ASCIIMathML.js is a JavaScript that translates simple ASCII mathematics expressions to MathML while the webpage is loading. It works in InternetExplorer+MathPlayer and Firefox. See for more details.\n
ASCIIsvg.js is a JavaScript that translates simple descriptions of diagrams into [[SVG]] while the webpage is loading. It works in InternetExplorer+SVGview and [[Firefox 1.5]]. See for more details.\n
<html>\n<center>\n<embed width="100" height="100" src="d.svg" script='\nxmin=0; xmax=4; xscl=4; ymin=0; noaxes()\nfill="white"\nr=1\nxc=r*2\nyc=r*2\nalpha=1\nbeta=2\ncircle([xc,yc],r)\nline([r*cos(alpha)+xc,r*sin(alpha)+yc],[r*cos(beta)+xc,r*sin(beta)+yc])\n'>\n</center>\n</html>\n
TriGonometria CordaCerchio HelloThere MathExamples Download MainFeatures
Let `N` be a subgroup of a group `G`. We say that `N` is //normal// if the representative operation on right cosets of `N`, defined by `(Nx)(Ny)=N(xy)` is well-defined, i.e. if `Nx=Nx'` and `Ny=Ny'` then `N(xy)=N(x'y')`.
You can download your own copy of this math notebook by right-clicking on This will allow you to add your own content.\n\nTo enable the math features you also should download the following ''three'' files into the same directory:\n\n\n\n\n\nOptionally you can also download [[jsMath]] from\n\n\n\nand unzip (or later version) in the same directory. This script parses a bigger subset of LaTeX and does not require MathML support.\n
Firefox 1.5 is a version of the Firefox browser that supports both MathML and [[SVG]] right out of the box. It's free, its easy to install, and its worth trying the current beta version:
This is an adaptation of Jeremy Ruston's TiddlyWiki (version 1.2.37) that is suitable for writing mathematics notes (e.g. lecture notes, homework, projects, research notes, ...). See MainFeatures for a brief description of the three scripts that enable the math formulas, diagrams and graphs.\n\nThese features require InternetExplorer+MathPlayer+SVGview or [[Firefox 1.5]]. Take a look at the MathExamples below to see if your browser is setup okay.
JavaScript is a computer language that is implemented in many webbrowsers and is used to create dynamic webpages. In InternetExplorer it is called ~JScript, and part of the language has been standardized as ~ECMAscript. It is a weakly typed language with a syntax based on c and a prototype mechnism for handling objects. JavaScript programs are downloaded with the webpage, cached and interpreted on the client machine.
LaTeX is a standard mathematical typesetting language used all over the world for publishing scientific texts. An introduction can be found at
''Lemma'' Let `H` be a subgroup of a group `G`. Then for all `x,y in G`\n`x in Hy` if and only if `Hx=Hy` and\n`x in yH` if and only if `xH=yH`.\n\n//Proof// (hint). Recall that the left and right cosets of `H` are defined by `xH={xh|h in H}` and `Hx={hx|h in H}`. Now the "if" part is obvious since the identity `e` is in `H`. For the forward implication we have to make use of some very basic properties of groups (try it).
The MathML and SVG features work in Firefox 1.5 (RC2) and Internet Explorer (but the latter requires some plugins).\n\nASCIIMathML is used to translate notation like {{{\s`x^2\s`}}} to MathML. InternetExplorer requires the MathPlayer plugin to display MathML. It can be downloaded free from For details about the syntax of ASCIIMathML see \nIf the [[jsMath]] scripts are installed, LaTeX formulas (enclosed in dollar-signs or double-dollar-signs) are translated to math notation by these scripts (and otherwise ASCIIMathML tries to convert them, but it handles only a small subset of LaTeX).\n\nASCIIsvg is used to display diagrams described by an <embed> tag. InternetExplorer requires the Adobe SVGview plugin to display SVG. It can be downloaded free from For details about the syntax of ASCIIsvg see
HelloThere\nTiddlyWiki\n[[Download]]\nMainFeatures\n[[Thanks]]\n\n<<newTiddler>>\n<<newJournal "DD MMM YYYY">>
A famous result due to [[Euler|]] in ASCIIMathML: `sum_(n=1)^oo 1/n^2=pi^2/6`\n\nand again in LaTeX: $\ssum_{n=1}^\sinfty \sfrac1{n^2}=\sfrac{\spi^2}6$\n\nHere is a graph of `x-x^3/6` and `sin(x)` (double click on this tiddler to see the <embed> tag that produced it).\n\n<html>\n<center>\n<embed width="400" height="200" src="d.svg" script='\nxmin=-2pi; xmax=2pi; xscl=1; axes(); stroke="red"\nplot(sin(x))\nstroke="blue"; strokedasharray="10,10"\nplot(x-x^3/6,-4,4)\n'>\n</center>\n</html>\n\nand a diagram illustrating why Pythagoras' Theorem holds:\n<html>\n<center>\n<embed width="400" height="200" src="d.svg" script='\nxmin=-4; xmax=4; xscl=1; ymin=1; noaxes()\nfill="yellow"\nrect([1,1],[4,4])\nrect([-4,1],[-1,4])\nfill="red"\nrect([-2,1],[-1,2])\nfill="blue"\nrect([-4,2],[-2,4])\nfill="green"\npath([[1,2],[2,4],[4,3],[3,1],[1,2]])\nfill="none"\npath([[-4,1],[-2,2],[-1,4]])'>\n</center>\n</html>\n\n
MathML is the W3C standard for mathematics notation on webpages. See for more details.
