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if at least one occurrence of it is bound. Since the same variable symbol may appear in multiple places in an expression, some occurrences of the variable symbol may be free while others are bound, hence "free" and "bound" are at first defined for occurrences and then generalized over all occurrences
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While the domain of discourse in many contexts is understood, when an explicit range of values for the bound variable has not been given, it may be necessary to specify the domain in order to properly evaluate the expression. For example, consider the following expression in which both variables are
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of said variable symbol in the expression. However it is done, the variable ceases to be an independent variable on which the value of the expression depends, whether that value be a truth value or the numerical result of a calculation, or, more generally, an element of an image set of a function.
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Variables bound at the top level of a program are technically free variables within the terms to which they are bound but are often treated specially because they can be compiled as fixed addresses. Similarly, an identifier bound to a
2537:) while true pronouns are free variables in some grammatical structures but variables that cannot be bound in other grammatical structures. The binding phenomena found in natural languages was particularly important to the syntactic
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The term "dummy variable" is also sometimes used for a bound variable (more commonly in general mathematics than in computer science), but this should not be confused with the identically named but unrelated concept of
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which act on functions of the bound variable. In more complicated contexts, such notations can become awkward and confusing. It can be useful to switch to notations which make the binding explicit, such as
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English does allow specifying coreference, but it is optional, as both interpretations of the previous example are valid (the ungrammatical interpretation is indicated with an asterisk):
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is an operator with two parameters—a one-parameter function, and a set to evaluate that function over. The other operators listed above can be expressed in similar ways; for example, the
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whose leaf nodes are variables, constants, function constants or predicate constants and whose non-leaf nodes are logical operators. This expression can then be determined by doing an
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Before stating a precise definition of free variable and bound variable, the following are some examples that perhaps make these two concepts clearer than the definition would:
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248:. This may be achieved through the use of logical quantifiers, variable-binding operators, or an explicit statement of allowed values for the variable (such as, "...where
1112:{\displaystyle \sum _{x\in S}\quad \quad \prod _{x\in S}\quad \quad \int _{0}^{\infty }\cdots \,dx\quad \quad \lim _{x\to 0}\quad \quad \forall x\quad \quad \exists x}
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More widely, in most proofs, bound variables are used. For example, the following proof shows that all squares of positive even integers are divisible by
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Variable-binding mechanisms occur in different contexts in mathematics, logic and computer science. In all cases, however, they are purely
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indicating both the semantic subject and the semantic object of sentence as being bound. This returns the semantic interpretation
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properties of expressions and variables in them. For this section we can summarize syntax by identifying an expression with a
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can only refer to a female that is not Ashley. This means that it can never have a reflexive meaning equivalent to
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The distinction is not purely of academic interest, as some languages do actually have different forms for
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and it may contain other variables. In this case we say that function definition binds the variables
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The first interpretation is impossible. Only the second interpretation is permitted by the grammar.
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may take place and is not a parameter of this or any container expression. The idea is related to a
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that occur in almost every formal language. A binding operator Q takes two arguments: a variable
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Thus, it can be seen that reflexives and reciprocals are bound variables (known technically as
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In this manner, function definition expressions of the kind shown above can be thought of as
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coindexed is permissible, but the other interpretation where they are not coindexed is
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is a bound variable; consequently the value of this expression depends on the value of
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is a bound variable; consequently the value of this expression depends on the value of
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is a bound variable; consequently the value of this expression depends on the value of
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To give an example from mathematics, consider an expression which defines a function
162:, a variable may be said to be either free or bound. Some older books use the terms
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2325:, and can never refer to a different female person. In this example, the variable
2085:) or to a book that belongs to a different female (e.g. Jane's book). Whoever the
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is also technically a free variable within its own body but is treated specially.
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is the predicate function (a lambda abstraction) with the lambda notation and
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position. Indicating the coindexation, the first interpretation with
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Pronouns can also behave in a different way. In the sentence below
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variable binding operator, analogous to the lambda expressions of
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330:{\displaystyle \forall y\,\exists x\,\left(x={\sqrt {y}}\right).}
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1211:{\displaystyle \sum _{1,\ldots ,10}\left(k\mapsto f(k,n)\right)}
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659:{\displaystyle \lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}},}
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as used in statistics, most commonly in regression analysis.
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have been used as bound variables as a whole in the proof.
2411:. The sentence with the reflexive could be represented as
2476:. The grammatical and ungrammatical interpretations are:
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is can be established according to the situational (i.e.
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1657:
applying to a function. So, for example, the expression
1405:). The meaning of binding operators is supplied by the
33:"Free variable" redirects here. Not to be confused with
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can be thought of as an operator that evaluates to the
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is rebound in this term. This nested, inner binding of
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for that variable in an expression and a non-leaf node
1363:{\displaystyle \forall x\,((\exists y\,A(x))\vee B(z))}
26:
For free variables in systems of linear equations, see
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is said to "shadow" the outer binding. Occurrences of
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is a positive integer".) A variable symbol overall is
2670:. Vol. 30. Mathematical Association of America.
