', equal to ''x''(1-''x'')/''n'', is bounded from above by 1/(4''n'') irrespective of ''x''.

Because ''&fnof;'', being continuous on a closed bounded interval, must be [[uniform continuity|uniformly continuous]] on that interval, one infers a statement of the form
: <math>\lim_{n \to \infty}{ P\left( \left| f\left( \frac{K}{n} \right) - f\left( x \right) \right| > \varepsilon \right) } = 0</math>
uniformly in ''x''. Taking into account that ''ƒ'' is bounded (on the given interval) one gets for the expectation
: <math>\lim_{n \to \infty}{ E\left( \left| f\left( \frac{K}{n} \right) - f\left( x \right) \right| \right) } = 0</math>
uniformly in ''x''. To this end one splits the sum for the expectation in two parts. On one part the difference does not exceed ε; this part cannot contribute more than ε.
On the other part the difference exceeds ε, but does not exceed 2''M'', where ''M'' is an upper bound for |''ƒ''(x)|; this part cannot contribute more than 2''M'' times the small probability that the difference exceeds ε.

Finally, one observes that the absolute value of the difference between expectations never exceeds the expectation of the absolute value of the difference, and that E(''&fnof;''(''K''/''n'')) is just the Bernstein polynomial ''B''''n''(''ƒ'', ''x'').

See for instance.<ref>L. Koralov and Y. Sinai, "Theory of probability and random processes" (second edition), Springer 2007; see page 29, Section "Probabilistic proof of the Weierstrass theorem".</ref>

==See also==
*[[Bézier curve]]
*[[Polynomial interpolation]]
*[[Newton polynomial|Newton form]]
*[[Lagrange polynomial|Lagrange form]]

==Notes==
<references />

==References==
* [http://www.idav.ucdavis.edu/education/CAGDNotes/Bernstein-Polynomials.pdf BERNSTEIN POLYNOMIALS by Kenneth I. Joy ]
* H. Caglar and A. N. Akansu, [http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=224242&userType=inst "A Generalized Parametric PR-QMF Design Technique Based on Bernstein Polynomial Approximation,"] IEEE Transactions on Signal Processing, vol. 41, no. 7, pp. 2314–2321, July 1993.
* [http://www.ams.org/featurecolumn/archive/bezier.html From Bézier to Bernstein]
* {{springer|title=Bernstein polynomials|id=B/b015730|last=Korovkin|first=P.P.}}
* {{mathworld|urlname=BernsteinPolynomial|title=Bernstein Polynomial}}
* {{PlanetMath attribution|id=9775|title=properties of Bernstein polynomial}}

{{DEFAULTSORT:Bernstein Polynomial}}
[[Category:Numerical analysis]]
[[Category:Polynomials]]
[[Category:Articles containing proofs]]

Band emission

2013-12-07T14:28:11Z

101.63.174.28: Undid revision 584994427 by 101.63.174.28 (talk)

The term "''''''homogeneous''''''" is used in more than one context in mathematics. Perhaps the most prominent are the following three distinct cases:

# Homogeneous functions
# Homogeneous type of first order differential equations
# Homogeneous differential equations (in contrast to "inhomogeneous" differential equations). This definition is used to define a property of certain linear differential equations—it is unrelated to the above two cases.

Each one of these cases will be briefly explained as follows.

== Homogeneous functions ==
{{main|Homogeneous function}}
'''Definition'''. A function  <math>f(x)</math>  is said to be homogeneous of degree   <math>n</math>   if, by introducing a constant parameter  <math>\lambda</math>, replacing the variable   <math>x</math>   with   <math>\lambda x</math>   we find:
:<math> f(\lambda x) = \lambda^n f(x)\,. </math>

This definition can be generalized to functions of more-than-one variables; for example, a function of two variables <math>f(x,y)</math> is said to be homogeneous of degree  <math>n</math>  if we replace both variables  <math>x</math>  and  <math>y</math>  by  <math>\lambda x</math>  and  <math>\lambda y</math>,  we find:

:<math>f(\lambda x, \lambda y) = \lambda^n f(x,y)\,. </math>

'''Example.''' The function  <math>f(x,y) = (2x^2-3y^2+4xy)</math>  is a homogeneous function of degree 2 because:
:<math>f(\lambda x, \lambda y) = [2(\lambda x)^2-3(\lambda y)^2+4(\lambda x \lambda y)] = (2\lambda^2x^2-3\lambda^2y^2+4\lambda^2 xy) = \lambda^2(2x^2-3y^2+4xy)=\lambda^2f(x,y).</math>
 
This definition of homogeneous functions has been used to classify certain types of first order differential equations.

