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{{short description|List of definitions of terms and concepts commonly used in calculus}}
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This '''glossary of calculus''' is a list of definitions about '''[[Calculus|calculus]]''', its sub-disciplines, and related fields.


{{Calculus}}

== A ==
{{glossary}}

{{term|[[Abel's test]]}}
{{defn|defn=A method of testing for the [[Convergent series|convergence]] of an [[Series (mathematics)|infinite series]].}}
{{term|[[Absolute convergence|absolute convergence]]}}
{{defn|defn=An [[Series (mathematics)|infinite series]] of numbers is said to '''converge absolutely''' (or to be '''absolutely convergent''') if the sum of the [[Absolute value|absolute value]]s of the summands is finite<!-- don't link to [[Finite set|finite set]], please -->. More precisely, a real or complex series <math>\textstyle\sum_{n=0}^\infty a_n</math> is said to '''converge absolutely''' if <math>\textstyle\sum_{n=0}^\infty \left|a_n\right| = L</math> for some real number <math>\textstyle L</math>. Similarly, an [[Improper integral|improper integral]] of a [[Function (mathematics)|function]], <math>\textstyle\int_0^\infty f(x)\,dx</math>, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if <math>\textstyle\int_0^\infty \left|f(x)\right|dx = L.</math>}}
{{term|[[Maxima and minima|absolute maximum]]}}
{{defn|defn=The highest value a function attains.}}
{{term|[[Maxima and minima|absolute minimum]]}}
{{defn|defn=The lowest value a function attains.}}
{{term|[[Absolute value|absolute value]]}}
{{defn|defn= The '''absolute value''' or '''modulus''' {{math|{{!}}''x''{{!}}}} of a [[Real number|real number]]&nbsp;{{mvar|x}} is the non-negative value of&nbsp;{{mvar|x}} without regard to its [[Sign (mathematics)|sign]]. Namely, {{math|1={{!}}''x''{{!}} = ''x''}} for a positive&nbsp;{{mvar|x}}, {{math|1={{!}}''x''{{!}} = −''x''}} for a [[Negative number|negative]]&nbsp;{{mvar|x}} (in which case {{math|−''x''}} is positive), and {{math|1={{!}}0{{!}} = 0}}. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its [[Distance|distance]] from zero.}}
{{term|[[Alternating series|alternating series]]}}
{{defn|defn= An infinite series whose terms alternate between positive and negative.}}
{{term|[[Alternating series test|alternating series test]]}}
{{defn|defn= Is the method used to prove that an [[Alternating series|alternating series]] with terms that decrease in absolute value is a [[Convergent series|convergent series]]. The test was used by Gottfried Leibniz and is sometimes known as '''Leibniz's test''', '''Leibniz's rule''', or the '''Leibniz criterion'''.}}
{{term|[[Annulus (mathematics)|annulus]]}}
{{defn|defn= A ring-shaped object, a region bounded by two concentric circles.}}
{{term|[[Antiderivative|antiderivative]]}}
{{term|[[Inverse trigonometric functions|arcsin]]}}
{{defn|defn=}}
{{term|area under a curve}}
{{defn|defn=}}
{{term|[[Asymptote|asymptote]]}}
{{defn|defn= In [[Analytic geometry|analytic geometry]], an '''asymptote''' of a [[Curve|curve]] is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates [[Limit of a function#Limits at infinity|tends to infinity]]. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors.<ref>[https://web.archive.org/web/*/http://rowdy.msudenver.edu/~talmanl/PDFs/APCalculus/OnAsymptotes.pdf "Asymptotes" by Louis A. Talman]</ref> In [[Projective geometry|projective geometry]] and related contexts, an asymptote of a curve is a line which is [[Tangent|tangent]] to the curve at a [[Point at infinity|point at infinity]].<ref>{{citation|title=An elementary treatise on the differential calculus|chapter=Asymptotes|first=Benjamin|last=Williamson|chapter-url=https://books.google.com/books?id=znsXAAAAYAAJ&pg=241|year=1899}}</ref><ref>{{citation|first=Jeffrey|last=Nunemacher|title=Asymptotes, Cubic Curves, and the Projective Plane|journal=Mathematics Magazine|volume=72|issue=3|year=1999|pages=183–192|jstor=2690881|doi=10.2307/2690881|citeseerx=10.1.1.502.72}}</ref>}}
{{term|[[Automatic differentiation|automatic differentiation]]}}
{{defn|defn=In [[Mathematics|mathematics]] and [[Computer algebra|computer algebra]], '''automatic differentiation''' ('''AD'''), also called '''algorithmic differentiation''' or '''computational differentiation''',<ref>{{cite journal|last=Neidinger|first=Richard D.|title=Introduction to Automatic Differentiation and MATLAB Object-Oriented Programming|journal=SIAM Review|year=2010|volume=52|issue=3|pages=545–563|url=http://academics.davidson.edu/math/neidinger/SIAMRev74362.pdf|doi=10.1137/080743627|s2cid=17134969 }}</ref><ref name="baydin2018automatic">{{cite journal|last1=Baydin|first1=Atilim Gunes|last2=Pearlmutter|first2=Barak|last3=Radul|first3=Alexey Andreyevich|last4=Siskind|first4=Jeffrey|title=Automatic differentiation in machine learning: a survey|journal=Journal of Machine Learning Research|year=2018|volume=18|pages=1–43|url=http://jmlr.org/papers/v18/17-468.html}}</ref> is a set of techniques to numerically evaluate the [[Derivative|derivative]] of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the [[Chain rule|chain rule]] repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.}}

