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2013-11-12T02:41:04Z

134.84.1.157:

The '''Hazen–Williams equation''' is an [[empirical relationship]] which relates the flow of water in a pipe with the physical properties of the pipe and the pressure drop caused by friction. It is used in the design of [[water pipe]] systems<ref>{{Cite web|url=http://docs.bentley.com/en/HMFlowMaster/FlowMasterHelp-06-05.html |title=Hazen–Williams Formula |accessdate=2008-12-06}}</ref> such as [[fire sprinkler system]]s,<ref>{{Cite web|url=http://www.canutesoft.com/index.php/Basic-Hydraulics-for-fire-protection-engineers/Hazen-Williams-formula-for-use-in-fire-sprinkler-systems.html|title=Hazen–Williams equation in fire protection systems|date=27 January 2009|publisher=Canute LLP|accessdate=2009-01-27}} {{Dead link|date=October 2010|bot=H3llBot}}</ref> [[water supply network]]s, and [[irrigation]] systems. It is named after [[Allen Hazen]] and Gardner Stewart Williams.

The Hazen–Williams equation has the advantage that the coefficient ''C'' is not a function of the [[Reynolds number]], but it has the disadvantage that it is only valid for [[water]]. Also, it does not account for the temperature or [[viscosity]] of the water.<ref>{{Cite book|last=Brater|first=Errest|coauthors=King Horace|others=Lindell E. James|title=Handbook of Hydraulics|publisher=Mc Graw Hill|location=New York|year=1996|edition=Seventh Edition|pages=6.29|chapter=6|isbn=0-07-007247-7}}</ref>

==General form==
[[Henri Pitot]] discovered that the velocity of a fluid was proportional to the square root of its head in the early 18th century. It takes energy to push a fluid through a pipe, and [[Antoine de Chézy]] discovered that the head loss was proportional to the velocity squared.<ref>{{Citation |last=Walski |first=Thomas M. |title=A history of water distribution |journal=Journal of the American Water Works Association |publisher=American Water Works Association |volume=98 |issue=3 |pages=110–121 |date=March 2006 |doi= }}, p. 112.</ref> Consequently, the [[Chézy formula]] relates hydraulic slope ''S'' (head loss per unit length) to the fluid velocity ''V'' and [[hydraulic radius]] ''R'':
:<math>V=C\sqrt{RS}=C\, R^{0.5}\, S^{0.5}</math>

The variable ''C'' expresses the proportionality, but the value of ''C'' is not a constant. In 1838 and 1839, [[Gotthilf Hagen]] and [[Jean Léonard Marie Poiseuille]] independently determined a head loss equation for [[laminar flow]], the [[Hagen–Poiseuille equation]]. Around 1845, [[Julius Weisbach]] and [[Henry Darcy]] developed the [[Darcy–Weisbach equation]].<ref>{{Harvnb|Walski|2006|p=112}}</ref>

The Darcy-Weisbach equation was difficult to use because the friction factor was difficult to estimate.<ref>{{Harvnb|Walski|2006|p=113}}</ref> In 1906, Hazen and Williams provided an [[empirical relationship|empirical formula]] that was easy to use. The general form of the equation relates the mean velocity of water in a pipe with the geometric properties of the pipe and slope of the energy line.

:<math>V = k\, C\, R^{0.63}\, S^{0.54}</math>

where:
* ''V'' is velocity
* ''k'' is a conversion factor for the unit system (k = 1.318 for US customary units, k = 0.849 for SI units)
* ''C'' is a roughness coefficient
* ''R'' is the [[hydraulic radius]]
* ''S'' is the slope of the energy line ([[head loss]] per length of pipe or hf/L)

The equation is similar to the Chézy formula but the exponents have been adjusted to better fit data from typical engineering situations. A result of adjusting the exponents is that the value of ''C'' appears more like a constant over a wide range of the other parameters.<ref>{{Harvnb|Williams|Hazen|1914|p=1}}, stating "Exponents can be selected, however, representing approximate average conditions, so that the value of ''c'' for a given condition of surface will vary so little as to be practically constant."</ref>

The conversion factor ''k'' was chosen so that the values for ''C'' were the same as in the Chézy formula for the typical hydraulic slope of ''S''=0.001.<ref>{{harvnb|Williams|Hazen|1914|p=1}}</ref> The value of ''k'' is 0.001-0.04.<ref>{{harvnb|Williams|Hazen|1914|pp=1–2}}</ref>

Typical ''C'' factors used in design, which take into account some increase in roughness as pipe ages are as follows:<ref name="ET">{{Citation |url=http://www.engineeringtoolbox.com/hazen-williams-coefficients-d_798.html |title=Hazen-Williams Coefficients |publisher=Engineering ToolBox |accessdate=7 October 2012 |doi= }}</ref>

{| class="wikitable sortable"
|-
! Material !! C Factor low !! C Factor high !! Reference
|-
| [[Eternit|Asbestos-cement]] || 140 || 140 || -
|-
| [[Cast iron]] new || 130 || 130 || <ref name="ET"/>
|-
| [[Cast iron]] 10 years || 107 || 113 || <ref name="ET"/>
|-
| [[Cast iron]] 20 years || 89|| 100 || <ref name="ET"/>
|-
| [[Cast iron]] 30 years || 75 || 90 || <ref name="ET"/>
|-
| [[Cast iron]] 40 years || 64 || 83 || <ref name="ET"/>
|-
| [[Cement-Mortar Lined Ductile Iron Pipe]] || 140 || 140 || -
|-
| [[Concrete]] || 100 || 140 || <ref name="ET"/>
|-
| [[Copper]] || 130 || 140 || <ref name="ET"/>
|-
| [[Steel]] || 90 || 110 || -
|-
| [[Galvanized iron]] || 120 || 120 || <ref name="ET"/>
|-
| [[Polyethylene]] || 140 || 140 || <ref name="ET"/>
|-
| [[Polyvinyl chloride]] (PVC) || 150 || 150 || <ref name="ET"/>
|-
| [[Fibre-reinforced plastic]] (FRP) || 150 || 150 || <ref name="ET"/>
|}

==Pipe equation==
The general form can be specialized for full pipe flows. Taking the general form
:<math>V = k\, C\, R^{0.63}\, S^{0.54}</math>
and exponentiating each side by <math>1/0.54</math> gives (rounding exponents to 2 decimals)
:<math>V^{1.85} = k^{1.85}\, C^{1.85}\, R^{1.17}\, S</math>
Rearranging gives
:<math>S = {V^{1.85} \over k^{1.85}\, C^{1.85}\, R^{1.17}} </math>
The flow rate ''Q'' = ''V'' ''A'', so
:<math>S = {V^{1.85} A^{1.85}\over k^{1.85}\, C^{1.85}\, R^{1.17}\, A^{1.85}} = {Q^{1.85}\over k^{1.85}\, C^{1.85}\, R^{1.17}\, A^{1.85}} </math>
The [[hydraulic radius]] ''R'' (which is different from the geometric radius ''r'') for a full pipe of geometric diameter ''d'' is ''d''/4; the pipe's cross sectional area ''A'' is <math>\pi d^2 / 4</math>, so
:<math>S = {4^{1.17}\, 4^{1.85}\,Q^{1.85}\over \pi^{1.85}\,k^{1.85}\, C^{1.85}\, d^{1.17}\, d^{3.70}}
= {4^{3.02}\,Q^{1.85}\over \pi^{1.85}\,k^{1.85}\, C^{1.85}\, d^{4.87}}
= { 4^{3.02} \over \pi^{1.85}\,k^{1.85}} {Q^{1.85}\over C^{1.85}\, d^{4.87}}
= { 7.916 \over k^{1.85}} {Q^{1.85}\over C^{1.85}\, d^{4.87}}
</math>