== Homogeneous type of first-order differential equations ==
{{Differential equations}}

A first-order [[ordinary differential equation]] in the form:

:<math>M(x,y)\,dx + N(x,y)\,dy = 0 </math>

is a homogeneous type if both functions ''M''(''x, y'') and ''N''(''x, y'') are [[homogeneous function]]s of the same degree ''n''.<ref>{{harvnb|Ince|1956|p=18}}</ref> That is, multiplying each variable by a parameter  <math>\lambda</math>, we find:

:<math>M(\lambda x, \lambda y) = \lambda^n M(x,y) </math>     and     <math> N(\lambda x, \lambda y) = \lambda^n N(x,y)\,. </math>

Thus,
:<math>\frac{M(\lambda x, \lambda y)}{N(\lambda x, \lambda y)} = \frac{M(x,y)}{N(x,y)}\,. </math>

===Solution method===
In the quotient   <math>\frac{M(tx,ty)}{N(tx,ty)} = \frac{M(x,y)}{N(x,y)}</math>,
we can let   <math>t = 1/x</math>   to simplify this quotient to a function <math>f</math> of the single variable <math>y/x</math>:

:<math>\frac{M(x,y)}{N(x,y)} = \frac{M(tx,ty)}{N(tx,ty)} = \frac{M(1,y/x)}{N(1,y/x)}=f(y/x)\,. </math>

Introduce the [[change of variables]] <math>y=ux</math>; differentiate using the [[product rule]]:

:<math>\frac{d(ux)}{dx} = x\frac{du}{dx} + u\frac{dx}{dx} = x\frac{du}{dx} + u,</math>

thus transforming the original differential equation into the [[Separation of variables|separable]] form:
: <math>x\frac{du}{dx} = f(u) - u\,; </math>

this form can now be integrated directly (see [[ordinary differential equation]]).

===Special case===

A first order differential equation of the form (''a'', ''b'', ''c'', ''e'', ''f'', ''g'' are all constants):
:<math> (ax + by + c) dx + (ex + fy + g) dy = 0\, , </math>

can be transformed into a homogeneous type by a linear transformation of both variables (<math>\alpha</math> and <math>\beta</math> are constants):
:<math>t = x + \alpha; \,\,\,\, z = y + \beta \,. </math>

==Homogeneous linear differential equations==

'''Definition.''' A linear differential equation is called '''homogeneous''' if the following condition is satisfied: If  <math>\phi(x)</math>  is a solution, so is  <math>c \phi(x)</math>, where <math>c</math> is an arbitrary (non-zero) constant. Note that in order for this condition to hold, each term in a linear differential equation of the dependent variable y must contain y or any derivative of y; a constant term breaks homogeneity. A linear differential equation that fails this condition is called '''inhomogeneous.'''

A [[linear differential equation]] can be represented as a [[linear operator]] acting on ''y(x)'' where ''x'' is usually the independent variable and ''y'' is the dependent variable. Therefore, the general form of a [[linear homogeneous differential equation]] is of the form:

:<math> L(y) = 0 \,</math>
<math>
</math>where ''L'' is a [[differential operator]], a sum of derivatives, each multiplied by a function  <math>f_i</math>  of ''x'':

:<math> L = \sum_{i=1}^n f_i(x)\frac{d^i}{dx^i} \,; </math>
where  <math>f_i</math>  may be constants, but not all  <math>f_i</math>  may be zero.

For example, the following differential equation is homogeneous

:<math> \sin(x) \frac{d^2y}{dx^2} + 4 \frac{dy}{dx} + y = 0 \,, </math>

whereas the following two are inhomogeneous:

:<math> 2 x^2 \frac{d^2y}{dx^2} + 4 x \frac{dy}{dx} + y = \cos(x) \,; </math>

:<math> 2 x^2 \frac{d^2y}{dx^2} - 3 x \frac{dy}{dx} + y = 2 \,. </math>

==See also==
* [[Method of separation of variables]]

==Notes==
{{Reflist}}

==References==
* {{citation | last1=Boyce | first1=William E. | last2=DiPrima | first2=Richard C. | title = Elementary differential equations and boundary value problems | year=2012 | publisher=Wiley | isbn=978-0470458310 | edition=10th}}. (This is a good introductory reference on differential equations.)
* {{citation | last1=Ince | first1=E. L. | title=Ordinary differential equations | url=http://archive.org/details/ordinarydifferen029666mbp | year=1956 | publisher=Dover Publications | location=New York | isbn=0486603490}}. (This is a classic reference on ODEs, first published in 1926.)

==External links==
*[http://mathworld.wolfram.com/HomogeneousOrdinaryDifferentialEquation.html Homogeneous differential equations at MathWorld]
*[http://en.wikibooks.org/wiki/Ordinary_Differential_Equations/Substitution_1 Wikibooks: Ordinary Differential Equations/Substitution 1]

[[Category:Differential equations]]

Graph labeling

2013-12-05T12:19:36Z

101.63.155.126: /* Edge-graceful labeling */

{{About|energy per unit volume|energy per unit mass or energy density of foods|specific energy}}

'''Energy density''' is the amount of [[energy]] stored in a given system or region of space per unit [[volume]] or [[mass]], though the latter is more accurately termed [[specific energy]]. Often only the ''useful'' or extractable energy is measured, which is to say that chemically inaccessible energy such as [[rest mass]] energy is ignored.<ref>{{cite web|url=http://physics.nist.gov/Pubs/SP811/sec04.html |title=The Two Classes of SI Units and the SI Prefixes |work=NIST Guide to the SI |publisher= |date= |accessdate=2012-01-25}}</ref> In [[physical cosmology|cosmological]] and other [[general relativity|general relativistic]] contexts, however, the energy densities considered are those that correspond to the elements of the [[stress-energy tensor]] and therefore do include mass energy as well as energy densities associated with the pressures described in the next paragraph.