MathPlayer is a free plugin for InternetExplorer that allows this browser to display MathML. The plugin is freely available from
/***\nASCIIMathML plugin: see for syntax help.\n\n!Code:\n***/\n//{{{\nwindow.createTiddlerViewer_original_MathPlugin = window.createTiddlerViewer;\nwindow.createTiddlerViewer=function(title,highlightText,highlightCaseSensitive) {\n window.createTiddlerViewer_original_MathPlugin(title,highlightText,highlightCaseSensitive);\n var theViewer = document.getElementById("viewer"+title);\n if (theViewer) {\n if (jsMath.ConvertTeX) {\n jsMath.ConvertTeX(theViewer);\n jsMath.ProcessBeforeShowing(theViewer);\n }\n if (AMprocessNode) AMprocessNode(theViewer,false);\n if (drawPictures) setTimeout('drawPictures()',100);\n }\n}\n//}}}\n\n
SVG is the W3C standard for scalable vector graphics on webpages. See for more details.
SVGview.exe is the filename of the Adobe SVG Viewer plugin for InternetExplorer. It is freely available from
a web notebook for mathematics
Many thanks to all who have contributed to making this math notebook possible.\n\nTiddlyWiki is due to Jeremy Ruston (with contributions from other volunteers).\nThe MathPlugin is adapted from [[Franco Bagnoli's version of TiddlyWiki|]]\n[[jsMath]] is due to Davide Cervione.\nASCIIMathML and ASCIIsvg are scripts I wrote in early 2004.\nThanks also to the many individuals who have worked on creating MathML, LaTeX, [[SVG]], JavaScript, Firefox 1.5, InternetExplorer, MathPlayer and Adobe SVGview.
TiddlyMath can be helpful for students to create and keep track of definitions, lemmas, theorems, corollaries and their proofs. The trick is to develop a system for naming results, and for writing their proofs.\n\nEvery definition and result should have a (preferably short) meaningful name. Wellknown results mostly have names already, so that is no problem (e.g. [[Fundamental Theorem of Calculus Part I]], [[Cayley's Theorem]], ...). If a good name cannot be found, one can of course resort to something like [[Defn 1.1]], [[Lemma 2]], [[Theorem 3]], [[Cor 4.4]].\n\nProofs can be part of the same tiddler that states the result, or they can be in a separate one, e.g. ProofOfThm3. The latter allows tiddlers like HintForThm3 or ProofOutlineThm3. Steps within proofs should be justified by referring to the definitions and results in other tiddlers. Of course the nonlinear structure of TiddlyWiki can make it harder to spot circular reasoning. On the other hand it makes it easier to provide lots of detail in small pieces that can be skipped by experienced readers.