===U.S. customary units (Imperial)===

When used to calculate the pressure drop using the [[US customary units]] system, the equation is:<ref>2007 version of NFPA 13: Standard for the Installation of Sprinkler Systems, page 13-213, eqn 22.4.2.1</ref>
:<math>S_{\mathrm{psi\ per\ foot}} = \frac{P_d}{L} = \frac{4.52\ Q^{1.85}}{C^{1.85}\ d^{4.87}}</math>

where:
*''S''psi per foot = frictional resistance (pressure drop per foot of pipe) in psig/ft ([[Pound-force per square inch|pounds per square inch gauge pressure]] per foot)
*''Pd = pressure drop over the length of pipe in psig ([[Pound-force per square inch|pounds per square inch gauge pressure]])
*''L'' = length of pipe in feet
*''Q'' = flow, gpm ([[gallons per minute]])
*''C'' = pipe roughness coefficient
*''d'' = inside pipe diameter, in (inches)

:'''Note:''' Caution with U S Customary Units is advised. The equation for head loss in pipes, also referred to as slope, S, expressed in "feet per foot of length" vs. in 'psi per foot of length' as described above, with the inside pipe diameter, d, being entered in feet vs. inches, and the flow rate, Q, being entered in cubic feet per second, cfs, vs. gallons per minute, gpm, appears very similar. However, the constant is 4.75 vs. the 4.52 constant as shown above in the formula as arranged by NFPA for sprinkler system design. The exponents and the Hazen-Williams "C" values are unchanged.

===SI units===
When used to calculate the head loss with the [[International System of Units]], the equation becomes:<ref name="urlrpitt.eng.ua.edu">{{Cite web|url=http://rpitt.eng.ua.edu/Class/Water%20Resources%20Engineering/M3e%20Comparison%20of%20methods.pdf |title=Comparison of Pipe Flow Equations and Head Losses in Fittings |format=PDF |accessdate=2008-12-06}}</ref>

:<math>S = \frac{h_f}{L} = \frac{10.67\ Q^{1.85}}{C^{1.85}\ d^{4.87}}</math>

where:

* ''S'' = Hydraulic slope
* hf = [[Hydraulic head#Head_loss|head loss]] in meters (water) over the length of pipe
* ''L'' = length of pipe in meters
* ''Q'' = volumetric flow rate, m3/s (cubic meters per second)
* ''C'' = pipe roughness coefficient
* ''d'' = inside pipe diameter, m (meters)

:Note: pressure drop can be computed from head loss as ''hf'' × the unit weight of water (e.g., 9810 N/m3 at 4 deg C)

==See also==
*[[Fluid dynamics]]
*[[Friction]]
*[[Pressure]]
*[[Prony equation]]
*[[Volumetric flow rate]]
*[[Water pipe]]

==References==
{{reflist|30em}}

*{{Citation
|last= Finnemore
|first= E. John
|last2= Franzini
|first2= Joseph B.
|title= Fluid Mechanics
|edition= 10th
|year= 2002
|publisher= McGraw Hill
|isbn=
|doi=}}
*{{Citation
|last= Mays
|first= Larry W.
|title= Hydraulic Design Handbook
|year= 1999
|publisher= McGraw Hill
|isbn=
|doi=}}
*{{Citation
|last= Watkins
|first= James A.
|title= Turf Irrigation Manual
|edition= 5th
|year= 1987
|publisher= Telsco
|isbn=
|doi=}}
* {{Citation
|last2= Hazen
|first2= Allen
|author-link=
|last1= Williams
|first1= Gardner Stewart
|author2-link=
|title= Hydraulic tables: showing the loss of head due to the friction of water flowing in pipes, aqueducts, sewers, etc. and the discharge over weirs
|edition=first
|year=1905
|location= New York
|publisher= John Wiley and Sons
|url=http://books.google.com/books?id=dE9DAAAAIAAJ&printsec=frontcover&hl=en&sa=X&ved=0CF0Q6AEwAg#v=onepage&f=false
|doi=}}
*[http://catalog.hathitrust.org/Record/005745859 Williams and Hazen, Second edition, 1909]
* {{Citation
|last2= Hazen
|first2= Allen
|author-link=
|last1= Williams
|first1= Gardner Stewart
|author2-link=
|title= Hydraulic tables: the elements of gagings and the friction of water flowing in pipes, aqueducts, sewers, etc., as determined by the Hazen and Williams formula and the flow of water over sharp-edged and irregular weirs, and the quantity discharged as determined by Bazin's formula and experimental investigations upon large models.
|edition= 2nd revised and enlarged
|year=1914
|location= New York
|publisher= John Wiley and Sons
|isbn= |oclc= |url=http://books.google.com/books?id=7m4gAAAAMAAJ&pg=PR1&lpg=PR1&source=bl&ots=DUCACHZeQU&hl=en&sa=X&ved=0CD8Q6AEwAg
|doi=}}
* {{Citation
|last2= Hazen
|first2= Allen
|author-link=
|last1= Williams
|first1= Gardner Stewart
|author2-link=
|title= Hydraulic tables: the elements of gagings and the friction of water flowing in pipes, aqueducts, sewers, etc., as determined by the Hazen and Williams formula and the flow of water over sharp-edged and irregular weirs, and the quantity discharged as determined by Bazin's formula and experimental investigations upon large models.
|edition= 3rd
|year=1920
|location= New York
|publisher= John Wiley and Sons
|isbn= |oclc=1981183
|doi=}}

==External links==
*[http://www.engineeringtoolbox.com/hazen-williams-water-d_797.html Engineering Toolbox reference]
*[http://www.engineeringtoolbox.com/hazen-williams-coefficients-d_798.html Engineering toolbox Hazen–Williams coefficients]
*[http://www.calctool.org/CALC/eng/civil/hazen-williams_g Online Hazen–Williams calculator for gravity-fed pipes.]
*[http://www.calctool.org/CALC/eng/civil/hazen-williams_p Online Hazen–Williams calculator for pressurized pipes.]
*http://books.google.com/books?id=DxoMAQAAIAAJ&pg=PA736&hl=en&sa=X&ved=0CEsQ6AEwAA#v=onepage&f=false
*http://books.google.com/books?id=RAMX5xuXSrUC&pg=PA145&lpg=PA145&source=bl&ots=RucWGKXVYx&hl=en&sa=X&ved=0CDkQ6AEwAjgU States pocket calculators and computers make calculations easier. H-W is good for smooth pipes, but Manning better for rough pipes (compared to D-W model).
{{Use dmy dates|date=September 2010}}

{{DEFAULTSORT:Hazen-Williams Equation}}
[[Category:Equations of fluid dynamics]]
[[Category:Piping]]
[[Category:Plumbing]]
[[Category:Hydraulics]]
[[Category:Irrigation]]

Interaction picture

2013-10-24T02:09:56Z

134.84.140.54:

In [[logic]], a '''logical framework''' provides a means to define (or present) a logic as a signature in a higher-order [[type theory]] in such a way that [[provability]] of a formula in the original logic reduces to a [[type inhabitation]] problem in the framework type theory.<ref name="Jacobs2001">{{cite book|author=Bart Jacobs|title=Categorical Logic and Type Theory|year=2001|publisher=Elsevier|isbn=978-0-444-50853-9|page=598}}</ref><ref name="Gabbay1994">{{cite book|editor=Dov M. Gabbay|title=What is a logical system?|url=http://books.google.com/books?id=XqCu4XjHrIQC&pg=PA382|year=1994|publisher=Clarendon Press|isbn=978-0-19-853859-2|page=382}}</ref> This approach has been used successfully for (interactive) [[automated theorem proving]]. The first logical framework was [[Automath]], however the name of the idea comes from the more widely known Edinburgh Logical Framework, '''LF'''. Several more recent proof tools like [[Isabelle (theorem prover)|Isabelle]] are based on this idea.<ref name="Jacobs2001"/> Unlike a direct embedding, the logical framework approach allows many logics to be embedded in the same type system.<ref name="BoveBarbosa2009">{{cite book|author1=Ana Bove|author2=Luis Soares Barbosa|author3=Alberto Pardo|title=Language Engineering and Rigourous (sic) Software Development: International LerNet ALFA Summer School 2008, Piriapolis, Uruguay, February 24 - March 1, 2008, Revised, Selected Papers|url=http://books.google.com/books?id=YOqHiA5MYpEC&pg=PA48|year=2009|publisher=Springer|isbn=978-3-642-03152-6|pages=48}}</ref>

A logical framework is based on a general treatment of syntax, rules and proofs by means of a [[dependent type theory|dependently typed lambda calculus]]. Syntax is treated in a style similar to, but more general than [[Per Martin-Löf]]'s system of arities.