Energy per unit volume has the same physical units as [[pressure]], and in many circumstances is a [[synonym]]: for example, the energy density of a magnetic field may be expressed as (and behaves as) a physical pressure, and the energy required to compress a compressed gas a little more may be determined by multiplying the difference between the gas pressure and the external pressure by the change in volume. In short, pressure is a measure of the [[enthalpy]] per unit volume of a system. A pressure gradient has a potential to perform work on the surroundings by converting enthalpy until equilibrium is reached.

==Introduction to energy density==

Energy can be stored in many different types of material, and there are several types of reactions that release energy. In order of the typical magnitude of the energy released, these types of reactions are: nuclear, chemical, electrochemical, and electrical.

Chemical reactions are used by animals to derive energy from food, and by automobiles to derive energy from gasoline. Electrochemical reactions are used by most mobile devices such as laptop computers and mobile phones to release the energy from batteries.

===Energy densities of common energy storage materials===
{{Unreferenced section|date=October 2013}}
The following is a list of the combustion energy densities of commonly used or well-known energy storage materials; it doesn't include uncommon or experimental materials. Note that this list does not consider the mass of reactants commonly available such as the oxygen required for combustion.

The following unit conversions may be helpful when considering the data in the table: 1 [[Joule|MJ]] ≈ 0.28 [[Kilowatt hour|kWh]] ≈ 0.37 [[Horsepower-hour|HPh]].

{| class="wikitable" style="text-align: center;"
! Storage material !! Energy type !! Specific energy (MJ/kg) !! Energy density (MJ/L) !! Direct uses
|-
| style="text-align:left;"| '''[[Uranium-235]]''' || [[Nuclear power|Nuclear]] fission || 83 140 000 || 1 546 000 000 || Electric power plants (nuclear reactors)
|-
| style="text-align:left;"| '''[[Compressed hydrogen|Hydrogen (compressed]] at 70 MPa)''' || [[Chemical energy#Chemical energy|Chemical]] || 123 || 5.6 || Experimental automotive engines
|-
| style="text-align:left;"| '''[[Gasoline]] (petrol) / [[Diesel fuel|Diesel]]''' || Chemical || ~46 || ~36 || Automotive engines
|-
| style="text-align:left;"| '''[[Propane]] (including [[Liquefied petroleum gas|LPG]])''' || Chemical || 46.4 || 26 || Cooking, home heating, automotive engines
|-
| style="text-align:left;"| '''[[Fat]] (animal/vegetable)''' || Chemical || 37 || || Human/animal nutrition
|-
| style="text-align:left;"| '''[[Coal]]''' || Chemical || 24 || || Electric power plants, home heating
|-
| style="text-align:left;"| '''[[Lithium-air battery]] (theoretical)''' || Electrochemical || 18.7<ref>{{cite web|title=A review of high energy density lithium-air battery technology|url=http://link.springer.com/article/10.1007%2Fs10800-013-0620-8#page-1|publisher=Springer}}</ref> || || Electronic devices, vehicles
|-
| style="text-align:left;"| '''[[Carbohydrate]]s (including sugars)''' || Chemical || 17 || || Human/animal nutrition
|-
| style="text-align:left;"| '''[[Protein in nutrition|Protein]]''' || Chemical || 16.8 || || Human/animal nutrition
|-
| style="text-align:left;"| '''[[Wood fuel|Wood]]''' || Chemical || 16.2 || || Heating, outdoor cooking
|-
| style="text-align:left;"| '''[[Trinitrotoluene|TNT]]''' || Chemical || 4.6 || || Explosives
|-
| style="text-align:left;"| '''[[Gunpowder]]''' || Chemical || 3 || || Explosives
|-
| style="text-align:left;"| '''[[Lithium battery]] (non-rechargeable)'''|| Electrochemical || 1.8 || 4.32 || Portable electronic devices, flashlights
|-
| style="text-align:left;"| '''[[Lithium-ion battery]]''' || Electrochemical || 0.36<ref>{{cite web|title=Overview of lithium ion batteries|url=http://www.panasonic.com/industrial/includes/pdf/Panasonic_LiIon_Overview.pdf|publisher=Panasonic|archiveurl=http://web.archive.org/web/20111107060525/http://www.panasonic.com/industrial/includes/pdf/Panasonic_LiIon_Overview.pdf|archivedate=Nov 7, 2011|date=Jan., 2007|deadurl=no}}</ref>–0.875 || 0.9–2.63 || Laptop computers, mobile devices, some modern electric vehicles
|-
| style="text-align:left;"| '''[[Alkaline battery]]''' || Electrochemical || 0.67 || 1.8 || Portable electronic devices, flashlights
|-
| style="text-align:left;"| '''[[Nickel-metal hydride battery]]''' || Electrochemical || 0.288 || 0.504–1.08 || Portable electronic devices, flashlights
|-
| style="text-align:left;"| '''[[Lead-acid battery]]''' || Electrochemical || 0.17 || 0.34 || Automotive engine ignition
|-
| style="text-align:left;"| '''[[Supercapacitor]]''' || Electrical ||0.018 || || Electronic circuits
|-
| style="text-align:left;"| '''Electrostatic [[capacitor]]''' || Electrical || 0.000036 || || Electronic circuits
|}