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be a positive even integer. Then there is an integer
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594:
557:{\displaystyle \int _{0}^{\infty }x^{y-1}e^{-x}\,dx,}
499:
416:
370:
350:
286:
254:
2403:The coreference binding can be represented using a
74:. Unsourced material may be challenged and removed.
2081:could be referring to Lisa's book (an instance of
1836:
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1412:Variable binding relates three things: a variable
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260:
2031:, natural languages can be seen to have free and
1278:{\displaystyle D\left(x\mapsto x^{2}+2x+1\right)}
748:
717:
202:that will later be replaced by some value), or a
19:For bound variables in computer programming, see
3863:
2605:(1981). Harvard University Press, 0-674-55451-5.
1078:
596:
2065:In the sentence above, the possessive pronoun
1409:of the language and does not concern us here.
1300:Tree summarizing the syntax of the expression
759:{\displaystyle \forall x\ \exists y\ {\Big },}
2811:
999:
2105:indicates a second referent (different from
1385:of the tree. Variable-binding operators are
2317:can only refer to the previously mentioned
1942:. Unsourced material may be challenged and
1901:Learn how and when to remove these messages
1755:{\displaystyle \sum _{S}{(x\mapsto x^{2})}}
788:of this expression depends on the value of
28:Free variables (system of linear equations)
2818:
2804:
16:Concept in mathematics or computer science
3846:Regiomontanus' angle maximization problem
2597:
2595:
2015:Learn how and when to remove this message
1962:Learn how and when to remove this message
1855:applied over the (possibly infinite) set
1326:
1313:
1068:
544:
300:
293:
134:Learn how and when to remove this message
3689:
2638:
2627:
2077:or to any other female. In other words,
1295:
3194:Differentiating under the integral sign
2408:
1513:
462:{\displaystyle \sum _{k=1}^{10}f(k,n),}
206:that stands for an unspecified symbol.
3864:
2714:The Princeton Companion to Mathematics
2643:. Wokingham, England: Addison-Wesley.
2641:Type theory and functional programming
2592:
388:if the domain is the complex numbers.
3070:Inverse functions and differentiation
2799:
2435:is the subject referent argument and
1698:{\displaystyle \sum _{x\in S}{x^{2}}}
1606:may contain some, all or none of the
1510:is one containing no free variables.
1291:
780:are bound variables, associated with
186:(symbol) that specifies places in an
150:, and in other disciplines involving
2665:
1973:
1940:adding citations to reliable sources
1907:
1866:
1837:{\displaystyle \forall x\in S\ P(x)}
1649:. Other binding operators, like the
236:An instance of a variable symbol is
233:is often a synonym in this context.
83:"Free variables and bound variables"
72:adding citations to reliable sources
43:
2113:has the following interpretations:
1862:
1708:could be treated as a notation for
1126:. Each of them binds the variable
213:, the term free variable refers to
13:
2868:Free variables and bound variables
2693:
2051:, etc. can act as free variables.
1807:
1320:
1307:
1103:
1095:
1060:
706:
697:
510:
294:
287:
14:
3898:
3673:The Method of Mechanical Theorems
2668:A Tour through Mathematical Logic
2616:A Tour through Mathematical Logic
1882:This section has multiple issues.
3228:Partial fractions in integration
3144:Stochastic differential equation
1977:
1912:
1871:
1491:are free occurrences of the new
1451:and a free variable in the term
1447:is a bound variable in the term
48:
3366:Jacobian matrix and determinant
3221:Tangent half-angle substitution
3189:Fundamental theorem of calculus
2271:= of a female that is not Lisa)
2162:= of a female that is not Lisa)
1890:or discuss these issues on the
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1101:
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59:needs additional citations for
3442:Arithmetico-geometric sequence
3134:Ordinary differential equation
2608:
2526:= a female that is not Ashley)
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792:, but there is nothing called
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680:, but there is nothing called
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578:, but there is nothing called
483:, but there is nothing called
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277:bound by logical quantifiers:
1:
3265:Integro-differential equation
3139:Partial differential equation
2585:
2539:government and binding theory
2407:as mentioned in the previous
340:This expression evaluates to
2825:
2399:= a female that is not Jane)
2073:to the previously mentioned
1790:{\displaystyle \sum _{S}{f}}
931:{\displaystyle n^{2}=4k^{2}}
7:
3419:Generalized Stokes' theorem
3206:Integration by substitution
2548:
2101:indicates one referent and
2069:is a free variable. It may
1653:sign, can be thought of as
399:
229:of that function. The term
39:Dummy variable (statistics)
10:
3905:
2948:(ε, δ)-definition of limit
2723:Princeton University Press
2555:Closure (computer science)
2409:Formal explanation section
1124:variable-binding operators
1000:Variable-binding operators
800:on which it could depend.
684:on which it could depend.
582:on which it could depend.
487:on which it could depend.
32:
25:
18:
3841:Proof that 22/7 exceeds π
3778:
3756:
3682:
3630:Gottfried Wilhelm Leibniz
3600:
3577:e (mathematical constant)
3562:
3434:
3341:
3273:
3154:
2956:
2911:
2833:
2618:(2005). 978-0-88385-036-7
384:is the real numbers, but
3592:Stirling's approximation
3065:Implicit differentiation
3013:Rules of differentiation
2666:Wolf, Robert S. (2005).