''Theorem'' Let `N` be a subgroup of a group `G`. Then the following are equivalent:\n\n(i) `N` is [[normal|DefnNormalSubgroup]].\n(ii) `Nx=xN` for all `x in G`.\n(iii) `N` is closed under conjugation, i.e. `x^-1ax in N` for all `x in G` and `a in N`.\n\n//Proof//: We will prove (i)`=>`(ii)`=>`(iii)`=>`(i).\n(i)`=>`(ii): Assume `N` is normal. This means the representative operation on right cosets is well-defined. So for all `x,u,y,v in G` if `Nx=Nu` and `Ny=Nv` then `N(xy)=N(uv)`. By LemmaEqualCosets this means `x in Nu` and `y in Nv` imply `xy in N(uv)`, or put another way, for all `a,b in N` if `x=au` and `y=bv` then `xy=cuv` for some `c in N`.\n\nSo we now know that for all `u,v in G` and all `a,b in N` there exists `c in N` such that `aubv=cuv`.\n\nWe want to show that `Nx=xN` for all `x in G`. This is equivalent to `Nx sube xN` and `xN sube Nx`.\nSo let `x in G`, and let `y in xN`. This means `y=xb` for some `b in N`. \nWe want to show that `y in Nx`, i.e. that `xb=cx` for some `c in N`. But this follows directly from the assumption if we let `a=v=e` and `u=x`. Hence we have proved `xN sube Nx`.\nSince this holds for all `x in G` we obtain `x^-1N sube Nx^-1`. Thus for all `b in N` there exists `c in N` such that `x^-1b=cx^-1`, or equivalently `bx=xc`. Therefore `Nx sube xN` also holds.\n\n(ii)`=>`(iii): Assume `Nx=xN` for all `x in G`. We want to show that `N` is closed under conjugation. Let `x in G` and `a in N`. Then `ax in Nx`, so by assumption `ax in xN`. Hence `ax=xb` for some `b in N`, and it follows that `x^-1ax=b in N`. Therefore `N` is closed under conjugation.\n\n(iii)`=>`(i): Assume `N` is closed under conjugation, and let `Nx=Nu`, `Ny=Nv` for some `x,y,u,v in G`, which means `x=au` and `y=bv` for some `a,b in N`. We want to show that `Nxy=Nuv`, i.e. `xy in Nuv`.\nNow `xy=aubv=a(u^-1)^-1bu^-1uv=acuv` where `c=(u^-1)^-1bu^-1 in N` since `N` is closed under conjugation. We know that `N` is a subgroup, so `ac in N`, and hence we have shown that `xy in Nuv`.
TiddlyMath is the current name for a version of Jeremy Ruston's TiddlyWiki that is suitable for writing mathematical tiddlers. It is freely available from
Take a look at for a description of TiddlyWiki and the latest version of Jeremy Ruston's fantastic script.
<html>\n<center>\n<embed width="500" height="600" src="d.svg" script='\nsetBorder(0)\ninitPicture(-2*pi-0.2,2*pi+0.2)\naxes(1, 1, "labels", "grid")\ndot([pi,0],"|",cpi,above)\ndot([-pi,0],"|","-"+cpi,above)\ndot([2*pi,0],"|","2"+cpi,above)\ndot([-2*pi,0],"|","-2"+cpi,above)\ndot([pi/2,0],"|",cpi+"/2",above)\ndot([-pi/2,0],"|","-"+cpi+"/2",above)\ndot([3*pi/2,0],"|","3"+cpi+"/2",above)\ndot([-3*pi/2,0],"|","-3"+cpi+"/2",above)\nstrokewidth = 1.5\n\nstroke = "blue"\nplot("cos(x)")\nfontfill = "blue"\ntext([-1.4,.8],"cos(x)")\n\nstroke = "green"\nplot("tan(x)",-2*pi-0.2,-3/2*pi-0.05)\nplot("tan(x)",-3/2*pi+0.05,-1/2*pi-0.05)\nplot("tan(x)",-1/2*pi+0.05,1/2*pi-0.05)\nplot("tan(x)",1/2*pi+0.05,3/2*pi-0.05)\nplot("tan(x)",3/2*pi+0.05,2*pi+0.2)\nfontfill = "green"\ntext([1.2,3],"tan(x)",aboveright)\n\nstroke = "black"\nplot("csc(x)",-2*pi+0.05,-pi-0.05)\nplot("csc(x)",-pi+0.05,-0.05)\nplot("csc(x)",0.05,pi-0.05)\nplot("csc(x)",pi+0.05,2*pi-0.05)\nfontfill = "none"\ntext([2.6,2],"csc(x)",aboveright)\n\nstroke = "red"\nplot("sin(x)")\nfontfill = "red"\ntext([2.9,.8],"sin(x)")\n'>\n</center>\n</html>\n\n