{| class="wikitable" style="text-align: center;"
|+ Energy capacities of common storage forms
|-
! Storage device !! Energy type !! Energy content (MJ) !! Typical mass !! W × H × D (mm)!! Uses
|-
| style="text-align:left;"| '''Automotive [[lead-acid battery]]''' || Electrochemical || 2.6 || 15 kg || 230 × 180 × 185 || Automotive starter motor and accessories
|-
| style="text-align:left;"| '''Sandwich ([[Subway (restaurant)|Subway]] 6 [[inch]] club)''' || Chemical || 1.3<ref>{{cite web|title=Subway Nutritional Information|url=http://www.foodinfodb.com/restaurants/s/subway|work=The Food Information Database|accessdate=27 July 2013}}</ref> || 240 grams || 150 × ? × ? || Human nutrition 
|-
| style="text-align:left;"| '''Alkaline [[AA battery]]''' || Electrochemical || 0.0154 || 23 g || 14.5 × 50.5 × 14.5 || Portable electronic equipment, flashlights
|-
| style="text-align:left;"| '''Lithium-ion battery (Nokia BL-5C)''' || Electrochemical || 0.0129 || 18.5 g || 54.2 × 33.8 × 5.8 || Mobile phones
|}

==Energy density in energy storage and in fuel==
[[File:Energy density.svg|thumb|400px|float|Selected energy densities plot]]

In [[energy storage]] applications the energy density relates the [[mass]] of an energy store to the volume of the storage facility, e.g. the [[fuel]] tank. The higher the energy density of the fuel, the more energy may be stored or transported for the same amount of volume. The energy density of a fuel per unit mass is called the [[specific energy]] of that fuel. In general an [[engine]] using that fuel will generate less [[kinetic energy]] due to [[inefficiency|inefficiencies]] and [[thermodynamics|thermodynamic]] considerations—hence the [[Thrust specific fuel consumption|specific fuel consumption]] of an engine will always be greater than its rate of production of the kinetic energy of motion.

The greatest energy source by far consists of mass itself. This energy, ''E = mc2,'' where ''m = ρV,'' ''ρ'' is the mass per unit volume, ''V'' is the volume of the mass itself and ''c'' is the speed of light. This energy, however, can be released only by the processes of [[nuclear fission]] (.1%), [[nuclear fusion]] (1%),{{Citation needed|date=September 2012}} or the annihilation of some or all of the matter in the volume ''V'' by matter-[[antimatter]] collisions (100%). Nuclear reactions cannot be realized by chemical reactions such as combustion. Although greater matter densities can be achieved, the density of a [[neutron star]] would approximate the most dense system capable of matter-antimatter annihilation possible. A [[black hole]], although denser than a neutron star, doesn't have an equivalent anti-particle form.

The highest density sources of energy aside from antimatter are [[nuclear fusion|fusion]] and [[Nuclear fission|fission]]. Fusion includes energy from the sun which will be available for billions of years (in the form of [[sunlight]]) but so far (2011), sustained [[fusion power]] production continues to be elusive. Fission of uranium and thorium in [[nuclear power]] plants will be available for a long time due to the vast supply of the element on earth,{{Citation needed|date=March 2013}} though the full potential of this source can only be realised through [[breeder reactor]]s, which are, apart from the [[BN-600 reactor]], not yet used commercially.<ref name="cohen">{{cite web|url=http://www-formal.stanford.edu/jmc/progress/cohen.html |title=Facts from Cohen |publisher=Formal.stanford.edu |date=2007-01-26 |accessdate=2010-05-07}}</ref> [[Coal]], [[gas]], and [[petroleum]] are the current primary energy sources in the U.S.<ref>{{cite web|url=http://www.eia.doe.gov/emeu/aer/pecss_diagram.html|archiveurl=http://web.archive.org/web/20100506022627/http://www.eia.doe.gov/emeu/aer/pecss_diagram.html|archivedate=2010-05-06 |title=U.S. Energy Information Administration (EIA) - Annual Energy Review |publisher=Eia.doe.gov |date=2009-06-26 |accessdate=2010-05-07}}</ref> but have a much lower energy density. Burning local [[biomass]] fuels supplies household energy needs ([[Biomass Cook Stoves|cooking fires]], [[oil lamp]]s, etc.) worldwide.