2639:Thompson, Simon (1991).
3826:Euler–Maclaurin formula
3731:trigonometric functions
3184:Constant of integration
2451:being the same person.
2035:variables. In English,
1850:Boolean-valued function
1588:{\displaystyle f=\left}
772:is a free variable and
672:is a free variable and
570:is a free variable and
475:is a free variable and
3795:Differential geometry
3640:Infinitesimal calculus
3343:Multivariable calculus
3291:Directional derivative
3097:Second derivative test
3075:Logarithmic derivative
3048:General Leibniz's rule
2943:Order of approximation
2109:). Thus, the sentence
1838:
1791:
1756:
1699:
1655:higher-order functions
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3872:Mathematical notation
3714:logarithmic functions
3709:exponential functions
3625:Generality of algebra
3503:Tests of convergence
3129:Differential equation
3113:Further applications
3102:Extreme value theorem
3092:First derivative test
2986:Differential operator
2958:Differential calculus
2731:10.1515/9781400830398
2719:Princeton, New Jersey
2543:Binding (linguistics)
2329:is bound to the noun
2192:translate coreferent
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1288:for differentiation.
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958:{\displaystyle n^{2}}
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3882:Computer programming
3779:Miscellaneous topics
3719:hyperbolic functions
3704:irrational functions
3582:Exponential function
3435:Sequences and series
3201:Integration by parts
2522:(interpretation #2:
2496:(interpretation #1:
2395:(interpretation #2:
2368:(interpretation #1:
2267:(interpretation #2:
2240:(interpretation #1:
2158:(interpretation #2:
2132:(interpretation #1:
1936:improve this section
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1799:universal quantifier
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1514:Function expressions
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888:{\displaystyle n=2k}
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211:computer programming
68:improve this article
3766:List of derivatives
3602:History of calculus
3517:Cauchy condensation
3414:Exterior derivative
3371:Lagrange multiplier
3107:Maximum and minimum
2938:Limit of a sequence
2926:Limit of a function
2873:Graph of a function
2853:Continuous function
2686:10.4169/j.ctt5hh94h
2575:Scope (programming)
2293:reciprocal pronouns
2111:Lisa found her book
1846:logical conjunction
1471:contains a subterm
1064:
784:; consequently the
782:logical quantifiers
514:
490:In the expression
242:domain of discourse
3699:rational functions
3666:Method of Fluxions
3512:Alternating series
3409:Differential forms
3391:Partial derivative
3351:Divergence theorem
3233:Quadratic integral
3001:Leibniz's notation
2991:Mean value theorem
2976:Partial derivative
2921:Indeterminate form
2725:. pp. 15–16.
2705:Barrow-Green, June
2603:Mathematical Logic
2474:Ashley hit herself
2277:reflexive pronouns
2205:and noncoreferent
1989:. You can help by
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1602:is an expression.
1585:
1501:recursive function
1393:and an expression
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1292:Formal explanation
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1137:Many of these are
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687:In the expression
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585:In the expression
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407:In the expression
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231:non-local variable
204:wildcard character
178:, respectively. A
156:mathematical logic
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3858:
3785:Complex calculus
3774:
3773:
3655:Law of Continuity
3587:Natural logarithm
3572:Bernoulli numbers
3563:Special functions
3522:Direct comparison
3386:Multiple integral
3260:Integral equation
3156:Integral calculus
3087:Stationary points
3061:Other techniques
3006:Newton's notation
2971:Second derivative
2863:Finite difference
2740:978-0-691-11880-2
2677:978-0-88385-042-8
2560:Combinatory logic
2405:lambda expression
2037:personal pronouns
2027:When analyzed in
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2024:
2017:
2007:
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1387:logical operators
1383:inorder traversal
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978:{\displaystyle 4}
859:{\displaystyle k}
839:{\displaystyle n}
816:{\displaystyle 4}
714:
705:
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377:{\displaystyle y}
357:{\displaystyle x}
344:if the domain of
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261:{\displaystyle n}
221:that are neither
168:apparent variable
144:
143:
136:
118:
3894:
3789:Contour integral
3687:
3686:
3537:Limit comparison
3446:Types of series
3405:Advanced topics
3396:Surface integral
3240:Trapezoidal rule
3179:Basic properties
3174:Riemann integral
3122:Taylor's theorem
2848:Concave function
2843:Binomial theorem
2820:
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2614:Robert S. Wolf,
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2601:W. V. O. Quine,
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2029:formal semantics
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954:
953:
937:
935:
934:
929:
927:
926:
911:
910:
894:
892:
891:
886:
865:
863:
862:
857:
845:
843:
842:
837:
822:
820:
819:
814:
765:
763:
762:
757:
752:
751:
721:
720:
712:
703:
665:
663:
662:
657:
652:
647:
612:
609:
563:
561:
560:
555:
543:
542:
530:
529:
513:
508:
468:
466:
465:
460:
436:
431:
383:
381:
380:
375:
363:
361:
360:
355:
336:
334:
333:
328:
323:
319:
318:
313:
267:
265:
264:
259:
160:computer science
152:formal languages
139:
132:
128:
125:
119:
117:
76:
52:
44:
3904:
3903:
3897:
3896:
3895:
3893:
3892:
3891:
3887:Predicate logic
3862:
3861:
3860:
3855:
3851:Steinmetz solid
3836:Integration Bee
3770:
3752:
3678:
3620:Colin Maclaurin
3596:
3564:
3558:
3430:
3424:Tensor calculus
3401:Volume integral
3337:
3312:Basic theorems
3275:Vector calculus
3269:
3150:
3117:Newton's method
2952:
2931:One-sided limit
2907:
2888:Rolle's theorem
2878:Linear function
2829:
2824:
2741:
2711:, eds. (2008).