The abbreviation for the World Wide Web Consortium, online at\n
$\smathbf{a} = (d\smathbf{v}) / dt $
Il '''teorema binomiale''' (o '''formula di Newton''', o ancora '''binomio di Newton''') è una formula\nche permette di ottenere lo sviluppo della [[potenza (matematica)|potenza]] ''n''-esima di un [[binomio]] qualsiasi. La forma algebrica di tale formula è:\n\n$(a+b)^n_{} = \ssum_{k=0}^n {n \schoose k} a^k b^{(n-k)}$\n\nin cui il fattore ${n \schoose k}$ rappresenta il [[coefficiente binomiale]] di ciascun [[monomio]] costituente lo sviluppo in questione.\n\nIl valore del generico coefficiente binomiale è infatti uguale all'elemento dell<nowiki>'</nowiki>''n''-esima riga in ''k''-esima posizione del [[triangolo di Tartaglia]], con $\s n=0,1,...,m$ e $\s k=0,...,n$\nInfatti detto $\s a_{n,k}$ il generico termine del triangolo, valgono le identità $a_{n,0}={n \schoose 0}=1$ ed $a_{n,n}={n \schoose n}=1$, da cui discende che essendo il triangolo di Tartaglia costruito secondo la ricorsione \n$\s a_{n,k}=a_{n-1,k}+a_{n-1,k-1}$, con $\s a_{n,0}=1$ ed $\s a_{n,n}=1$ e in forza della proprietà dei coefficienti binomiali ${n-1 \schoose k}+{n-1 \schoose k-1}={n \schoose k}$, vale l'uguaglianza\n\n$a_{n,k}={n \schoose k}$.\n\nCome esempio di applicazione della formula, riportiamo i casi piccoli, ''n''&nbsp;=&nbsp;2, ''n''&nbsp;=&nbsp;3 ed ''n''&nbsp;=&nbsp;4::\n\n:$(x + y)^2 = x^2 + 2xy + y^2\s,$\n:$(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\s,$\n:$(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4.\s,$\n\n== Dimostrazione ==\nIl Teorema binomiale può essere dimostrato per [[induzione matematica|induzione]].\n\nInfatti è possibile introdurre per tale teorema un passo base per cui esso risulta banalmente vero\n\n:$(a+b)^1_{} = \ssum_{k=0}^1 {1 \schoose k} a^{(1-k)} b^{k} = a+b$\n\ne provare con il passo induttivo la veridicità del teorema per un esponente n qualsiasi. Infatti presa per corretta l'espressione \n\n:$(a+b)^n_{} = \ssum_{k=0}^n {n \schoose k} a^{(n-k)} b^{k}$\n\nsicuramente vera per n=1\n\nsi ha \n{|\n|$(a+b)^{n+1}\s,$\n|$=(a+b)(a+b)^n\s,$\n|-\n|\n|$=(a+b)\ssum_{k=0}^n\s,{n \schoose k}a^{n-k}b^{k}$\n|-\n|\n|$=\ssum_{k=0}^n\s,{n \schoose k} a^{n+1-k}b^{k}+\ssum_{k=0}^n\s,{n \schoose k}a^{n-k}\nb^{k+1}$\n|}\n\nda cui, essendo \n{|\n|$\s \ssum_{k=0}^n\s,{n \schoose k} a^{n+1-k}b^{k}$\n|$= {n \schoose 0} a^{n+1} + \ssum_{k=1}^{n}\s,{n \schoose k} a^{n+1-k}b^{k}$\n|-\n|\n|$= {n \schoose 0} a^{n+1} + \ssum_{k=0}^{n-1}\s,{n \schoose k+1} a^{n+1-(k+1)}b^{k+1}$\n|-\n|\n|$= {n \schoose 0} a^{n+1} + \ssum_{k=0}^{n-1}\s,{n \schoose k+1} a^{n-k}b^{k+1}$\n|}\n \ned\n\n{|\n|<math>\s \ssum_{k=0}^n\s,{n \schoose k} a^{n-k}b^{k+1}</math>\n|<math>= \ssum_{k=0}^{n-1}\s,{n \schoose k} a^{n-k}b^{k+1}+ {n \schoose n} b^{n+1} </math>\n|}\n\nsi ha che\n\n{|\n|<math>(a+b)^{n+1} \s,</math>\n|<math>={n \schoose 0} a^{n+1}+\ssum_{k=0}^{n-1}\s,\sleft({n \schoose k} + {n \schoose k+1}\sright)a^{n-k}b^{k+1}+{n \schoose n} b^{n+1} \s,</math>\n|-\n|\n|<math>={n \schoose 0} a^{n+1}+\ssum_{k=0}^{n-1}\s,{n+1 \schoose k+1} a^{n-k}b^{k+1}+ {n \schoose n} b^{n+1}</math>\n|-\n|\n|<math>={n \schoose 0} a^{n+1}+\ssum_{k=1}^{n}\s,{n+1 \schoose k} a^{n+1-k}b^{k}+ {n \schoose