{{term|[[Rate (mathematics)#rate of change|average rate of change]]}}
{{defn|defn=}}

{{glossary end}}

==B==
{{glossary}}

{{term|[[Binomial coefficient|binomial coefficient]]}}
{{defn|Any of the positive [[Integer|integer]]s that occurs as a [[Coefficient|coefficient]] in the [[Binomial theorem|binomial theorem]] is a '''binomial coefficient'''. Commonly, a binomial coefficient is indexed by a pair of integers {{math|''n'' ≥ ''k'' ≥ 0}} and is written <math>\tbinom{n}{k}.</math> It is the [[Coefficient|coefficient]] of the {{math|''x''<sup>''k''</sup>}} term in the [[Polynomial expansion|polynomial expansion]] of the [[Binomial (polynomial)|binomial]] [[Exponentiation|power]] {{math|(1 + ''x'')<sup>''n''</sup>}}, and it is given by the formula

:<math>\binom{n}{k} = \frac{n!}{k! (n-k)!}.</math>}}
{{term|[[Binomial theorem|binomial theorem]]''' (or '''binomial expansion)}}
{{defn| Describes the algebraic expansion of [[Exponentiation|powers]] of a [[Binomial (polynomial)|binomial]].}}
{{term|[[Bounded function|bounded function]]}}
{{defn|A [[Function (mathematics)|function]] ''f'' defined on some [[Set (mathematics)|set]] ''X'' with [[Real number|real]] or [[Complex number|complex]] values is called '''bounded''', if the set of its values is [[Bounded set|bounded]]. In other words, there exists a real number ''M'' such that 
:<math>|f(x)|\le M</math>
for all ''x'' in ''X''. A function that is ''not'' bounded is said to be '''unbounded'''.

Sometimes, if ''f''(''x'') ≤ ''A'' for all ''x'' in ''X'', then the function is said to be '''bounded above''' by ''A''. On the other hand, if ''f''(''x'') ≥ ''B'' for all ''x'' in ''X'', then the function is said to be '''bounded below''' by ''B''.}}
{{term|[[Bounded function#bounded sequence|bounded sequence]]}}
{{defn| .}}