Energy density (how much energy you can carry) does not tell you about [[energy conversion efficiency]] (net output per input) or [[embodied energy]] (what the energy output costs to provide, as [[energy industry|harvesting]], [[refinery|refining]], distributing, and dealing with [[pollution]] all use energy). Like any process occurring on a large scale, intensive energy use impacts the world. For example, [[climate change]], [[nuclear waste]] storage, and [[deforestation]] may be some of the consequences of supplying our growing energy demands from carbohydrate fuels, nuclear fission, or biomass.

No single energy storage method boasts the best in [[Power-to-weight ratio|specific power]], [[specific energy]], and energy density. [[Peukert's Law]] describes how the amount of useful energy that can be obtained (for a lead-acid cell) depends on how quickly we pull it out. To maximize both specific energy and energy density, one can compute the [[specific energy density]] of a substance by multiplying the two values together, where the higher the number, the better the substance is at storing energy efficiently.

Gravimetric and volumetric energy density of some fuels and storage technologies (modified from the [[Gasoline]] article):
:Note: Some values may not be precise because of [[isomers]] or other irregularities. See [[Heating value]] for a comprehensive table of specific energies of important fuels.
:Note: Also it is important to realise that generally the density values for chemical fuels do not include the weight of oxygen required for combustion. This is typically two oxygen atoms per carbon atom, and one per two hydrogen atoms. The [[atomic weight]] of carbon and oxygen are similar, while hydrogen is much lighter than oxygen. Figures are presented this way for those fuels where in practice air would only be drawn in locally to the burner. This explains the apparently lower energy density of materials that already include their own oxidiser (such as gunpowder and TNT), where the mass of the oxidiser in effect adds dead weight, and absorbs some of the energy of combustion to dissociate and liberate oxygen to continue the reaction. This also explains some apparent anomalies, such as the energy density of a sandwich appearing to be higher than that of a stick of dynamite.

{{cleanup|section|date=October 2008}}



===Energy densities ignoring external components===
This table lists energy densities of systems that require external components, such as oxidisers or a heat sink or source. These figures do not take into account the mass and volume of the required components as they are assumed to be freely available and present in the atmosphere. Such systems cannot be compared with self-contained systems. These values may not be computed at the same reference conditions. Most of them seem to be higher heating value (HHV).