n}\nb^{n+1}</math>\n|}\n\nessendo infine <math>\s {n \schoose 0} = {n+1 \schoose 0} = 1</math> e <math>\s {n \schoose n} = {n+1 \schoose n+1} = 1</math>\n\nsi ha che\n\n<math>\s {n \schoose 0} a^{n+1}+\ssum_{k=1}^{n}\s,{n+1 \schoose k} a^{n+1-k}b^{k}+ {n \schoose n}\nb^{n+1} = {n+1 \schoose 0} a^{n+1}+\ssum_{k=1}^{n}\s,{n+1 \schoose k} a^{n+1-k}b^{k}+ {n+1 \schoose n+1}\nb^{n+1}</math>\n\nsi ottiene l'espressione formale dello sviluppo della potenza successiva del binomio\n\n<math>\s (a+b)^{n+1} = \ssum_{k=0}^{n+1}\s,{n+1 \schoose k} a^{(n+1)-k}b^{k}</math>\n\nche conferma la tesi.\n\n==Voci correlate==\n\n* [[Triangolo di Tartaglia]]\n* [[Teorema multinomiale]]\n\n[[Categoria:Combinatorica]]\n\n[[bn:দ্বিপদী উপপাদ্য]]\n[[de:Binomischer Lehrsatz]]\n[[en:Binomial theorem]]\n[[es:Teorema del binomio]]\n[[fr:Formule du binôme de Newton]]\n[[he:הבינום של ניוטון]]\n[[ja:二項定理]]\n[[ko:이항정리]]\n[[nl:Binomium van Newton]]\n[[ru:Бином Ньютона]]\n[[sv:Binomialsatsen]]\n[[zh:二项式定理]]\n
<html>\n<center>\n<embed width="400" height="200" src="d.svg" script='\nxmin=-2pi; xmax=2pi; xscl=1; axes(); stroke="red"\nplot(ln(x))\n'>\n</center>\n</html>\n
<html>\n<center>\n\n<embed width="400" height="200" src="d.svg" script='\nxmin=-3; xmax=3; xscl=1; axes(); stroke="red"\nplot(-x^2 +3/2)\n'>\n</center>\n</html>\n
<html>\n<center>\n<embed width="400" height="200" src="d.svg" script='\nxmin=-2pi; xmax=2pi; xscl=1; axes(); stroke="red"\nplot(sin(x))\n'>\n</center>\n</html>\n
circonferenza(O,r)$:={\sforall(A):d(A,O}=r}$\ncerchio(O,r)$:={\sforall(A):d(A,O}\sler}$\nasse(A,B)$:={\sforall(P):d(P,A}=d(P,B) }$\n
jsMath is a collection of ~JavaScripts, written by Davide Cervone that convert LaTeX formulas to typeset expressions within a webpage while the page is loading. The scripts can be freely downloaded from
<html>\n<center>\n<embed width="300" height="300" src="/d.svg" \nscript='\nfunction update() {\n p = []\n with (Math) \n for (var t = 0; less(t,2.1*PI); t += 0.01)\n p[p.length] = [sin(a*t), sin(b*t)]\n path(p,"Lissajous")\n setText(a,"aval")\n setText(b,"bval")\n}\n initPicture(-1,1,-1,1)\n\n stroke = "red"\n\n axes()\n\n a = 2\n\n b = 3\n\n update()'>\n\n\n\n</script>\n</center>\n</html>
`sum_(n=1)^oo 1/n^2=pi^2/6`\n
{{Calcolo letterale}}\n\nIn [[matematica]] un '''monomio''' è un particolare tipo di [[polinomio]] composto semplicemente da un termine solo. Monomi quindi sono tutte le potenze della variabile ''x'', ovvero $x^n$, con ''n'' [[numero intero|intero]]. Se poi si hanno differenti variabili ''x'', ''y'', ''z'', un monomio prodotto di queste sarà della forma $x^ay^bz^c$, con ''a'', ''b'' e ''c'' numeri interi. Monomio è considerata anche un'espressione simile con un coefficiente diverso da 1 (solitamente sottinteso), come ad esempio ''Cx<sup>a</sup>y<sup>b</sup>z<sup>c</sup>'', con ''C'' un qualsiasi [[numero reale]]. Le [[operazioni coi monomi]] si eseguono così come avviene coi numeri: le differenze sono solo apparenti.\n\nOvviamente si può definire un qualsiasi polinomio come una [[combinazione lineare]] di monomi, che così possono essere presi come [[base]] dello [[spazio vettoriale]] dei polinomi. Dall'[[analisi funzionale]] deriva poi il fatto che un [[insieme]] completo di monomi non serve per misurare un [[sottospazio]] lineare di ''C''[0,1], che è denso per la [[norma uniforme]]: è sufficiente che la somma dei reciproci ''n''<sup>-1</sup> diverga.