{{glossary end}}

==C==
{{glossary}}

{{term|[[Calculus|calculus]]}}
{{defn|(From [[Social:Latin|Latin]] ''calculus'', literally 'small pebble', used for counting and calculations, as on an [[Abacus|abacus]])<ref name="oxdic">{{cite web|title=Calculus|url=http://www.oxforddictionaries.com/us/definition/american_english/calculus|archive-url=https://web.archive.org/web/20130430073245/http://oxforddictionaries.com/us/definition/american_english/calculus|url-status=dead|archive-date=April 30, 2013|website=OxfordDictionaries|access-date=15 September 2017}}</ref> is the [[Mathematics|mathematical]] study of <!-- Please, do not link "continuous", it has the common-language meaning, and does not refer to the technical mathematical concept -->continuous change, in the same way that [[Geometry|geometry]] is the study of shape and [[Algebra|algebra]] is the study of generalizations of arithmetic operations.}}
{{term|[[Cavalieri's principle]]}}
{{defn| '''Cavalieri's principle''', a modern implementation of the '''method of indivisibles''', named after Bonaventura Cavalieri, is as follows:<ref>Howard Eves, "Two Surprising Theorems on Cavalieri Congruence", ''The College Mathematics Journal'', volume 22, number 2, March, 1991), pages 118&ndash;124</ref>
* '''2-dimensional case''': Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas.
* '''3-dimensional case''': Suppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in [[Cross section (geometry)|cross-sections]] of equal area, then the two regions have equal volumes.}}
{{term|[[Chain rule|chain rule]]}}
{{defn|The '''chain rule''' is a [[Formula|formula]] for computing the [[Derivative|derivative]] of the [[Function composition|composition]] of two or more [[Function (mathematics)|functions]]. That is, if ''f'' and ''g'' are functions, then the chain rule expresses the derivative of their composition {{math|''f'' {{large|∘}} ''g''}} (the function which maps ''x'' to ''f''(''g''(''x'')) ) in terms of the derivatives of ''f'' and ''g'' and the [[Pointwise product|product of functions]] as follows:

:<math>(f\circ g)'=(f'\circ g)\cdot g'.</math>

This may equivalently be expressed in terms of the variable. Let {{math|1=''F'' = ''f'' {{large|∘}} ''g''}}, or equivalently, {{math|1=''F''(''x'') = ''f''(''g''(''x''))}} for all ''x''. Then one can also write
:<math>F'(x) = f'(g(x)) g'(x).</math>

The chain rule may be written in [[Leibniz's notation]] in the following way. If a variable ''z'' depends on the variable ''y'', which itself depends on the variable ''x'', so that ''y'' and ''z'' are therefore dependent variables, then ''z'', via the intermediate variable of ''y'', depends on ''x'' as well. The chain rule then states,

:<math>\frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx}. </math>

The two versions of the chain rule are related; if <math>z=f(y)</math> and <math>y=g(x)</math>, then

:<math>\frac{dz}{dx}=\frac{dz}{dy}\cdot\frac{dy}{dx} = f'(y)g'(x) = f'(g(x))g'(x).</math>