{|class="wikitable sortable" style="text-align: right;"
|+ Energy densities of energy media
!Storage type
!Specific energy (MJ/kg)
!Energy density (MJ/L)
!Peak recovery efficiency %
!Practical recovery efficiency %
|-
|align=left | [[Antimatter]] || {{nowrap|1.80e11}} || {{nowrap|9.266032e104}} || ||
|-
|align=left | [[Planck energy|Planck energy density]] || {{nowrap|8.99e10}} || {{nowrap|4.633016e104}} || ||
|-
|align=left | [[Liquid hydrogen|Hydrogen, liquid]]<ref name="H2">Hydrogen properties [http://www1.eere.energy.gov/hydrogenandfuelcells/tech_validation/pdfs/fcm01r0.pdf Hydrogen Properties]. Retrieved 2011-11-30.</ref> || 141.86 || 8.491 || ||
|-
|align=left | [[Compressed gaseous hydrogen|Hydrogen, at 690 bar and 15°C]]<ref name="H2"/> || 141.86|| 4.5 || ||
|-
|align=left|[[Gaseous hydrogen|Hydrogen, gas]]<ref name="H2"/> || 141.86|| 0.01005 || ||
|-
|align=left | [[Diborane]]<ref>Greenwood, Norman N.; Earnshaw, Alan (1997), Chemistry of the Elements (2nd ed) (page 164)</ref> || 78.2 || || ||
|-
|align=left|[[Beryllium]] ||67.6||125.1|| ||
|-
|align=left|[[Lithium borohydride]] ||65.2||43.4|| ||
|-
|align=left|[[Boron]]<ref>{{cite web|url=http://www.eagle.ca/~gcowan/boron_blast.html#TOC |title=Boron: A Better Energy Carrier than Hydrogen? (28 February 2009) |publisher=Eagle.ca |date= |accessdate=2010-05-07}}</ref> ||58.9||137.8|| ||
|-
|align=left|[[Methane]] (1.013 bar, 15°C) ||55.6||0.0378 || ||
|-
|align=left|[[Natural gas]] |||53.6<ref name="ngau">Envestra Limited. [http://www.natural-gas.com.au/about/references.html Natural Gas]. Retrieved 2008-10-05.</ref>||0.0364|| ||
|-
|align=left|[[Liquefied natural gas|LNG]] (NG at −160°C)|||53.6<ref name="ngau"/>||22.2|| ||
|-
|align=left|[[Compressed natural gas|CNG]] (NG compressed to 250 bar/~3,600 psi) || 53.6<ref name="ngau"/> || 9 ||
|-
|align=left | [[Liquefied petroleum gas|LPG]] [[propane]]<ref name="IOR">IOR Energy. [http://web.archive.org/web/20100924142555/http://www.ior.com.au/ecflist.html List of common conversion factors (Engineering conversion factors)]. Retrieved 2008-10-05.</ref> || 49.6 || 25.3 || ||
|-
|align=left| [[Liquefied petroleum gas|LPG]] [[butane]]<ref name="IOR"/> || 49.1 || 27.7 || ||
|-
|align=left|[[Gasoline|Gasoline (petrol)]]<ref name="IOR"/> || 46.4 || 34.2 || ||
|-
|align=left|[[Polypropylene]] plastic||46.4<ref name="aquafoam"/>||41.7|| ||
|-
|align=left|[[Polyethylene]] plastic||46.3<ref name="aquafoam">{{cite web|url=http://www.aquafoam.com/papers/selection.pdf |title=ALTERNATE DAILY COVER MATERIALS AND SUBTITLE D - THE SELECTION TECHNIQUE |author=Paul A. Kittle, Ph.D |publisher= |date= |accessdate=2012-01-25}}</ref>||42.6|| ||
|-
|align=left|[[Crude oil]] (according to the definition of [[ton of oil equivalent]])||46.3||37<ref name="ngau"/>|| ||
|-
|align=left|[[Diesel fuel]]/residential [[heating oil]] <ref name="IOR"/>||46.2||37.3|| ||
|-
|align=left|[[100LL]] Avgas ||44.0<ref>{{cite web|url=http://www-static.shell.com/static/aus/downloads/aviation/avgas_100ll_pds.pdf |title=537.PDF |format=PDF |date=June 1993 |accessdate=2012-01-25}}</ref>||31.59|| ||
|-
|align=left|[[Gasohol]] E10 (10% ethanol 90% gasoline by volume)||43.54||33.18|| ||
|-
|align=left|[[Lithium]] ||43.1||23.0|| ||
|-
|align=left|[[Jet fuel|Jet A]] [[aviation fuel]]<ref>{{cite web|url=http://hypertextbook.com/facts/2003/EvelynGofman.shtml |title=Energy Density of Aviation Fuel |publisher=Hypertextbook.com |date= |accessdate=2010-05-07}}</ref>/[[kerosene]]||42.8||33|| ||
|-
|align=left|[[Biodiesel]] oil (vegetable oil)||42.20||33|| ||
|-
|align=left|[[2,5-Dimethylfuran|DMF]] (2,5-dimethylfuran){{Clarify|date=February 2009|need a quote from the cite containing "42" and "37.8" or equivalent in wh/kg and wh/litre}} ||42<ref>{{cite web|author=Nature |url=http://www.nature.com/nature/journal/v447/n7147/abs/nature05923.html |title=Production of dimethylfuran for liquid fuels from biomass-derived carbohydrates : Abstract |publisher=Nature |date= |accessdate=2010-05-07}}</ref> ||37.8|| ||
|-
|align=left|[[Polystyrene]] plastic||41.4<ref name="aquafoam"/>||43.5|| ||
|-
|align=left|[[Fatty acid metabolism|Body fat metabolism]]||38||35|||22<ref name="JLE5">{{cite web |author=Justin Lemire-Elmore |title=The Energy Cost of Electric and Human-Powered Bicycles |url=http://www.ebikes.ca/sustainability/Ebike_Energy.pdf |page=5 |quote=properly trained athlete will have efficiencies of 22 to 26% |date=2004-04-13 |accessdate=2009-02-26}}</ref>||
|-
|align=left|[[Butanol fuel|Butanol]]||36.6||29.2|| ||
|-
|align=left|Gasohol [[E85]] (85% ethanol 15% gasoline by volume)||33.1||25.65|| ||
|-
|align=left|[[Graphite]] ||32.7||72.9|| ||
|-
|align=left|[[Coal]], [[anthracite]]<ref name='fisher'>{{cite web | last = Fisher | first = Juliya | title = Energy Density of Coal | work = The Physics Factbook | url = http://hypertextbook.com/facts/2003/JuliyaFisher.shtml|year=2003|accessdate = 2006-08-25 }}</ref>||32.5||72.4{{dubious|date=January 2012}}||36||
|-
|align=left|[[Silicon]] <ref>[http://web.archive.org/web/20090327030002/http://www.dbresearch.com/PROD/DBR_INTERNET_EN-PROD/PROD0000000000079095.pdf Silicon as an intermediary between renewable energy and hydrogen]</ref> ||32.2||75.1|| ||
|-
|align=left|[[Aluminum]] ||31.0||83.8|| ||
|-
|align=left|[[Ethanol]]||30||24|| ||
|-