To describe a logical framework, one must provide the following:

# A characterization of the class of object-logics to be represented;
# An appropriate meta-language;
# A characterization of the mechanism by which object-logics are represented.

This is summarized by:

''‘Framework = Language + Representation’''.

== LF ==

In the case of the '''LF logical framework''', the meta-language is the [[λΠ-calculus|<math>\lambda\Pi</math>-calculus]]. This is a system of first-order dependent function types which are related by the [[propositions as types principle]] to first-order minimal logic. The key features of the <math>\lambda\Pi</math>-calculus are that it consists of entities of three levels: objects, types and families of types. It is [[Impredicativity | predicative]], all well-typed terms are [[strongly normalizing]] and [[Church-Rosser]] and the property of being well-typed is [[Decidability (logic)|decidable]]. However, [[type inference]] is [[Decidability (logic)|undecidable]].

A logic is represented in the '''LF logical framework''' by the judgements-as-types representation mechanism. This is inspired by [[Per Martin-Löf]]'s development of [[Immanuel Kant|Kant]]'s notion of [[judgement (mathematical logic)|judgement]], in the 1983 Siena Lectures. The two higher-order judgements, the hypothetical <math>J\vdash K</math> and the general, <math>\Lambda x\in J. K(x)</math>, correspond to the ordinary and dependent function space, respectively. The methodology of judgements-as-types is that judgements are represented as the types of their proofs. A [[logical system]] <math>{\mathcal L}</math> is represented by its signature which assigns kinds and types to a finite set of constants that represents its syntax, its judgements and its rule schemes. An object-logic's rules and proofs are seen as primitive proofs of hypothetico-general judgements <math>\Lambda x\in C. J(x)\vdash K</math>.

An implementation of the LF logical framework is provided by the [[Twelf]] system at [[Carnegie Mellon University]]. Twelf includes
:* a logic programming engine
:* meta-theoretic reasoning about logic programs (termination, coverage, etc.)
:* an inductive [[meta-logic]]al theorem prover

==References==
{{reflist}}

==See also==

* [[Grammatical Framework]]

==Further reading==
* {{cite book|editor=Helmut Schwichtenberg, Ralf Steinbrüggen|title=Proof and system-reliability|chapter=Logical frameworks—a brief introduction|year=2002|publisher=Springer|isbn=978-1-4020-0608-1|author=[[Frank Pfenning]]|url=http://www.cs.cmu.edu/~fp/papers/mdorf01.pdf}}
*[[Robert Harper (computer scientist)|Robert Harper]], Furio Honsell and [[Gordon Plotkin]]. ''A Framework For Defining Logics''. Journal of the Association for Computing Machinery, 40(1):143-184, 1993
*Arnon Avron, Furio Honsell, Ian Mason and Randy Pollack. ''Using typed lambda calculus to implement formal systems on a machine''. Journal of Automated Reasoning, 9:309-354, 1992.
*Robert Harper. ''An Equational Formulation of LF''. Technical Report, University of Edinburgh, 1988. LFCS report ECS-LFCS-88-67.
*Robert Harper, Donald Sannella and Andrzej Tarlecki. ''Structured Theory Presentations and Logic Representations''. Annals of Pure and Applied Logic, 67(1-3):113-160, 1994.
*Samin Ishtiaq and David Pym. ''A Relevant Analysis of Natural Deduction''. Journal of Logic and Computation 8, 809-838, 1998.
* Samin Ishtiaq and David Pym. ''Kripke Resource Models of a Dependently-typed, Bunched <math>\lambda</math>-calculus''. Journal of Logic and Computation 12(6), 1061-1104, 2002.
* Per Martin-Löf. "[http://www.hf.uio.no/filosofi/njpl/vol1no1/meaning/meaning.html On the Meanings of the Logical Constants and the Justifications of the Logical Laws]." "[[Nordic Journal of Philosophical Logic]]", 1(1): 11-60, 1996.
* Bengt Nordström, Kent Petersson, and Jan M. Smith. ''Programming in Martin-Löf's Type Theory''. Oxford University Press, 1990. (The book is out of print, but [http://www.cs.chalmers.se/Cs/Research/Logic/book/ a free version] has been made available.)
*David Pym. ''A Note on the Proof Theory of the <math>\lambda\Pi</math>-calculus''. Studia Logica 54: 199-230, 1995.
*David Pym and Lincoln Wallen. ''Proof-search in the <math>\lambda\Pi</math>-calculus''. In: G. Huet and G. Plotkin (eds), Logical Frameworks, Cambridge University Press, 1991.
*Didier Galmiche and David Pym. ''Proof-search in type-theoretic languages:an introduction''. Theoretical Computer Science 232 (2000) 5-53.
*Philippa Gardner. ''Representing Logics in Type Theory''. Technical Report, University of Edinburgh, 1992. LFCS report ECS-LFCS-92-227.
*Gilles Dowek. ''The undecidability of typability in the lambda-pi-calculus''. In M. Bezem, J.F. Groote (Eds.), Typed Lambda Calculi and Applications. Volume 664 of ''Lecture Notes in Computer Science'', 139-145, 1993.
*David Pym. ''Proofs, Search and Computation in General Logic''. Ph.D. thesis, University of Edinburgh, 1990.
*David Pym. ''A Unification Algorithm for the <math>\lambda\Pi</math>-calculus.'' Int. J. of Foundations of Computer Science 3(3), 333-378, 1992.

== External links ==
* [http://www.cs.cmu.edu/~fp/lfs-impl.html Specific Logical Frameworks and Implementations] (a list maintained by [[Frank Pfenning]])

[[Category:Logic in computer science]]
[[Category:Type theory]]
[[Category:Proof assistants]]
[[Category:Dependently typed programming]]

Ordered exponential

2013-10-24T01:56:03Z

134.84.140.54: /* Definition */ alginment of punctuation

In [[mathematics]], '''Knuth's up-arrow notation''' is a method of notation for [[large number|very large]] [[integer]]s, introduced by [[Donald Knuth]] in 1976.<ref>{{cite journal |author=Knuth, Donald E. |title=Mathematics and Computer Science: Coping with Finiteness |journal=Science |volume=194 |issue=4271 |year=1976 |pages=1235–1242 |doi=10.1126/science.194.4271.1235 |pmid=17797067}}</ref> It is closely related to the [[Ackermann function]] and especially to the [[hyperoperation]] sequence. The idea is based on the fact that [[multiplication]] can be viewed as [[iteration|iterated]] [[addition]] and [[exponentiation]] as iterated [[multiplication]]. Continuing in this manner leads to [[Tetration|iterated exponentiation]] (tetration) and to the remainder of the hyperoperation sequence, which is commonly denoted using Knuth arrow notation.

==Introduction==

The ordinary arithmetical operations of [[addition]], [[multiplication]] and [[exponentiation]] are naturally extended into a sequence of [[hyperoperations]] as follows.