In [[Integral|integration]], the counterpart to the chain rule is the substitution rule.}}
{{term|[[Change of variables|change of variables]]}}
{{defn| Is a basic technique used to simplify problems in which the original [[Variable (mathematics)|variable]]s are replaced with [[Function (mathematics)|functions]] of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem.}}
{{term|[[Cofunction|cofunction]]}}
{{defn|defn=A [[Function (mathematics)|function]] ''f'' is '''cofunction''' of a function ''g'' if ''f''(''A'') = ''g''(''B'') whenever ''A'' and ''B'' are complementary angles.<ref name="Hall_1909">{{cite book |title=Trigonometry |volume=Part I: Plane Trigonometry |author-first1=Arthur Graham |author-last1=Hall |author-first2=Fred Goodrich |author-last2=Frink |date=January 1909 |location=Ann Arbor, Michigan, USA |chapter=Chapter II. The Acute Angle [10] Functions of complementary angles |publisher=[[Company:Henry Holt and Company|Henry Holt and Company]] / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA |publication-place=New York, USA |pages=11–12 |url=https://archive.org/stream/planetrigonometr00hallrich#page/n26/mode/1up |access-date=2017-08-12 }}</ref> This definition typically applies to [[Trigonometric functions|trigonometric functions]].<ref name="Aufmann_Nation_2014">{{cite book |title=Algebra and Trigonometry |author-first1=Richard |author-last1=Aufmann |author-first2=Richard |author-last2=Nation |edition=8 |publisher=Cengage Learning |year=2014 |isbn=978-128596583-3 |page=528 |url=https://books.google.com/books?id=JEDAAgAAQBAJ&pg=PA528 |access-date=2017-07-28}}</ref><ref name="Bales_2012">{{cite web |title=5.1 The Elementary Identities |work=Precalculus |author-first=John W. |author-last=Bales |date=2012 |orig-year=2001 |url=http://jwbales.home.mindspring.com/precal/part5/part5.1.html |access-date=2017-07-30 |url-status=dead |archive-url=https://web.archive.org/web/20170730201433/http://jwbales.home.mindspring.com/precal/part5/part5.1.html |archive-date=2017-07-30 }}</ref> The prefix "co-" can be found already in [[Biography:Edmund Gunter|Edmund Gunter]]'s ''Canon triangulorum'' (1620).<ref name="Gunter_1620">{{cite book |author-first=Edmund |author-last=Gunter |title=Canon triangulorum |date=1620}}</ref><ref name="Roegel_2010">{{cite web |title=A reconstruction of Gunter's Canon triangulorum (1620) |editor-first=Denis |editor-last=Roegel |type=Research report |publisher=HAL |date=2010-12-06 |id=inria-00543938 |url=https://hal.inria.fr/inria-00543938/document |access-date=2017-07-28 |url-status=live |archive-url=https://web.archive.org/web/20170728192238/https://hal.inria.fr/inria-00543938/document |archive-date=2017-07-28}}</ref>}}
{{term|[[Concave function|concave function]]}}
{{defn| Is the [[Additive inverse|negative]] of a [[Convex function|convex function]]. A concave function is also synonymously called '''concave downwards''', '''concave down''', '''convex upwards''', '''convex cap''' or '''upper convex'''.}}
{{term|[[Constant of integration|constant of integration]]}}