2701:Gowers, Timothy
2696:
2694:Further reading
2678:
2651:
2635:
2634:
2626:
2622:
2613:
2609:
2600:
2593:
2588:
2551:
2519:
2513:
2493:
2487:
2392:
2386:
2365:
2359:
2333:that occurs in
2321:, in this case
2264:
2258:
2237:
2231:
2212:
2199:
2184:: for example,
2182:
2173:
2155:
2149:
2129:
2123:
2021:
2010:
2009:
2008:
2003:
1997:
1994:
1987:needs expansion
1968:
1957:
1951:
1948:
1933:
1917:
1876:
1872:
1865:
1805:
1802:
1801:
1782:
1776:
1770:
1767:
1766:
1742:
1738:
1728:
1722:
1716:
1713:
1712:
1688:
1684:
1683:
1671:
1665:
1662:
1661:
1647:lambda calculus
1637:
1628:
1621:
1612:
1565:
1561:
1546:
1542:
1538:
1534:
1526:
1523:
1522:
1516:
1492:
1488:
1484:
1480:
1476:
1472:
1468:
1464:
1460:
1456:
1452:
1448:
1444:
1441:lambda calculus
1305:
1302:
1301:
1294:
1249:
1245:
1238:
1234:
1229:
1226:
1225:
1178:
1174:
1156:
1150:
1147:
1146:
1081:
1059:
1054:
1036:
1018:
1012:
1009:
1008:
1002:
970:
967:
966:
949:
945:
943:
940:
939:
922:
918:
906:
902:
900:
897:
896:
871:
868:
867:
851:
848:
847:
831:
828:
827:
808:
805:
804:
747:
746:
716:
715:
695:
692:
691:
613:
611:
599:
593:
590:
589:
535:
531:
519:
515:
509:
504:
498:
495:
494:
432:
421:
415:
412:
411:
402:
369:
366:
365:
349:
346:
345:
312:
305:
301:
285:
282:
281:
253:
250:
249:
223:local variables
140:
129:
123:
120:
77:
75:
65:
53:
42:
31:
24:
17:
12:
11:
5:
3902:
3901:
3890:
3889:
3884:
3879:
3874:
3857:
3856:
3854:
3853:
3848:
3843:
3838:
3833:
3831:Gabriel's horn
3828:
3823:
3822:
3821:
3816:
3811:
3806:
3801:
3793:
3792:
3791:
3782:
3780:
3776:
3775:
3772:
3771:
3769:
3768:
3763:
3761:List of limits
3757:
3754:
3753:
3751:
3750:
3749:
3748:
3743:
3738:
3728:
3727:
3726:
3716:
3711:
3706:
3701:
3695:
3693:
3684:
3680:
3679:
3677:
3676:
3669:
3662:
3660:Leonhard Euler
3657:
3652:
3647:
3642:
3637:
3632:
3627:
3622:
3617:
3612:
3606:
3604:
3598:
3597:
3595:
3594:
3589:
3584:
3579:
3574:
3568:
3566:
3560:
3559:
3557:
3556:
3555:
3554:
3549:
3544:
3539:
3534:
3529:
3524:
3519:
3514:
3509:
3501:
3500:
3499:
3494:
3493:
3492:
3487:
3477:
3472:
3467:
3462:
3457:
3452:
3444:
3438:
3436:
3432:
3431:
3429:
3428:
3427:
3426:
3421:
3416:
3411:
3403:
3398:
3393:
3388:
3383:
3378:
3373:
3368:
3363:
3361:Hessian matrix
3358:
3353:
3347:
3345:
3339:
3338:
3336:
3335:
3334:
3333:
3328:
3323:
3318:
3316:Line integrals
3310:
3309:
3308:
3303:
3298:
3293:
3288:
3279:
3277:
3271:
3270:
3268:
3267:
3262:
3257:
3256:
3255:
3250:
3242:
3237:
3236:
3235:
3225:
3224:
3223:
3218:
3213:
3203:
3198:
3197:
3196:
3186:
3181:
3176:
3171:
3166:
3164:Antiderivative
3160:
3158:
3152:
3151:
3149:
3148:
3147:
3146:
3141:
3136:
3126:
3125:
3124:
3119:
3111:
3110:
3109:
3104:
3099:
3094:
3084:
3083:
3082:
3077:
3072:
3067:
3059:
3058:
3057:
3052:
3051:
3050:
3040:
3035:
3030:
3025:
3020:
3010:
3009:
3008:
3003:
2993:
2988:
2983:
2978:
2973:
2968:
2962:
2960:
2954:
2953:
2951:
2950:
2945:
2940:
2935:
2934:
2933:
2923:
2917:
2915:
2909:
2908:
2906:
2905:
2900:
2895:
2890:
2885:
2880:
2875:
2870:
2865:
2860:
2855:
2850:
2845:
2839:
2837:
2831:
2830:
2823:
2822:
2815:
2808:
2800:
2794:
2793:
2739:
2695:
2692:
2691:
2690:
2676:
2663:
2649:
2633:
2632:
2620:
2607:
2590:
2589:
2587:
2584:
2583:
2582:
2577:
2572:
2567:
2565:Lambda lifting
2562:
2557:
2550:
2547:
2528:
2527:
2515:
2509:
2505:
2489:
2483:
2466:
2465:
2445:JANE hurt JANE
2429:
2428:
2401:
2400:
2388:
2382:
2377:
2361:
2355:
2313:the reflexive
2311:
2310:
2273:
2272:
2260:
2254:
2249:
2233:
2227:
2208:
2195:
2178:
2169:
2164:
2163:
2151:
2145:
2141:
2125:
2119:
2063:
2062:
2023:
2022:
2005:
2004:
1984:
1982:
1970:
1969:
1920:
1918:
1911:
1906:
1880:
1879:
1877:
1870:
1864:
1861:
1833:
1830:
1827:
1824:
1818:
1815:
1812:
1809:
1785:
1779:
1775:
1763:
1762:
1750:
1745:
1741:
1737:
1734:
1731:
1725:
1721:
1706:
1705:
1691:
1687:
1680:
1677:
1674:
1670:
1633:
1626:
1617:
1610:
1596:
1595:
1583:
1579:
1576:
1573:
1568:
1564:
1560:
1557:
1554:
1549:
1545:
1541:
1537:
1533:
1530:
1515:
1512:
1424:of the form Q(
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1325:
1322:
1319:
1316:
1312:
1309:
1293:
1290:
1286:
1285:
1273:
1269:
1266:
1263:
1260:
1257:
1252:
1248:
1244:
1241:
1237:
1233:
1219:
1218:
1206:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1177:
1171:
1168:
1165:
1162:
1159:
1155:
1120:
1119:
1108:
1105:
1100:
1097:
1090:
1087:
1084:
1080:
1074:
1071:
1067:
1062:
1057:
1053:
1045:
1042:
1039:
1035:
1027:
1024:
1021:
1017:
1004:The following
1001:
998:
986:
985:
974:
952:
948:
925:
921:
917:
914:
909:
905:
884:
881:
878:
875:
855:
835:
812:
767:
766:
755:
750:
745:
742:
739:
736:
733:
730:
727:
724:
719:
711:
708:
702:
699:
667:
666:
655:
650:
646:
643:
640:
637:
634:
631:
628:
625:
622:
619:
616:
608:
605:
602:
598:
565:
564:
553:
550:
547:
541:
538:
534:
528:
525:
522:
518:
512:
507:
503:
470:
469:
458:
455:
452:
449:
446:
443:
440:
435:
430:
427:
424:
420:
401:
398:
394:dummy variable
373:
353:
338:
337:
326:
322:
316:
311:
308:
304:
299:
296:
292:
289:
257:
176:bound variable
142:
141:
56:
54:
47:
35:Free parameter
15:
9:
6:
4:
3:
2:
3900:
3899:
3888:
3885:
3883:
3880:
3878:
3877:Logic symbols
3875:
3873:
3870:
3869:
3867:
3852:
3849:
3847:
3844:
3842:
3839:
3837:
3834:
3832:
3829:
3827:
3824:
3820:
3817:
3815:
3812:
3810:
3807:
3805:
3802:
3800:
3797:
3796:
3794:
3790:
3787:
3786:
3784:
3783:
3781:
3777:
3767:
3764:
3762:
3759:
3758:
3755:
3747:
3744:
3742:
3739:
3737:
3734:
3733:
3732:
3729:
3725:
3722:
3721:
3720:
3717:
3715:
3712:
3710:
3707:
3705:
3702:
3700:
3697:
3696:
3694:
3692:
3688:
3685:
3681:
3675:
3674:
3670:
3668:
3667:
3663:
3661:
3658:
3656:
3653:
3651:
3648:
3646:
3643:
3641:
3638:
3636:
3635:Infinitesimal
3633:
3631:
3628:
3626:
3623:
3621:
3618:
3616:
3613:
3611:
3608:
3607:
3605:
3603:
3599:
3593:
3590:
3588:
3585:
3583:
3580:
3578:
3575:
3573:
3570:
3569:
3567:
3561:
3553:
3550:
3548:
3545:
3543:
3540:
3538:
3535:
3533:
3530:
3528:
3525:
3523:
3520:
3518:
3515:
3513:
3510:
3508:
3505:
3504:
3502:
3498:
3495:
3491:
3488:
3486:
3483:
3482:
3481:
3478:
3476:
3473:
3471:
3468:
3466:
3463:
3461:
3458:
3456:
3453:
3451:
3448:
3447:
3445:
3443:
3440:
3439:
3437:
3433:
3425:
3422:
3420:
3417:
3415:
3412:
3410:
3407:
3406:
3404:
3402:
3399:
3397:
3394:
3392:
3389:
3387:
3384:
3382:
3379:
3377:
3376:Line integral
3374:
3372:
3369:
3367:
3364:
3362:
3359:
3357:
3354:
3352:
3349:
3348:
3346:
3344:
3340:
3332:
3329:
3327:
3324:
3322:
3319:
3317:
3314:
3313:
3311:
3307:
3304:
3302:
3299:
3297:
3294:
3292:
3289:
3287:
3284:
3283:
3281:
3280:
3278:
3276:
3272:
3266:
3263:
3261:
3258:
3254:
3251:
3249:
3248:Washer method
3246:
3245:
3243:
3241:
3238:
3234:
3231:
3230:
3229:
3226:
3222:
3219:
3217:
3214:
3212:
3211:trigonometric
3209:
3208:
3207:
3204:
3202:
3199:
3195:
3192:
3191:
3190:
3187:
3185:
3182:
3180:
3177:
3175:
3172:
3170:
3167:
3165:
3162:
3161:
3159:
3157:
3153:
3145:
3142:
3140:
3137:
3135:
3132:
3131:
3130:
3127:
3123:
3120:
3118:
3115:
3114:
3112:
3108:
3105:
3103:
3100:
3098:
3095:
3093:
3090:
3089:
3088:
3085:
3081:
3080:Related rates
3078:
3076:
3073:
3071:
3068:
3066:
3063:
3062:
3060:
3056:
3053:
3049:
3046:
3045:
3044:
3041:
3039:
3036:
3034:
3031:
3029:
3026:
3024:
3021:
3019:
3016:
3015:
3014:
3011:
3007:
3004:
3002:
2999:
2998:
2997:
2994:
2992:
2989:
2987:
2984:
2982:
2979:
2977:
2974:
2972:
2969:
2967:
2964:
2963:
2961:
2959:
2955:
2949:
2946:
2944:
2941:
2939:
2936:
2932:
2929:
2928:
2927:
2924:
2922:
2919:
2918:
2916:
2914:
2910:
2904:
2901:
2899:
2896:
2894:
2891:
2889:
2886:
2884:
2881:
2879:
2876:
2874:
2871:
2869:
2866:
2864:
2861:
2859:
2856:
2854:
2851:
2849:
2846:
2844:
2841:
2840:
2838:
2836:
2832:
2828:
2821:
2816:
2814:
2809:
2807:
2802:
2801:
2798:
2790:
2786:
2782:
2778:
2774:
2770:
2766:
2762:
2758:
2754:
2750:
2746:
2742:
2736:
2732:
2728:
2724:
2720:
2716:
2715:
2710:
2706:
2702:
2698:
2697:
2687:
2683:
2679:
2673:
2669:
2664:
2660:
2656:
2652:
2646:
2642:
2637:
2636:
2630:, p. 33.
2629:
2628:Thompson 1991
2624:
2617:
2611:
2604:
2598:
2596:
2591:
2581:
2580:Scope (logic)
2578:
2576:
2573:
2571:
2568:
2566:
2563:
2561:
2558:
2556:
2553:
2552:
2546:
2544:
2540:
2536:
2531:
2525:
2521:
2518:
2512:
2506:
2503:
2499:
2495:
2492:
2486:
2479:
2478:
2477:
2475:
2471:
2464:
2462:
2457:
2456:
2455:
2452:
2450:
2446:
2442:
2438:
2434:
2426:
2422:
2418:
2414:
2413:
2412:
2410:
2406:
2398:
2394:
2391:
2385:
2378:
2375:
2371:
2367:
2364:
2358:
2352:
2351:
2350:
2348:
2347:ungrammatical
2344:
2340:
2336:
2332:
2328:
2324:
2320:
2316:
2309:
2307:
2302:
2301:
2300:
2298:
2294:
2290:
2286:
2282:
2278:
2270:
2266:
2263:
2257:
2250:
2247:
2243:
2239:
2236:
2230:
2224:
2223:
2222:
2219:
2217:
2213:
2211:
2204:
2200:
2198:
2191:
2187:
2183:
2181:
2174:
2172:
2161:
2157:
2154:
2148:
2142:
2139:
2135:
2131:
2128:
2122:
2116:
2115:
2114:
2112:
2108:
2104:
2100:
2096:
2092:
2088:
2084:
2080:
2076:
2072:
2068:
2061:
2059:
2054:
2053:
2052:
2050:
2046:
2042:
2038:
2034:
2030:
2019:
2016:
2001:
1998:December 2008
1992:
1988:
1985:This section
1983:
1980:
1976:
1975:
1966:
1963:
1955:
1945:
1941:
1937:
1931:
1930:
1926:
1921:This section
1919:
1915:
1910:
1909:
1904:
1902:
1895:
1894:
1889:
1888:
1883:
1878:
1869:
1868:
1860:
1858:
1854:
1851:
1847:
1828:
1822:
1816:
1813:
1810:
1800:
1783:
1777:
1773:
1743:
1739:
1732:
1723:
1719:
1711:
1710:
1709:
1689:
1685:
1678:
1675:
1672:
1668:
1660:
1659:
1658:
1656:
1652:
1648:
1644:
1639:
1636:
1632:
1625:
1620:
1616:
1609:
1605:
1601:
1581:
1577:
1566:
1562:
1558:
1555:
1552:
1547:
1543:
1535:
1531:
1528:
1521:
1520:
1519:
1511:
1509:
1504:
1502:
1496:
1442:
1437:
1435:
1431:
1427:
1423:
1419:
1416:, a location
1415:
1410:
1408:
1404:
1400:
1396:
1392:
1388:
1384:
1380:
1376:
1351:
1345:
1342:
1333:
1327:
1323:
1310:
1298:
1289:
1271:
1267:
1264:
1261:
1258:
1255:
1250:
1246:
1239:
1235:
1231:
1224:
1223:
1222:
1204:
1197:
1194:
1191:
1185:
1179:
1175:
1169:
1166:
1163:
1160:
1157:
1153:
1145:
1144:
1143:
1140:
1135:
1133:
1130:for some set
1129:
1125:
1106:
1098:
1088:
1082:
1072:
1069:
1065:
1055:
1051:
1043:
1040:
1037:
1033:
1025:
1022:
1019:
1015:
1007:
1006:
1005:
997:
995:
991:
972:
965:divisible by
950:
946:
923:
919:
915:
912:
907:
903:
882:
879:
876:
873:
853:
833:
825:
824:
823:
810:
801:
799:
795:
791:
787:
786:logical value
783:
779:
775:
771:
753:
740:
737:
734:
731:
728:
722:
709:
700:
690:
689:
688:
685:
683:
679:
675:
671:
653:
648:
641:
635:
632:
626:
623:
620:
614:
606:
600:
588:
587:
586:
583:
581:
577:
573:
569:
551:
548:
545:
539:
536:
532:
526:
523:
520:
516:
505:
501:
493:
492:
491:
488:
486:
482:
478:
474:
456:
450:
447:
444:
438:
433:
428:
425:
422:
418:
410:
409:
408:
405:
397:
395:
389:
387:
371:
351:
343:
324:
320:
314:
309:
306:
302:
297:
290:
280:
279:
278:
274:
271:
255:
247:
243:
239:
234:
232:
228:
224:
220:
216:
212:
207:
205:
201:
197:
193:
189:
185:
181:
180:free variable
177:
173:
172:free variable
169:
165:
164:real variable
161:
157:
153:
149:
138:
135:
127:
124:December 2008
116:
113:
109:
106:
102:
99:
95:
92:
88:
85: –
84:
80:
79:Find sources:
73:
69:
63:
62:
57:This article
55:
51:
46:
45:
40:
36:
29:
22:
3746:Secant cubed
3671:
3664:
3645:Isaac Newton
3615:Brook Taylor
3282:Derivatives
3253:Shell method
2981:Differential
2867:
2712:
2709:Leader, Imre
2667:
2640:
2623:
2615:
2610:
2602:
2570:Name binding
2532:
2529:
2523:
2516:
2510:
2507:
2501:
2497:
2490:
2484:
2481:
2473:
2469:
2468:the pronoun
2467:
2460:
2458:
2453:
2448:
2444:
2440:
2436:
2432:
2430:
2424:
2420:
2416:
2402:
2396:
2389:
2387:hurt herself
2383:
2380:
2373:
2369:
2362:
2360:hurt herself
2356:
2353:
2342:
2338:
2330:
2326:
2322:
2314:
2312:
2305:
2303:
2296:
2291:, etc., and
2288:
2284:
2280:
2274:
2268:
2261:
2255:
2252:
2245:
2241:
2234:
2228:
2225:
2220:
2215:
2209:
2206:
2202:
2196:
2193:
2179:
2176:
2170:
2167:
2165:
2159:
2152:
2146:
2143:
2137:
2133:
2126:
2120:
2117:
2110:
2106:
2102:
2098:
2090:
2078:
2074:
2066:
2064:
2057:
2055:
2048:
2044:
2040:
2026:
2011:
1995:
1991:adding to it
1986:
1958:
1949:
1934:Please help
1922:
1898:
1891:
1885:
1884:Please help
1881:
1856:
1852:
1764:
1707:
1642:
1640:
1634:
1630:
1623:
1618:
1614:
1607:
1603:
1599:
1597:
1517:
1507:
1505:
1497:
1463:and free in
1459:is bound in
1438:
1433:
1429:
1425:
1421:
1417:
1413:
1411:
1402:
1398:
1394:
1390:
1372:
1287:
1221:for sums or
1220:
1136:
1131:
1127:
1123:
1121:
1003:
993:
989:
987:
802:
797:
793:
789:
777:
773:
769:
768:
686:
681:
677:
673:
669:
668:
584:
579:
575:
571:
567:
566:
489:
484:
480:
476:
472:
471:
406:
403:
390:
385:
341:
339:
275:
269:
237:
235:
208:
195:
192:substitution
179:
175:
171:
167:
163:
154:, including
145:
130:
121:
111:
104:
97:
90:
78:
66:Please help
61:verification
58:
21:Name binding
3814:of surfaces
3565:and numbers
3527:Dirichlet's
3497:Telescoping
3450:Alternating
3038:L'HĂ´pital's
2835:Precalculus
2541:(see also:
2459:Ashley hit
2437:λx.x hurt x
2083:coreference
2056:Lisa found
1508:closed term
196:placeholder
148:mathematics
3866:Categories
3610:Adequality
3296:Divergence
3169:Arc length
2966:Derivative
2789:1242.00016
2757:2008020450
2749:j.ctt7sd01
2650:0201416670
2586:References
2319:antecedent
2304:Jane hurt
2297:each other
2295:, such as
2289:themselves
2279:, such as
1952:March 2023
1887:improve it
938:, we have
866:such that
227:parameters
217:used in a
188:expression
94:newspapers
3809:of curves
3804:Curvature
3691:Integrals
3485:Maclaurin
3465:Geometric
3356:Geometric
3306:Laplacian
3018:linearity
2858:Factorial
2781:19327100M
2773:227205932
2431:in which
2275:However,
2265:own book.