[[Multiplication]] by a [[natural number]] is defined as iterated [[addition]]:

:<math>
\begin{matrix}
a\times b & = & \underbrace{a+a+\dots+a} \\
& & b\mbox{ copies of }a
\end{matrix}
</math>

For example,

:<math>
\begin{matrix} 4\times 3 & = & \underbrace{4+4+4} & = & 12\\
& & 3\mbox{ copies of }4
\end{matrix}
</math>

[[Exponentiation]] for a natural power <math>b</math> is defined as iterated multiplication, which Knuth denoted by a single up-arrow:

:<math>
\begin{matrix}
a\uparrow b= a^b = & \underbrace{a\times a\times\dots\times a}\\
& b\mbox{ multiplied copies of }a
\end{matrix}
</math>

For example,

:<math>
\begin{matrix}
4\uparrow 3= 4^3 = & \underbrace{4\times 4\times 4} & = & 64\\
& 3\mbox{ multiplied copies of }4
\end{matrix}
</math>

To extend the sequence of operations beyond exponentiation, Knuth defined a “double arrow” [[Operator (mathematics)|operator]] to denote iterated exponentiation ([[tetration]]):

:<math>
\begin{matrix}
a\uparrow\uparrow b & = {\ ^{b}a} = & \underbrace{a^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} &
= & \underbrace{a\uparrow (a\uparrow(\dots\uparrow a))}
\\
& & b\mbox{ multiplied copies of }a\uparrow
& & b\mbox{ multiplied copies of }a\uparrow
\end{matrix}
</math>

For example,

:<math>
\begin{matrix}
4\uparrow\uparrow 3 & = {\ ^{3}4} = & \underbrace{4^{4^4}} &
= & \underbrace{4\uparrow (4\uparrow 4)} & = & 4^{256} & \approx & 1.34078079\times 10^{154}&
\\
& & 3\mbox{ multiplied copies of }4\uparrow
& & 3\mbox{ multiplied copies of }4\uparrow
\end{matrix}
</math>

Here and below evaluation is to take place from right to left, as Knuth's arrow operators (just like exponentiation) are defined to be [[Right associative operator|right-associative]].

According to this definition,
:<math>3\uparrow\uparrow 2=3^3=27 </math>
:<math>3\uparrow\uparrow 3=3^{3^3}=3^{27}=7625597484987 </math>
:<math>3\uparrow\uparrow 4=3^{3^{3^3}}=3^{3^{27}}=3^{7625597484987}\approx 1.2580143\times 10^{3638334640024}
</math>
:<math>3\uparrow\uparrow 5=3^{3^{3^{3^3}}}=3^{3^{3^{27}}}=3^{3^{7625597484987}} </math>
:etc.

This already leads to some fairly large numbers, but Knuth extended the notation. He went on to define a “triple arrow” operator for iterated application of the “double arrow” operator (also known as [[pentation]]):

:<math>
\begin{matrix}
a\uparrow\uparrow\uparrow b= &
\underbrace{a_{}\uparrow\uparrow (a\uparrow\uparrow(\dots\uparrow\uparrow a))}\\
& b\mbox{ multiplied copies of }a
\end{matrix}
</math>

followed by a 'quadruple arrow' operator (also known as hexation):

:<math>
\begin{matrix}
a\uparrow\uparrow\uparrow\uparrow b= &
\underbrace{a_{}\uparrow\uparrow\uparrow (a\uparrow\uparrow\uparrow(\dots\uparrow\uparrow\uparrow a))}\\
& b\mbox{ multiplied copies of }a
\end{matrix}
</math>

and so on. The general rule is that an <math>n</math>-arrow operator expands into a right-associative series of (<math>n - 1</math>)-arrow operators. Symbolically,

:<math>
\begin{matrix}
a\ \underbrace{\uparrow_{}\uparrow\!\!\dots\!\!\uparrow}_{n}\ b=
\underbrace{a\ \underbrace{\uparrow\!\!\dots\!\!\uparrow}_{n-1}
\ (a\ \underbrace{\uparrow_{}\!\!\dots\!\!\uparrow}_{n-1}
\ (\dots
\ \underbrace{\uparrow_{}\!\!\dots\!\!\uparrow}_{n-1}
\ a))}_{b\text{ multiplied copies of }a}
\end{matrix}
</math>

Examples:

:<math>3\uparrow\uparrow\uparrow2 = 3\uparrow\uparrow3 = 3^{3^3} = 3^{27}=7,625,597,484,987</math>

:<math>
\begin{matrix}
3\uparrow\uparrow\uparrow3 = 3\uparrow\uparrow3\uparrow\uparrow3 = 3\uparrow\uparrow(3\uparrow3\uparrow3) = &
\underbrace{3_{}\uparrow 3\uparrow\dots\uparrow 3} \\
& 3\uparrow3\uparrow3\mbox{ multiplied copies of }3
\end{matrix}
\begin{matrix}
= & \underbrace{3_{}\uparrow 3\uparrow\dots\uparrow 3} \\
& \mbox{7,625,597,484,987 multiplied copies of 3}
\end{matrix}=\underbrace{3^{3^{3^{3^{\cdot^{\cdot^{\cdot^{\cdot^{3}}}}}}}}}_{7,625,597,484,987}
</math>

The notation <math>a \uparrow^n b</math> is commonly used to denote <math>a \uparrow\uparrow \dots \uparrow b</math> with ''n'' arrows.

==Notation==
In expressions such as <math>a^b</math>, the notation for exponentiation is usually to write the exponent <math>b</math> as a superscript to the base number <math>a</math>. But many environments — such as [[programming language]]s and plain-text [[e-mail]] — do not support superscript typesetting. People have adopted the linear notation <math>a \uparrow b</math> for such environments; the up-arrow suggests 'raising to the power of'. If the character set doesn't contain an up arrow, the [[caret]] ^ is used instead.

The superscript notation <math>a^b</math> doesn't lend itself well to generalization, which explains why Knuth chose to work from the inline notation <math>a \uparrow b</math> instead.

<math>a \uparrow^n b</math> is a shorter alternative notation for n uparrows. Thus <math>a \uparrow^4 b = a \uparrow \uparrow \uparrow \uparrow b</math>.

===Writing out up-arrow notation in terms of powers===
Attempting to write <math>a \uparrow \uparrow b</math> using the familiar superscript notation gives a power tower.

:For example: <math>a \uparrow \uparrow 4 = a \uparrow (a \uparrow (a \uparrow a)) = a^{a^{a^a}}</math>

If ''b'' is a variable (or is too large), the power tower might be written using dots and a note indicating the height of the tower.

:<math>a \uparrow \uparrow b = \underbrace{a^{a^{.^{.^{.{a}}}}}}_{b}</math>

Continuing with this notation, <math>a \uparrow \uparrow \uparrow b</math> could be written with a stack of such power towers, each describing the size of the one above it.

:<math>a \uparrow \uparrow \uparrow 4 = a \uparrow \uparrow (a \uparrow \uparrow (a \uparrow \uparrow a)) =
\underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{a} }}</math>

Again, if ''b'' is a variable or is too large, the stack might be written using dots and a note indicating its height.

:<math>a \uparrow \uparrow \uparrow b =
\left. \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\} b</math>

Furthermore, <math>a \uparrow \uparrow \uparrow \uparrow b</math> might be written using several columns of such stacks of power towers, each column describing the number of power towers in the stack to its left:

:<math>a \uparrow \uparrow \uparrow \uparrow 4 = a \uparrow \uparrow \uparrow (a \uparrow \uparrow \uparrow (a \uparrow \uparrow \uparrow a)) =
\left.\left.\left. \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\}
\underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\}
\underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\}
a</math>

And more generally:

:<math>a \uparrow \uparrow \uparrow \uparrow b =
\underbrace{
\left.\left.\left. \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\}
\underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\}
\cdots \right\}
a
}_{b}</math>

This might be carried out indefinitely to represent <math>a \uparrow^n b</math> as iterated exponentiation of iterated exponentiation for any ''a'', ''n'' and ''b'' (although it clearly becomes rather cumbersome).

====Using tetration====

The [[tetration]] notation <math>^{b}a</math> allows us to make these diagrams slightly simpler while still employing a geometric representation (we could call these ''tetration towers'').