{{defn|defn= The indefinite integral of a given function (i.e., the [[Set (mathematics)|set]] of all [[Antiderivative|antiderivative]]s of the function) on a connected domain is only defined [[Up to|up to]] an additive constant, the '''constant of integration'''.<ref>{{cite book | last=Stewart | first=James |  title=Calculus: Early Transcendentals | publisher=Brooks/Cole | edition=6th | year=2008 | isbn=978-0-495-01166-8 | url=https://archive.org/details/calculusearlytra00stew_1 }}</ref><ref>{{cite book | last1=Larson | first1=Ron | author-link=Ron Larson (mathematician)| last2=Edwards | first2=Bruce H. | title=Calculus | publisher=Brooks/Cole | edition=9th | year=2009 | isbn=978-0-547-16702-2}}</ref> This constant expresses an ambiguity inherent in the construction of antiderivatives. If a function <math>f(x)</math> is defined on an [[Interval (mathematics)|interval]] and <math>F(x)</math> is an antiderivative of <math>f(x)</math>, then the set of ''all'' antiderivatives of <math>f(x)</math> is given by the functions <math>F(x) + C</math>, where ''C'' is an arbitrary constant (meaning that ''any'' value for ''C'' makes <math>F(x) + C</math> a valid antiderivative). The constant of integration is sometimes omitted in [[Lists of integrals|lists of integrals]] for simplicity.}}
{{term|[[Continuous function|continuous function]]}}
{{defn|defn= Is a [[Function (mathematics)|function]] for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a ''discontinuous'' function. A continuous function with a continuous [[Inverse function|inverse function]] is called a [[Homeomorphism|homeomorphism]].}}
{{term|[[Differentiable function#Differentiability classes|continuously differentiable]]}}
{{defn|defn=A function ''f'' is said to be ''continuously differentiable'' if the derivative ''{{prime|f}}''(''x'') exists and is itself a continuous function.}}
{{term|contour integration}}
{{defn|defn= In the mathematical field of [[Complex analysis|complex analysis]], '''contour integration''' is a method of evaluating certain [[Integral|integral]]s along paths in the complex plane.<ref name=Stalker>{{Cite book|title=Complex Analysis: Fundamentals of the Classical Theory of Functions |first=John |last=Stalker |page=77 | url=https://books.google.com/books?id=yl3GIXd3dFIC&q=%22calculus+of+residues%22&pg=PP12|isbn=0-8176-4038-X |publisher=Springer |year=1998}}</ref><ref name=Bak>{{Cite book|title=Complex Analysis |first1=Joseph |last1=Bak |first2=Donald J. |last2=Newman |chapter=Chapters 11 & 12 |pages=130–156 |chapter-url=https://books.google.com/books?id=JX2YSgfZwbYC&q=%22contour+integral%22&pg=PA130 |isbn=0-387-94756-6 |year=1997 |publisher=Springer}}</ref><ref name=Krantz>{{Cite book|title=Handbook of Complex Variables |first=Steven George |last=Krantz |chapter= Chapter 2 |chapter-url=https://books.google.com/books?id=aYU2AdF_0dIC&q=Calculus++Residues+inauthor:krantz&pg=PT13 |isbn=0-8176-4011-8 |year=1999 |publisher=Springer }}</ref>}}
{{term|convergence tests}}
{{defn|defn=Are methods of testing for the [[Convergent series|convergence]], [[Conditional convergence|conditional convergence]], [[Absolute convergence|absolute convergence]], interval of convergence or divergence of an [[Series (mathematics)|infinite series]] <math>\sum_{n=1}^\infty a_n</math>.}}
{{term|[[Convergent series|convergent series]]}}
{{defn|defn= In [[Mathematics|mathematics]], a [[Series (mathematics)|series]] is the [[Summation|sum]] of the terms of an infinite sequence of numbers.