|align=left|[[Polyester]] plastic||26.0 <ref name="aquafoam"/>||35.6|| ||
|-
|align=left|[[Magnesium]] ||24.7||43.0|| ||
|-
|align=left|[[Coal]], [[Bitumen|bituminous]]<ref name='fisher'/> ||24||20|| ||
|-
|align=left|[[Polyethylene terephthalate|PET]] plastic||23.5 (impure)<ref>{{cite web|url=http://www.payne-worldwide.com/documents/cms/Elite_bloc_msds.pdf |title=Elite_bloc.indd |format=PDF |date= |accessdate=2010-05-07}}</ref> || || ||
|-
|align=left|[[Methanol]]||19.7||15.6|| ||
|-
|align=left|[[Hydrazine]] (toxic) combusted to N2+H2O||19.5||19.3|| ||
|-
|align=left|Liquid [[ammonia]] (combusted to N2+H2O)||18.6||11.5|| ||
|-
|align=left|[[PVC]] plastic ([[Polyvinyl chloride#Dioxins|improper combustion toxic]]){{Clarify|date=October 2008}}||18.0<ref name="aquafoam"/>||25.2|| ||
|-
|align=left|[[Wood]]<ref>{{cite web|url=http://www.woodgas.com/fuel_densities.htm|archiveurl=http://web.archive.org/web/20100110042311/http://www.woodgas.com/fuel_densities.htm|archivedate=2010-01-10 |title=Biomass Energy Foundation: Fuel Densities |publisher=Woodgas.com |date= |accessdate=2010-05-07}}</ref> ||18.0 || || ||
|-
|align=left|[[Peat]] [[briquette]] <ref>{{cite web|url=http://www.bnm.ie/files/20061124040716_peat_for_energy.pdf|archiveurl=http://web.archive.org/web/20071119083231/http://www.bnm.ie/files/20061124040716_peat_for_energy.pdf|archivedate=2007-11-19 |title=Bord na Mona, Peat for Energy |publisher=Bnm.ie |date= |accessdate=2012-01-25}}</ref> ||17.7|| || ||
|-
|align=left|[[Fatty acid metabolism|Sugars, carbohydrates, and protein metabolism]]{{Citation needed|date=February 2009|reason=Justin Lemire-Elmore PDF does not specify type of food nor fatty acids nor dextrose, so specific cite needed with page number and precise quotes}}||17||26.2([[dextrose]])|||2222<ref>{{cite web|url=http://www.ebikes.ca/sustainability/Ebike_Energy.pdf |title=The Energy Cost of Electric and Human-Powered Bicycle |author=Justin Lemire-Elmor |publisher= |date=April 13, 2004 |accessdate=2012-01-25}}</ref> ||
|-
|align=left|[[Calcium]]{{Citation needed|date=November 2008}}||15.9||24.6|| ||
|-
|align=left|[[Glucose]]||15.55||23.9|| ||
|-
|align=left|Dry [[cow dung]] and [[Manure#Uses of manure|cameldung]]||15.5<ref>{{cite web|url=http://www.davdata.nl/math/energy.html |title=energy buffers |publisher=Home.hccnet.nl |date= |accessdate=2010-05-07}}</ref> || || ||
|-
|align=left|[[Coal]], [[lignite]]{{Citation needed|date=November 2008}}||14.0|| || ||
|-
|align=left|[[Sodium]] (burned to wet [[sodium hydroxide]])||13.3||12.8|| ||
|-
|align=left|Sod [[peat]] ||12.8|| || ||
|-
|align=left|[[Nitromethane]] ||11.3|| || ||
|-
|align=left|[[Sulfur]] (burned to [[sulfur dioxide]])<ref name='Wignall'>Anne Wignall and Terry Wales. [http://www.wignallandwales.co.nz/Chem-12-WB/Sample-chapter.pdf Chemistry 12 Workbook, page 138]. Pearson Education NZ ISBN 978-0-582-54974-6</ref> ||9.23||19.11 ||
|-
|align=left|[[Sodium]] (burned to dry [[sodium oxide]])||9.1||8.8|| ||
|-
|align=left | [[Lithium air battery|Battery, lithium-air rechargeable]]||9.0<ref>{{cite journal |url=http://pubs.rsc.org/en/content/articlelanding/2011/ee/c1ee01496j |title=All-carbon-nanofiber electrodes for high-energy rechargeable Li–O2 batteries |first=Robert R. |last=Mitchell |coauthors=Betar M. Gallant; Carl V. Thompson; Yang Shao-Horn |journal=Energy & Environmental Science |year=2011 |volume=4 |pages=2952–2958 |doi=10.1039/C1EE01496J}}</ref> || || ||
|-
|align=left|[[Household waste]]|||8.0<ref>David E. Dirkse. [http://www.davdata.nl/math/energy.html energy buffers]. "household waste 8..11 MJ/kg"</ref>|| || ||
|-
|align=left|[[Zinc]] ||5.3||38.0|| ||
|-
|align=left|[[Iron]] (burned to [[iron(III) oxide]])||5.2||40.68|| ||
|-
|align=left|[[PTFE|Teflon]] plastic (combustion toxic, but flame retardant)||5.1||11.2|| ||
|-
|align=left|[[Iron]] (burned to [[iron(II) oxide]])||4.9||38.2|| ||
|-
| align=left | [[ANFO]] || 3.7 || ||
|-
|align=left | [[Zinc-air battery|Battery, zinc-air]]<ref name="duracell-za-tech">{{cite web|url=http://www.duracell.com/oem/primary/Zinc/zinc_air_tech.asp|archiveurl=http://web.archive.org/web/20090127030703/http://www.duracell.com/oem/primary/Zinc/zinc_air_tech.asp|archivedate=2009-01-27|accessdate=2009-04-21|publisher=[[Duracell]]|title=Technical bulletin on Zinc-air batteries}}</ref> || 1.59 || 6.02 || ||
|-
|align=left | [[Liquid nitrogen economy|Liquid nitrogen]]{{Clarify|date=November 2008}}||0.77<ref name="Knowlen">C. Knowlen, A.T. Mattick, A.P. Bruckner and A. Hertzberg, [http://web.archive.org/web/20081217082655/http://www.aa.washington.edu/AERP/cryocar/Papers/sae98.pdf "High Efficiency Conversion Systems for Liquid Nitrogen Automobiles"], Society of Automotive Engineers Inc, 1988.</ref> || 0.62 || ||
|-
|align=left | [[Compressed air]] at 300 bar (potential energy) || 0.5 || 0.2 || || >50%{{Citation needed|date=February 2010}}
|-
|align=left|[[Enthalpy of fusion|Latent heat of fusion]] of ice{{Citation needed|date=June 2009}} (thermal)||0.335||0.335|| ||
|-
|align=left|[[Hydroelectricity|Water at 100 m dam height]] (potential energy)||0.001||0.001|| ||8585-90%{{Citation needed|date=May 2009}}
|- class="sortbottom"
!Storage type
!Energy density by mass (MJ/kg)
!Energy density by volume (MJ/[[Liter|L]])
!Peak recovery efficiency %
!Practical recovery efficiency %
|}