: <math> a \uparrow \uparrow b = { }^{b}a</math>

: <math> a \uparrow \uparrow \uparrow b = \underbrace{^{^{^{^{^{a}.}.}.}a}a}_{b}</math>

: <math> a \uparrow \uparrow \uparrow \uparrow b =
\left. \underbrace{^{^{^{^{^{a}.}.}.}a}a}_{ \underbrace{^{^{^{^{^{a}.}.}.}a}a}_{ \underbrace{\vdots}_{a} }} \right\} b</math>

Finally, as an example, the fourth Ackermann number <math>4 \uparrow^4 4</math> could be represented as:
: <math>\underbrace{^{^{^{^{^{4}.}.}.}4}4}_{ \underbrace{^{^{^{^{^{4}.}.}.}4}4}_{ \underbrace{^{^{^{^{^{4}.}.}.}4}4}_{4} }} =
\underbrace{^{^{^{^{^{4}.}.}.}4}4}_{ \underbrace{^{^{^{^{^{4}.}.}.}4}4}_{ ^{^{^{4}4}4}4 }}</math>

==Generalizations==
Some numbers are so large that multiple arrows of Knuth's up-arrow notation become too cumbersome; then an ''n''-arrow operator <math>\uparrow^n</math> is useful (and also for descriptions with a variable number of arrows), or equivalently, [[hyper operator]]s.

Some numbers are so large that even that notation is not sufficient. [[Graham's number]] is an example. The [[Conway chained arrow notation]] can then be used: a chain of three elements is equivalent with the other notations, but a chain of four or more is even more powerful.

:<math>
\begin{matrix}
a\uparrow^n b & = & \mbox{hyper}(a,n+2,b) & = & a\to b\to n \\
\mbox{(Knuth)} & & & & \mbox{(Conway)}
\end{matrix}
</math>

It is generally suggested that Knuth's arrow should be used for smaller magnitude numbers, and the chained arrow or hyper operators for larger ones.

==Definition==

The up-arrow notation is formally defined by

:<math>
a\uparrow^n b=
\left\{
\begin{matrix}
a\times b, & \mbox{if }n=0; \\
a^b, & \mbox{if }n=1; \\
1, & \mbox{if }b=0; \\
a\uparrow^{n-1}(a\uparrow^n(b-1)), & \mbox{otherwise}
\end{matrix}
\right.
</math>
for all integers <math>a, b, n</math> with <math>b \ge 0, n \ge 1</math>.

All up-arrow operators (including normal exponentiation, <math>a \uparrow b</math>) are [[Associativity#Non-associativity|right associative]], i.e. evaluation is to take place from right to left in an expression that contains two or more such operators. For example, <math>a \uparrow b \uparrow c = a \uparrow (b \uparrow c)</math>, not <math>(a \uparrow b) \uparrow c</math>; for example
 <math>3\uparrow\uparrow 3=3^{3^3}</math> is <math>3^{(3^3)}=3^{27}=7625597484987</math> not <math>\left(3^3\right)^3=27^3=19683.</math>

There is good reason for the choice of this right-to-left order of evaluation. If we used left-to-right evaluation, then <math>a \uparrow\uparrow b</math> would equal
<math>a \uparrow (a \uparrow (b - 1))</math>, so that <math>\uparrow\uparrow</math> would not be an essentially new operation.
Right associativity is also natural because we can rewrite the iterated arrow expression <math>a\uparrow^n\cdots\uparrow^na</math> that appears
in the expansion of <math>a \uparrow^{n + 1}b</math> as
<math>a\uparrow^n\cdots\uparrow^na\uparrow^n1</math>, so that all the <math>a</math>s appear
as left operands of arrow operators. This is significant since the arrow operators are not [[commutativity|commutative]].

Writing <math>(a\uparrow ^m)^b</math> for the ''b''th [[functional power]] of the function <math>f(x)=a\uparrow ^m x</math> we have <math>a\uparrow ^n b = (a\uparrow ^{n-1})^b 1</math>.

The definition could be extrapolated one step, starting with <math>a\uparrow^n b= ab</math> if ''n'' = 0, because exponentiation is repeated multiplication starting with 1. Extrapolating one step more, writing multiplication as repeated addition, is not as straightforward because multiplication is repeated addition starting with 0 instead of 1. "Extrapolating" again one step more, writing addition of ''n'' as repeated addition of 1, requires starting with the number ''a''. Compare the [[Hyperoperation#Definition|definition of the hyper operator]], where the starting values for addition and multiplication are also separately specified.

==Tables of values==

===Computing 2↑''m''''n''===

Computing <math>2\uparrow^m n</math> can be restated in terms of an infinite table. We place the numbers <math>2^n</math> in the top row, and fill the left column with values 2. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.

{| class="wikitable"
|+ Values of <math>2\uparrow^m n</math> = [[Hyper operator|hyper]](2, ''m'' + 2, ''n'') = [[Conway chained arrow notation|2 → n → m]]
|-
! ''m''\''n''
! 1
! 2
! 3
! 4
! 5
! 6
! formula
|-
! 1
| 2 || 4 || 8 || 16 || 32 || 64 || <math>2^n</math>
|-
! 2
| 2 || 4 || 16 || 65536 || <math>2^{65\,536}\approx 2.0 \times 10^{19\,728}</math> || <math>2^{2^{65\,536}}\approx 10^{6.0 \times 10^{19\,727}}</math> || <math>2\uparrow\uparrow n</math>
|-
! 3
| 2 || 4 || 65536 || <math>
\begin{matrix}
\underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^2}}}}}} \\
65536\mbox{ multiplied copies of }2 \end{matrix}
</math> ||   ||   || <math>2\uparrow\uparrow\uparrow n</math>
|-
! 4
| 2 || 4 || <math>
\begin{matrix}
\underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^2}}}}}}\\
65536\mbox{ multiplied copies of }2
\end{matrix}</math> ||   ||   ||  
| <math>2\uparrow\uparrow\uparrow\uparrow n</math>
|}

The table is the same as [[Ackermann function#Table of values|that of the Ackermann function]], except for a shift in <math>m</math> and <math>n</math>, and an addition of 3 to all values.

===Computing 3↑''m''''n''===

We place the numbers <math>3^n</math> in the top row, and fill the left column with values 3. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.

{| class="wikitable"
|+ Values of <math>3\uparrow^m n</math> = [[Hyper operator|hyper]](3, ''m'' + 2, ''n'') = [[Conway chained arrow notation|3 → n → m]]
|-
! ''m''\''n''
! 1
! 2
! 3
! 4
! 5
! formula
|-
! 1
| 3 || 9 || 27 || 81 || 243 || <math>3^n</math>
|-
! 2
| 3 || 27 || 7,625,597,484,987 || <math>3^{7{,}625{,}597{,}484{,}987}</math> ||   || <math>3\uparrow\uparrow n</math>
|-
! 3
| 3 || 7,625,597,484,987 || <math>
\begin{matrix}
\underbrace{3_{}^{3^{{}^{.\,^{.\,^{.\,^3}}}}}}\\
7{,}625{,}597{,}484{,}987\mbox{ multiplied copies of }3
\end{matrix}</math> ||   ||   || <math>3\uparrow\uparrow\uparrow n</math>
|-
! 4
| 3 || <math>\begin{matrix}
\underbrace{3_{}^{3^{{}^{.\,^{.\,^{.\,^3}}}}}}\\
7{,}625{,}597{,}484{,}987\mbox{ multiplied copies of }3
\end{matrix}</math> ||   ||   ||  
| <math>3\uparrow\uparrow\uparrow\uparrow n</math>
|}

===Computing 10↑''m''''n''===

We place the numbers <math>10^n</math> in the top row, and fill the left column with values 10. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.