Given an infinite sequence <math>\left ( a_1,\ a_2,\ a_3,\dots \right )</math>, the ''n''th partial sum <math>S_n</math> is the sum of the first ''n'' terms of the sequence, that is,

:<math>S_n = \sum_{k=1}^n a_k.</math>

A series is '''convergent''' if the sequence of its partial sums <math>\left \{ S_1,\ S_2,\ S_3,\dots \right \}</math> tends to a [[Limit of a sequence|limit]]; that means that the partial sums become closer and closer to a given number when the number of their terms increases. More precisely, a series converges, if there exists a number <math>\ell</math> such that for any arbitrarily small positive number <math>\varepsilon</math>, there is a (sufficiently large) [[Integer|integer]] <math>N</math> such that for all <math>n \ge \ N</math>,

:<math>\left | S_n - \ell \right \vert \le \ \varepsilon.</math>
If the series is convergent, the number <math>\ell</math> (necessarily unique) is called the '''sum of the series'''.

Any series that is not convergent is said to be [[Divergent series|divergent]].
}}
{{term|[[Convex function|convex function]]}}
{{defn|defn= In [[Mathematics|mathematics]], a [[Real-valued function|real-valued function]] defined on an [[Interval (mathematics)#Multi-dimensional intervals|''n''-dimensional interval]] is called '''convex''' (or '''convex downward''' or '''concave upward''') if the [[Line segment|line segment]] between any two points on the [[Graph of a function|graph of the function]] lies above or on the graph, in a [[Euclidean space]] (or more generally a [[Vector space|vector space]]) of at least two dimensions. Equivalently, a function is convex if its [[Epigraph (mathematics)|epigraph]] (the set of points on or above the graph of the function) is a [[Convex set|convex set]]. For a twice differentiable function of a single variable, if the second derivative is always greater than or equal to zero for its entire domain then the function is convex.<ref>{{Cite web|url=http://www.stat.cmu.edu/~larry/=stat705/Lecture2.pdf|title=Lecture Notes 2|website=www.stat.cmu.edu|access-date=3 March 2017}}</ref> Well-known examples of convex functions include the [[Quadratic function|quadratic function]] <math>x^2</math> and the [[Exponential function|exponential function]] <math>e^x</math>.}}
{{term|[[Cramer's rule]]}}
{{defn| In [[Linear algebra|linear algebra]], '''Cramer's rule''' is an explicit formula for the solution of a [[System of linear equations|system of linear equations]] with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the [[Determinant|determinant]]s of the (square) coefficient [[Matrix (mathematics)|matrix]] and of matrices obtained from it by replacing one column by the column vector of right-hand-sides of the equations. It is named after [[Biography:Gabriel Cramer|Gabriel Cramer]] (1704&ndash;1752), who published the rule for an arbitrary number of unknowns in 1750,<ref>{{cite web 
  | title = Introduction à l'Analyse des lignes Courbes algébriques
  | author = Cramer, Gabriel
  | year = 1750
  | location = Geneva
  | language = fr
  | url = https://www.europeana.eu/resolve/record/03486/E71FE3799CEC1F8E2B76962513829D2E36B63015
  | access-date = 2012-05-18
  | publisher = Europeana
  | pages = 656–659
}}</ref><ref>
{{Cite journal
 | last = Kosinski
 | first = A. A.
 | title = Cramer's Rule is due to Cramer
 | journal = Mathematics Magazine
 | volume = 74
 | issue = 4
 | pages = 310–312
 | year = 2001
 | doi = 10.2307/2691101
| jstor = 2691101
 }}</ref> although Colin Maclaurin also published special cases of the rule in 1748<ref>{{Cite book
  | last = MacLaurin
  | first = Colin
  | title = A Treatise of Algebra, in Three Parts.
  | url = https://archive.org/details/atreatisealgebr03maclgoog
  | year = 1748
| publisher = Printed for A. Millar & J. Nourse
 }}</ref> (and possibly knew of it as early as 1729).<ref>
{{Cite book
  | last = Boyer
  | first = Carl B.
  | title = A History of Mathematics
  | edition = 2nd
  | publisher = Wiley
  | year = 1968
  | pages = 431
}}
</ref><ref>
{{cite book
 | last = Katz
 | first = Victor
 | title = A History of Mathematics
 | publisher = Pearson Education
 | edition = Brief
 | year = 2004
 | pages = 378–379
}}
</ref><ref>
{{Cite journal
 | last = Hedman
 | first = Bruce A.
 | title = An Earlier Date for "Cramer's Rule"
 | journal = Historia Mathematica
 | volume = 26
 | issue =4
 | pages = 365–368
 | year = 1999
 | url = http://professorhedman.com/Cramers.Rule.pdf
 | doi = 10.1006/hmat.1999.2247
 | s2cid = 121056843
 }}
</ref>}}
{{term|[[Critical point (mathematics)|critical point]]}}
{{defn|defn=A '''critical point''' or '''stationary point''' of a [[Differentiable function|differentiable function]] of a [[Function of a real variable|real]] or complex variable is any value in its [[Domain of a function|domain]] where its [[Derivative|derivative]] is 0.<ref>{{cite book | last=Stewart | first=James |  title=Calculus: Early Transcendentals | publisher=Brooks/Cole | edition=6th | year=2008 | isbn=978-0-495-01166-8 | url=https://archive.org/details/calculusearlytra00stew_1 }}</ref><ref>{{cite book | last1=Larson | first1=Ron | author-link=Ron Larson (mathematician)| last2=Edwards | first2=Bruce H. | title=Calculus | publisher=Brooks/Cole | edition=9th | year=2009 | isbn=978-0-547-16702-2}}</ref>}}
{{term|[[Curve|curve]]}}
{{defn|defn= A '''curve''' (also called a '''curved line''' in older texts) is, generally speaking, an object similar to a [[Line (geometry)|line]] but that need not be [[Linearity|straight]].}} 
{{term|[[Curve sketching|curve sketching]]}}
{{defn|defn= 
 In [[Geometry|geometry]], '''curve sketching''' (or '''curve tracing''') includes techniques that can be used to produce a rough idea of overall shape of a [[Plane curve|plane curve]] given its equation without computing the large numbers of points required for a detailed plot. It is an application of the theory of curves to find their main features. Here input is an equation.
 In [[Digital geometry|digital geometry]] it is a method of drawing a curve pixel by pixel. Here input is an array (digital image).}}