\n\nPer campi come le [[equazioni differenziali parziali]] è sempre richiesta una notazione a monomi. La ''notazione a multi-indici'' è molto utile: se si scrive ''&alpha;'' = (''a'', ''b'', ''c''), possiamo allora definire ''x<sup>&alpha;</sup>'' = ''x''<sub>1</sub>''<sup>a</sup>x''<sub>2</sub>''<sup>b</sup>x''<sub>3</sub>''<sup>c</sup>'', guadagnando in spazio e semplicità di scrittura.\n\n----\n\nIn [[geometria algebrica]] le varietà definite da equazioni di monomi del tipo ''x<sup>&alpha;</sup>'' = 0 per un certo insieme di ''&alpha;'' hanno speciali proprietà di omogeneità. Ciò può essere tradotto nel linguaggio dei [[gruppo algebrico|gruppi algebrici]] in termini di esistenza di un '''gruppo azione''' di un '''toro algebrico''' (ch è l'equivalente del gruppo moltiplicatico per le [[matrice diagonale|matrici diagonali]].\n\n----\n\nIn [[teoria dei gruppi]] si parla di [[rappresentazione monomio]], che è un particolare genere di [[rappresentazione indotta]].\n\n[[Categoria:Algebra]]\n\n[[de:monom]]\n[[en:monomial]]\n[[fr:monôme]]\n[[pl:Jednomian]]\n
$\slim_{N \sto \sinfty} \ssum_{k=1}^N f(t_k) \sDelta t.$\n$AA(A,B,C) : bar{sdgd}$
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The first law of thermodynamics \n{{Laws of thermodynamics}}\n{{main|First law of thermodynamics}}\n\nFor a thermodynamic system with a fixed number of particles, the first law of thermodynamics may be stated as:\n\n:$\sdelta Q = dU + \sdelta W$\n\nwhere $\sdelta Q$ is the amount of energy added to the system by a heating process, $\sdelta W$ is the amount of energy lost by the system due to work done by the system on its surroundings and $dU$ is the increase in the internal energy of the system.\n\nThe &delta;'s before the heat and work terms are used to indicate that they describe an increment of energy which is to be interpreted somewhat differently than the ''dU'' increment of internal energy. Work and heat are ''processes'' which add or subtract energy, while the internal energy ''U'' is a particular ''form'' of energy associated with the system. Thus the term "heat energy" for $\sdelta Q$ means "that amount of energy added as the result of heating" rather than referring to a particular form of energy. Likewise, the term "work energy" for $\sdelta w$ means "that amount of energy lost as the result of work". The most significant result of this distinction is the fact that one can clearly state the amount of internal energy posessed by a thermodynamic system, but one cannot tell how much energy has flowed into or out of the system as a result of its being heated or cooled, nor as the result of work being performed on or by the system.\n\nThe first law can be written exclusively in terms of system variables. The work done by the system may be written\n$ \sdelta w = P dV $\n\nwhere ''P'' is the [[pressure]] and ''dV'' is a small change in the [[volume]] of the system, each of which are system variables. The heat energy may be written\n\n:$\sdelta q = T\sdS$\n\nwhere ''T'' is the [[temperature]] and ''dS'' is a small change in the [[entropy]] of the system. Temperature and entropy are also system variables.\n
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