{| class="wikitable"
|+ Values of <math>10\uparrow^m n</math> = [[Hyper operator|hyper]](10, ''m'' + 2, ''n'') = [[Conway chained arrow notation|10 → n → m]]
|-
! ''m''\''n''
! 1
! 2
! 3
! 4
! 5
! formula
|-
! 1
| 10 || 100 || 1,000 || 10,000 || 100,000 || <math>10^n</math>
|-
! 2
| 10 || 10,000,000,000 || <math>10^{10,000,000,000}</math> || <math>10^{10^{10,000,000,000}}</math> || <math>10^{10^{10^{10,000,000,000}}}</math> || <math>10\uparrow\uparrow n</math>
|-
! 3
| 10 || <math>
\begin{matrix}
\underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
10\mbox{ multiplied copies of }10
\end{matrix}</math> || <math>
\begin{matrix}
\underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
\underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
10\mbox{ multiplied copies of }10
\end{matrix}</math> || <math>
\begin{matrix}
\underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
\underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
\underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
10\mbox{ multiplied copies of }10
\end{matrix}</math> ||   || <math>10\uparrow\uparrow\uparrow n</math>
|-
! 4
| 10 || <math>
\begin{matrix}
\underbrace{^{^{^{^{^{10}.}.}.}10}10}\\
10\mbox{ multiplied copies of }10
\end{matrix}</math> || <math>
\begin{matrix}
\underbrace{^{^{^{^{^{10}.}.}.}10}10}\\
\underbrace{^{^{^{^{^{10}.}.}.}10}10}\\
10\mbox{ multiplied copies of }10
\end{matrix}</math> ||   ||  
| <math>10\uparrow\uparrow\uparrow\uparrow n</math>
|}

Note that for 2 ≤ ''n'' ≤ 9 the numerical order of the numbers <math>10\uparrow^m n</math> is the [[lexicographical order]] with ''m'' as the most significant number, so for the numbers of these 8 columns the numerical order is simply line-by-line. The same applies for the numbers in the 97 columns with 3 ≤ ''n'' ≤ 99, and if we start from ''m'' = 1 even for 3 ≤ ''n'' ≤ 9,999,999,999.

==Numeration systems based on the hyperoperation sequence==

[[Reuben Goodstein|R. L. Goodstein]],<ref>{{cite journal |author=Goodstein, R. L.|title=Transfinite ordinals in recursive number theory |jstor=2266486 |journal=Journal of Symbolic Logic |volume=12 |year=1947 |issue=4 |doi=10.2307/2266486 |pages=123–129}}</ref> with a system of notation different from Knuth arrows, used the sequence of [[hyperoperation|hyperoperators]] here denoted by <math>( +, \ \times, \ \uparrow, \ \uparrow\uparrow, \ \dots)\,\!</math> to create systems of numeration for the nonnegative integers. Letting superscripts <math> \quad ( ^{(1)}, \ ^{(2)}, \ ^{(3)}, \ ^{(4)}, \ \dots )\,\!</math> denote the respective hyperoperators <math>( +, \ \times, \ \uparrow, \ \uparrow\uparrow, \ \dots)\,\!</math>, the so-called ''complete hereditary representation'' of integer ''n'', at level ''k'' and base ''b'', can be expressed as follows using only the first ''k'' hyperoperators and using as digits only 0, 1, ..., ''b''-1:

* For 0 ≤ ''n'' ≤ ''b''-1, ''n'' is represented simply by the corresponding digit.
* For ''n'' > ''b''-1, the representation of ''n'' is found recursively, first representing ''n'' in the form
:<math>b^{(k)}{x_k}^{(k-1)}{x_{k-1}}^{(k-2)} \dots {x_2}^{(1)}x_1</math>
:where ''x''''k'', ..., ''x''1 are the largest integers satisfying (in turn)

:<math>b^{(k)}x_k \le n</math>

:<math>b^{(k)}{x_k}^{(k-1)}x_{k-1} \le n</math>

:...

:<math>b^{(k)}{x_k}^{(k-1)}{x_{k-1}}^{(k-2)} \dots {x_2}^{(1)}x_1 \le n</math>.

:Any ''x''''i'' exceeding ''b''-1 is then re-expressed in the same manner, and so on, repeating this procedure until the resulting form contains only the digits 0, 1, ..., ''b''-1.

The remainder of this section will use <math>( +, \ \times, \ \uparrow, \ \uparrow\uparrow, \ \uparrow\uparrow\uparrow, \ \dots)</math>, rather than superscripts, to denote the hyperoperators.

Unnecessary parentheses can be avoided by giving higher-level operators higher precedence in the order of evaluation; thus,

level-1 representations have the form <math>b + X</math>, with ''X'' also of this form;

level-2 representations have the form <math>b \times X + Y</math>, with ''X'',''Y'' also of this form;

level-3 representations have the form <math>b \uparrow X \times Y + Z</math>, with ''X'',''Y'',''Z'' also of this form;

level-4 representations have the form <math>b \uparrow\uparrow X \uparrow Y \times Z + T</math>, with ''X'',''Y'',''Z'',''T'' also of this form;

and so on.

The representations can be abbreviated by omitting any instances of <math>+0, \ \times1, \uparrow1, \ \uparrow\uparrow1, </math> etc.; for example, the level-3 base-2 representation of the number 6 is <math>2\uparrow(2\uparrow1\times1+0)\times1+(2\uparrow1\times1+0)</math>, which abbreviates to <math>2 \uparrow 2 + 2</math>.

Examples:
The unique base-2 representations of the number 266, at levels 1, 2, 3, 4, and 5 are as follows:

:<math>\text{Level 1:} \ \ 266 = 2 + 2 + \dots + 2 \ \ \text{(with 133 2s)}</math>
:<math>\text{Level 2:} \ \ 266 = 2 \times (2 \times (2 \times (2 \times 2 \times 2 \times 2 \times 2 + 1)) + 1)</math>
:<math>\text{Level 3:} \ \ 266 = 2 \uparrow 2 \uparrow (2 + 1) + 2 \uparrow (2 + 1) + 2</math>
:<math>\text{Level 4:} \ \ 266 = 2 \uparrow\uparrow (2 + 1) \uparrow 2 + 2 \uparrow\uparrow 2 \times 2 + 2</math>
:<math>\text{Level 5:} \ \ 266 = 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 2 + 2 \uparrow\uparrow\uparrow 2 \times 2 + 2</math>.

==See also==

*[[Primitive recursion]]
*[[Hyper operator]]
*[[Busy beaver]]
*[[Cutler's bar notation]]
*[[Tetration]]

==References==
{{reflist}}

==External links==
* {{mathworld|urlname=ArrowNotation|title=Arrow Notation}}
* Robert Munafo, ''[http://www.mrob.com/pub/math/largenum.html Large Numbers]''

[[Category:Mathematical notation]]
[[Category:Large numbers]]
[[Category:Donald Knuth]]

Sauter mean diameter

2013-08-06T23:56:42Z

134.84.0.173: grammar

{{about|the term as used in mathematics|other uses|derivative (disambiguation)}}
{{Calculus |Differential}}

In [[mathematics]], the [[derivative]] is a fundamental construction of [[differential calculus]] and admits many possible generalizations within the fields of [[mathematical analysis]], [[combinatorics]], [[algebra]], and [[geometry]].

== Derivatives in analysis ==
In real, complex, and functional analysis, derivatives are generalized to functions of several real or complex variables and functions between [[topological vector spaces]]. An important case is the [[functional derivative|variational derivative]] in the [[calculus of variations]]. Repeated application of differentiation leads to derivatives of higher order and differential operators.

=== Multivariable calculus ===
{{main|Fréchet derivative}}
The '''derivative''' is often met for the first time as an operation on a single real function of a single real variable. One of the simplest settings for generalizations is to vector valued functions of several variables (most often the domain forms a vector space as well). This is the field of [[multivariable calculus]].

In one-variable calculus, we say that a function <math>f: \R \to \R</math> is '''differentiable''' at a point ''x'' if the limit
:<math>\lim_{h \to 0}\frac{f(x+h) - f(x)}{h}</math>
exists. Its value is then the derivative ƒ'(''x''). A function is differentiable on an [[Interval (mathematics)|interval]] if it is differentiable at every point within the interval. Since the line <math>L(z) = f'(x)z - f'(x)x + f(x)</math> is tangent to the original function at the point <math>(x, f(x))</math>, the derivative can be seen as a way to find the ''best linear approximation'' of a function. If one ignores the constant term, setting <math>L(z) = f'(x)z</math>, ''L''(''z'') becomes an actual [[linear operator]] on '''R''' considered as a vector space over itself.