2259:found her
2238:own book.
2232:found her
2186:Norwegian
2150:found her
2124:found her
2095:pragmatic
1923:does not
1893:talk page
1814:∈
1808:∀
1774:∑
1736:↦
1720:∑
1676:∈
1669:∑
1651:summation
1575:↦
1556:…
1455:. We say
1449:M = λx. T
1407:semantics
1375:syntactic
1343:∨
1321:∃
1308:∀
1243:↦
1183:↦
1164:…
1154:∑
1139:operators
1104:∃
1096:∀
1086:→
1066:⋯
1061:∞
1052:∫
1041:∈
1034:∏
1023:∈
1016:∑
992:but also
988:not only
723:φ
707:∃
698:∀
633:−
604:→
537:−
524:−
511:∞
502:∫
419:∑
295:∃
288:∀
215:variables
3799:Manifold
3532:Integral
3475:Infinite
3470:Harmonic
3455:Binomial
3301:Gradient
3244:Volumes
3055:Quotient
2996:Notation
2827:Calculus
2659:23287456
2549:See also
2535:anaphors
2087:referent
2079:her book
895:. Since
400:Examples
246:universe
219:function
184:notation
3736:inverse
3724:inverse
3650:Fluxion
3460:Fourier
3326:Stokes'
3321:Green's
3043:Product
2903:Tangent
2765:2467561
2514:hit her
2488:hit her
2397:herself
2370:herself
2343:herself
2335:subject
2327:herself
2315:herself
2306:herself
2285:herself
2281:himself
2190:Swedish
1944:removed
1929:sources
1848:of the
1439:In the
108:scholar
3819:Tensor
3741:Secant
3507:Abel's
3490:Taylor
3381:Matrix
3331:Gauss'
2913:Limits
2893:Secant
2883:Radian
2787:
2779:
2771:
2763:
2755:
2747:
2737:
2684:
2674:
2657:
2647:
2508:Ashley
2502:Ashley
2482:Ashley
2216:hennes
1820:
1765:where
1598:where
713:
704:
200:symbol
190:where
110:
103:
96:
89:
81:
3683:Lists
3542:Ratio
3480:Power
3216:Euler
3033:Chain
3023:Power
2898:Slope
2745:JSTOR
2682:JSTOR
2447:with
2427:)Jane
2423:hurt
2244:= of
2156:book.
2136:= of
2130:book.
2071:refer
2060:book.
2039:like
2033:bound
1629:, …,
1613:, …,
1475:then
1473:λx. U
1467:. If
342:false
270:bound
238:bound
182:is a
115:JSTOR
101:books
3552:Term
3547:Root
3286:Curl
2769:OCLC
2753:LCCN
2735:ISBN
2672:ISBN
2655:OCLC
2645:ISBN
2449:JANE
2433:Jane
2381:Jane
2374:Jane
2354:Jane
2341:and
2339:Jane
2331:Jane
2323:Jane
2253:Lisa
2246:Lisa
2226:Lisa
2188:and
2175:and
2144:Lisa
2138:Lisa
2118:Lisa
2075:Lisa
2049:they
1927:any
1925:cite
1379:tree
826:Let
776:and
386:true
364:and
225:nor
174:and
170:for
166:and
158:and
87:news
3028:Sum
2785:Zbl
2727:doi
2545:).
2524:her
2498:her
2470:her
2461:her
2269:her
2242:her
2214:as
2207:her
2203:sin
2201:as
2194:her
2177:her
2168:her
2160:her
2134:her
2091:her
2089:of
2067:her
2058:her
2045:she
1993:.
1938:by
1643:the
1487:in
1079:lim
796:or
597:lim
244:or
209:In
198:(a
146:In
70:by
37:or
3868::
2783:.
2777:OL
2775:.
2767:.
2761:MR
2759:.
2751:.
2743:.
2733:.
2721::
2717:.
2707:;
2703:;
2680:.
2653:.
2594:^
2500:=
2415:(λ
2372:=
2349::
2287:,
2283:,
2218:.
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2041:he
1896:.
1859:.
1638:.
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1401:,
1170:10
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