Divide [[joule]] [[metre]]−3 with 109 to get MJ [[Liter|L]]−1.

==Energy density of electric and magnetic fields==

[[Electric field|Electric]] and [[magnetic field]]s store energy. In a vacuum, the (volumetric) energy density (in SI units) is given by

:<math> U = \frac{\varepsilon_0}{2} \mathbf{E}^2 + \frac{1}{2\mu_0} \mathbf{B}^2 </math>

where '''E''' is the [[electric field]] and '''B''' is the [[magnetic field]]. The solution will be in Joules per cubic metre. In the context of [[magnetohydrodynamics]], the physics of conductive fluids, the magnetic energy density behaves like an additional [[pressure]] that adds to the [[kinetic theory of gas|gas pressure]] of a [[plasma (physics)|plasma]].

In normal (linear) substances, the energy density (in SI units) is

:<math> U = \frac{1}{2} ( \mathbf{E} \cdot \mathbf{D} + \mathbf{H} \cdot \mathbf{B} ) </math>

where '''D''' is the [[electric displacement field]] and '''H''' is the [[Effective magnetic field|magnetizing field]].

==See also==
{{Portal|Energy}}
* [[Energy density Extended Reference Table]]
* [[Power density]] and specifically
** [[Power-to-weight ratio]]
* [[Orders of magnitude (specific energy)]]
* [[Figure of merit]]
* [[Energy content of biofuel]]
* [[Heat of combustion]]
* [[Heating value]]
* [[Rechargeable battery]]
* [[Specific impulse]]

==Footnotes==
{{Reflist|colwidth=30em}}

==External references==

===Density data===
*{{note|att}} "Aircraft Fuels." ''Energy, Technology and the Environment'' Ed. Attilio Bisio. Vol. 1. New York: John Wiley and Sons, Inc., 1995. 257–259

*"[http://www1.eere.energy.gov/vehiclesandfuels/pdfs/deer_2002/session1/2002_deer_eberhardt.pdf Fuels of the Future for Cars and Trucks]" - Dr. James J. Eberhardt - Energy Efficiency and Renewable Energy, U.S. Department of Energy - 2002 Diesel Engine Emissions Reduction (DEER) Workshop San Diego, California - August 25–29, 2002

===Energy storage===
*[http://www.tinaja.com/h2gas01.asp energy fundamentals]

===Books===
*''The Inflationary Universe: The Quest for a New Theory of Cosmic Origins'' by Alan H. Guth (1998) ISBN 0-201-32840-2
*''Cosmological Inflation and Large-Scale Structure'' by Andrew R. Liddle, David H. Lyth (2000) ISBN 0-521-57598-2
*Richard Becker, "Electromagnetic Fields and Interactions", Dover Publications Inc., 1964

{{DEFAULTSORT:Energy Density}}
[[Category:Energy]]
[[Category:Density]]