This motivates the following generalization to functions mapping '''R'''''m'' to '''R'''''n'': ƒ is differentiable at ''x'' if there exists a [[linear operator]] ''A''(''x'') (depending on ''x'') such that
:<math>\lim_{\|h\| \to 0}\frac{\|f(x+h) - f(x) - A(x)h\|}{\|h\|} = 0.</math>
Although this definition is perhaps not as explicit as the above, if such an operator exists, then it is unique, and in the one-dimensional case coincides with the original definition. (In this case the derivative is represented by a 1-by-1 matrix consisting of the sole entry ''f'''(''x'').) Note that, in general, we concern ourselves mostly with functions being differentiable in some open [[neighbourhood (mathematics)|neighbourhood]] of <math>x</math> rather than at individual points, as not doing so tends to lead to many [[Pathological (mathematics)|pathological]] [[counterexamples]].

An ''m'' by ''n'' [[matrix (mathematics)|matrix]], of the [[linear operator]] ''A''(''x'') is known as '''[[Jacobian matrix and determinant|Jacobian]]''' matrix '''J'''''x''(ƒ) of the mapping ƒ at point ''x''. Each entry of this matrix represents a '''[[partial derivative]]''', specifying the rate of change of one range coordinate with respect to a change in a domain coordinate. Of course, the Jacobian
matrix of the composition ''g°f'' is a product of corresponding Jacobian matrices:
'''J'''''x''(''g°f'') ='''J'''ƒ(''x'')(''g'')'''J'''''x''(ƒ). This is a higher-dimensional statement of the [[chain rule]].

For real valued functions from '''R'''''n'' to '''R''' ([[scalar field]]s), the total derivative can be interpreted as a [[vector field]] called the '''[[gradient]]'''. An intuitive interpretation of the gradient is that it points "up": in other words, it points in the direction of fastest increase of the function. It can be used to calculate '''[[directional derivative]]s''' of [[Scalar (mathematics)|scalar]] functions or normal directions.

Several linear combinations of partial derivatives are especially useful in the context of differential equations defined by a vector valued function '''R'''''n'' to '''R'''''n''. The '''[[divergence]]''' gives a measure of how much "source" or "sink" near a point there is. It can be used to calculate [[flux]] by [[divergence theorem]]. The '''[[curl (mathematics)|curl]]''' measures how much "[[rotation]]" a vector field has near a point.

For [[vector-valued functions]] from '''R''' to '''R'''''n'' (i.e., [[parametric curve]]s), one can take the derivative of each component separately. The resulting derivative is another vector valued function. This is useful, for example, if the vector-valued function is the position vector of a particle through time, then the derivative is the velocity vector of the particle through time.

The '''[[convective derivative]]''' takes into account changes due to time dependence and motion through space along vector field.

=== Convex analysis ===

The [[subderivative]] and [[subgradient]] are generalizations of the derivative to [[convex function]]s.

=== Higher-order derivatives and differential operators ===
One can iterate the differentiation process, that is, apply derivatives more than once, obtaining derivatives of second and higher order. A more sophisticated idea is to combine several derivatives, possibly of different orders, in one algebraic expression, a [[differential operator]]. This is especially useful in considering ordinary [[linear differential equation]]s with constant coefficients. For example, if ''f''(''x'') is a twice differentiable function of one variable, the differential equation

: <math>f''+2f'-3f=4x-1\,</math>

may be rewritten in the form

: <math>L(f)=4x-1,\,</math> &ensp;&ensp; where &ensp;&ensp; <math> L=\frac{d^2}{dx^2}+2\frac{d}{dx}-3</math>

is a ''second order linear constant coefficient differential operator'' acting on functions of ''x''. The key idea here is that we consider a particular [[linear combination]] of zeroth, first and second order derivatives "all at once". This allows us to think of the set of solutions of this differential equation as a "generalized antiderivative" of its right hand side 4''x'' − 1, by analogy with ordinary [[Integral|integration]], and formally write

: <math>f(x)=L^{-1}(4x-1).\, </math>

Higher derivatives can also be defined for functions of several variables, studied in [[multivariable calculus]]. In this case, instead of repeatedly applying the derivative, one repeatedly applies [[partial derivative]]s with respect to different variables. For example, the second order partial derivatives of a scalar function of ''n'' variables can be organized into an ''n'' by ''n'' matrix, the '''[[Hessian matrix]]'''. One of the subtle points is that the higher derivatives are not intrinsically defined, and depend on the choice of the coordinates in a complicated fashion (in particular, the Hessian matrix of a function is not a [[tensor]]). Nevertheless, higher derivatives have important applications to analysis of [[maxima and minima|local extrema]] of a function at its [[critical point (mathematics)|critical points]]. For an advanced application of this analysis to topology of [[manifold]]s, see [[Morse theory]].

As in the case of functions of one variable, we can combine first and higher order partial derivatives to arrive at a notion of a [[partial differential operator]]. Some of these operators are so important that they have their own names:

*The [[Laplace operator]] or '''Laplacian''' on '''R'''3 is a second-order partial differential operator ''Δ'' given by the [[divergence]] of the [[gradient]] of a scalar function of three variables, or explicitly as

:: <math> \Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}. </math>
Analogous operators can be defined for functions of any number of variables.

*The [[d'Alembertian]] or '''wave operator''' is similar to the Laplacian, but acts on functions of four variables. Its definition uses the indefinite [[metric tensor]] of [[Minkowski space]], instead of the [[Euclidean space|Euclidean]] [[dot product]] of '''R'''''3'':

:: <math> \square=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}. </math>

=== Analysis on fractals ===
Laplacians and differential equations can be defined on [[analysis on fractals|fractals]].

=== Fractional derivatives ===
In addition to ''n''-th derivatives for any natural number ''n'', there are various ways to define derivatives of fractional or negative orders, which are studied in '''[[fractional calculus]]'''. The -1 order derivative corresponds to the integral, whence the term '''[[differintegral]]'''.

=== Complex analysis ===

In [[complex analysis]], the central objects of study are [[holomorphic function]]s, which are complex-valued functions on the [[complex numbers]] satisfying a [[Fréchet derivative|suitably extended definition of differentiability]].

The '''[[Schwarzian derivative]]''' describes how a complex function is approximated by a [[fractional-linear map]], in much the same way that a normal derivative describes how a function is approximated by a linear map.

=== Functional analysis ===

In [[functional analysis]], the '''[[functional derivative]]''' defines the derivative with respect to a function of a functional on a space of functions. This is an extension of the directional derivative to an infinite [[dimension]]al vector space.

The '''[[Fréchet derivative]]''' allows the extension of the directional derivative to a general [[Banach space]]. The '''[[Gâteaux derivative]]''' extends the concept to [[locally convex]] [[topological vector space]]s. Fréchet differentiability is a strictly stronger condition than Gâteaux differentiability, even in finite dimensions. Between the two extremes is the '''[[quasi-derivative]]'''.

In [[measure theory]], the '''[[Radon–Nikodym derivative]]''' generalizes the [[Jacobian matrix and determinant|Jacobian]], used for changing variables, to measures. It expresses one measure μ in terms of another measure ν (under certain conditions).

In the theory of [[abstract Wiener space]]s, the [[H-derivative|''H''-derivative]] defines a derivative in certain directions corresponding to the Cameron-Martin [[Hilbert space]].

The derivative also admits a generalization to the space of '''[[distribution (mathematics)|distributions]]''' on a space of functions using [[integration by parts]] against a suitably well-behaved subspace.

On a [[function space]], the [[linear operator]] which assigns to each function its derivative is an example of a '''[[differential operator]]'''. General differential operators include higher order derivatives. By means of the [[Fourier transform]], '''[[pseudo-differential operator]]s''' can be defined which allow for fractional calculus.

===Analogues of derivatives in fields of positive characteristic===
The [[Carlitz derivative]] is an operation similar to usual differentiation have been devised with the usual context of real or complex numbers changed to [[local fields]] of positive [[Characteristic_(algebra)|characteristic]] in the form of [[formal Laurent series]] with coefficients in some [[finite field]] '''F'''''q'' (it is known that any local field of positive characteristic is isomorphic to a Laurent series field).

Along with suitably defined analogs to the [[exponential function]], [[logarithms]] and others the derivative can be used to develop notions of smoothness, analycity, integration, Taylor series as well as a theory of differential equations.<ref>{{cite book |title=Analysis in Positive Characteristic |last=Kochubei |first= Anatoly N.|year=2009 |publisher= Cambridge University Press |location= New York |isbn= 978-0-521-50977-0}}</ref>

==Difference operator, q-analogues and time scales==

* The '''[[q-derivative]]''' of a function is defined by the formula

: <math> D_q f(x)=\frac{f(qx)-f(x)}{(q-1)x}.</math>

For ''x'' nonzero, if ''f'' is a differentiable function of ''x'' then in the limit as ''q''&ensp;→ 1 we obtain the ordinary derivative, thus the ''q''-derivative may be viewed as its [[q-deformation]]. A large body of results from ordinary differential calculus, such as [[binomial formula]] and [[Taylor expansion]], have natural ''q''-analogues that were discovered in the 19th century, but remained relatively obscure for a big part of the 20th century, outside of the theory of [[special functions]]. The progress of [[combinatorics]] and the discovery of [[quantum group]]s have changed the situation dramatically, and the popularity of ''q''-analogues is on the rise.

* The '''[[difference operator]]''' of [[difference equations]] is another discrete analog of the standard derivative.
:<math>\Delta f(x)=f(x+1)-f(x)\,</math>

* The '''q-derivative''', the '''difference operator''' and the '''standard derivative''' can all be viewed as the same thing on different [[time scale calculus|time scales]].

== Derivatives in algebra ==
In algebra, generalizations of the derivative can be obtained by imposing the [[product rule|Leibniz rule of differentiation]] in an algebraic structure, such as a [[ring (mathematics)|ring]] or a [[Lie algebra]].

=== Derivations ===

A '''[[derivation (abstract algebra)|derivation]]''' is a linear map on a ring or [[algebra over a field|algebra]] which satisfies the Leibniz law (the product rule). Higher derivatives and [[algebraic differential equation|algebraic differential operators]] can also be defined. They are studied in a purely algebraic setting in [[differential Galois theory]] and the theory of [[D-module]]s, but also turn up in many other areas, where they often agree with less algebraic definitions of derivatives.

For example, the '''formal derivative''' of a [[polynomial]] over a commutative ring ''R'' is defined by
:<math>(a_dx^d + a_{d-1}x^{d-1} + \cdots+a_1x+a_0)' = da_dx^{d-1}+(d-1)a_{d-1}x^{d-2} + \cdots+a_1.</math>
The mapping <math>f\mapsto f'</math> is then a derivation on the [[polynomial ring]] ''R''[''X'']. This definition can be extended to [[rational function]]s as well.

The notion of derivation applies to noncommutative as well as commutative rings, and even to non-associative algebraic structures, such as Lie algebras.

Also see [[Pincherle derivative]].

=== Commutative algebra ===

In [[commutative algebra]], '''[[Kähler differential]]s''' are universal derivations of a [[commutative ring]] or [[module (algebra)|module]]. They can be used to define an analogue of exterior derivative
from differential geometry that applies to arbitrary [[algebraic varieties]], instead of just smooth manifolds.

=== Number theory ===

In [[p-adic analysis]], the usual definition of derivative is not quite strong enough, and one requires [[strictly differentiable|strict differentiability]] instead.

Also see [[arithmetic derivative]] and [[Hasse derivative]].

=== Type theory ===
Many [[abstract data type]]s in mathematics and [[computer science]] can be described as the [[universal algebra|algebra]] generated by a transformation that maps structures based on the type back into the type. For example, the type T of [[binary tree]]s containing values of type A can be represented as the algebra generated by the transformation 1+A×T2→T. The "1" represents the construction of an empty tree, and the second term represents the construction of a tree from a value and two subtrees. The "+" indicates that a tree can be constructed either way.

The derivative of such a type is the type that describes the context of a particular substructure with respect to its next outer containing structure. Put another way, it is the type representing the "difference" between the two. In the tree example, the derivative is a type that describes the information needed, given a particular subtree, to construct its parent tree. This information is a tuple that contains a binary indicator of whether the child is on the left or right, the value at the parent, and the sibling subtree. This type can be represented as 2×A×T, which looks very much like the derivative of the transformation that generated the tree type.

This concept of a derivative of a type has practical applications, such as the [[zipper (data structure)|zipper]] technique used in [[functional programming language]]s.

== Derivatives in geometry ==
Main types of derivatives in geometry are Lie derivatives along a vector field, exterior differential, and covariant derivatives.

=== Differential topology ===

In [[differential topology]], a '''[[vector field]]''' may be defined as a derivation on the ring of [[smooth function]]s on a [[manifold]], and a '''[[tangent vector]]''' may be defined as a derivation at a point. This allows the abstraction of the notion of a [[directional derivative]] of a scalar function to general manifolds. For manifolds that are [[subset]]s of '''R'''''n'', this tangent vector will agree with the directional derivative defined above.

The '''[[pushforward (differential)|differential or pushforward]]''' of a map between manifolds is the induced map between tangent spaces of those maps. It abstracts the [[Jacobian matrix]].

On the [[exterior algebra]] of [[differential forms]] over a [[smooth manifold]], the '''[[exterior derivative]]''' is the unique linear map which satisfies a [[graded]] version of the Leibniz law and squares to zero. It is a grade 1 derivation on the exterior algebra.

The '''[[Lie derivative]]''' is the rate of change of a vector or tensor field along the flow of another vector field. On vector fields, it is an example of a [[Lie bracket]] (vector fields form the [[Lie algebra]] of the [[diffeomorphism group]] of the manifold). It is a grade 0 derivation on the algebra.

Together with the '''[[interior product]]''' (a degree -1 derivation on the exterior algebra defined by contraction with a vector field), the exterior derivative and the Lie derivative form a [[Lie superalgebra]].

=== Differential geometry ===

In [[differential geometry]], the '''[[covariant derivative]]''' makes a choice for taking directional derivatives of vector fields along [[curve]]s. This extends the directional derivative of scalar functions to sections of [[vector bundle]]s or [[principal bundle]]s. In [[Riemannian geometry]], the existence of a metric chooses a unique preferred [[Torsion tensor|torsion]]-free covariant derivative, known as the [[Levi-Civita connection]]. See also [[gauge covariant derivative]] for a treatment oriented to physics.

The '''[[exterior covariant derivative]]''' extends the exterior derivative to vector valued forms.

== Other generalizations ==

It may be possible to combine two or more of the above different notions of extension or abstraction of the original derivative. For example, in [[Finsler geometry]], one studies spaces which look [[locally]] like [[Banach space]]s. Thus one might want a derivative with some of the features of a [[functional derivative]] and the [[covariant derivative]].

The study of [[stochastic processes]] requires a form of calculus known as the [[Malliavin calculus]]. One notion of derivative in this setting is the [[H-derivative|''H''-derivative]] of a function on an [[abstract Wiener space]].

== See also ==
*[[Arithmetic derivative]]
*[[Non-classical analysis]]

== Notes ==

{{reflist}}

[[Category:Differential calculus]]
[[Category:Generalizations of the derivative| ]]
[[Category:Real analysis]]
[[Category:Functional analysis]]
[[Category:Algebra]]
[[Category:Q-analogs]]
[[Category:Differential geometry]]

Surgery obstruction

2011-10-18T20:27:32Z

134.84.140.207: dab

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