{{glossary end}}

==D==
{{glossary}}
{{term|term=[[Damped sine wave|damped sine wave]]}}
{{defn|defn= Is a [[Physics:Sine wave|sinusoidal function]] whose amplitude approaches zero as time increases.<ref>Douglas C. Giancoli (2000). [''Physics for Scientists and Engineers with Modern Physics (3rd Edition)'']. Prentice Hall. {{ISBN|0-13-021517-1}}</ref>}}
{{term|term=[[Degree of a polynomial|degree of a polynomial]]}}
{{defn|defn=Is the highest degree of its [[Monomial|monomial]]s (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the [[Variable (mathematics)|variables]] that appear in it, and thus is a non-negative integer.}}
{{term|term=[[Derivative|derivative]]}}
{{defn|defn=The '''derivative''' of a [[Function of a real variable|function of a real variable]] measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of [[Calculus|calculus]]. For example, the derivative of the position of a moving object with respect to [[Time|time]] is the object's [[Physics:Velocity|velocity]]: this measures how quickly the position of the object changes when time advances.}}
{{term|term=[[Derivative test|derivative test]]}}
{{defn|defn=A '''derivative test''' uses the [[Derivative|derivative]]s of a function to locate the [[Stationary point|critical point]]s of a function and determine whether each point is a local maximum, a local minimum, or a [[Saddle point|saddle point]]. Derivative tests can also give information about the [[Concave function|concavity]] of a function.}}
{{term|term=[[Differentiable function|differentiable function]]}}
{{defn|defn=A '''differentiable function''' of one [[Real number|real]] variable is a function whose [[Derivative|derivative]] exists at each point in its [[Domain of a function|domain]]. As a result, the [[Graph of a function|graph]] of a differentiable function must have a (non-[[Vertical tangent|vertical]]) tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or [[Cusp (singularity)|cusps]].}}
{{term|term=[[Differential (infinitesimal)|differential (infinitesimal)]]}}
{{defn|defn=The term '''differential''' is used in [[Calculus|calculus]] to refer to an [[Infinitesimal|infinitesimal]] (infinitely small) change in some [[Variable (mathematics)|varying quantity]]. For example, if ''x'' is a [[Variable (mathematics)|variable]], then a change in the value of ''x'' is often denoted Δ''x'' (pronounced ''delta x''). The differential ''dx'' represents an infinitely small change in the variable ''x''. The idea of an infinitely small or infinitely slow change is extremely useful intuitively, and there are a number of ways to make the notion mathematically precise.

Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using [[Derivative|derivative]]s. If ''y'' is a function of ''x'', then the differential ''dy'' of ''y'' is related to ''dx'' by the formula
:<math>dy = \frac{dy}{dx} \,dx,</math>
where ''dy''/''dx'' denotes the [[Derivative|derivative]] of ''y'' with respect to ''x''. This formula summarizes the intuitive idea that the derivative of ''y'' with respect to ''x'' is the limit of the ratio of differences Δ''y''/Δ''x'' as Δ''x'' becomes infinitesimal.}}
{{term|term=[[Differential calculus|differential calculus]]}}
{{defn|defn=Is a subfield of calculus<ref>{{Cite web|url=https://www.merriam-webster.com/dictionary/differential%20calculus|title=Definition of DIFFERENTIAL CALCULUS|website=www.merriam-webster.com|language=en|access-date=2018-09-26}}</ref> concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus, the study of the area beneath a curve.<ref>{{Cite web|url=https://www.merriam-webster.com/dictionary/integral%20calculus|title=Integral Calculus - Definition of Integral calculus by Merriam-Webster|website=www.merriam-webster.com|language=en|access-date=2018-05-01}}</ref>}}
{{term|term=[[Differential equation|differential equation]]}}
{{defn|defn=Is a [[Mathematics|mathematical]] [[Equation|equation]] that relates some [[Function (mathematics)|function]] with its [[Derivative|derivative]]s. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two.}}
{{term|term=[[Differential operator|differential operator]]}}
{{defn|defn=.}}
{{term|term=[[Differential of a function|differential of a function]]}}
{{defn|defn=In [[Calculus|calculus]], the '''differential''' represents the [[Principal part#Calculus|principal part]] of the change in a function ''y''&nbsp;=&nbsp;''f''(''x'') with respect to changes in the independent variable. The differential ''dy'' is defined by
:<math>dy = f'(x)\,dx,</math>
where <math>f'(x)</math> is the [[Derivative|derivative]] of ''f'' with respect to ''x'', and ''dx'' is an additional real [[Variable (mathematics)|variable]] (so that ''dy'' is a function of ''x'' and ''dx''). The notation is such that the equation

:<math>dy = \frac{dy}{dx}\, dx</math>

holds, where the derivative is represented in the Leibniz notation ''dy''/''dx'', and this is consistent with regarding the derivative as the quotient of the differentials. One also writes

:<math>df(x) = f'(x)\,dx.</math>

The precise meaning of the variables ''dy'' and ''dx'' depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular [[Differential form|differential form]], or analytical significance if the differential is regarded as a [[Linear approximation|linear approximation]] to the increment of a function. Traditionally, the variables ''dx'' and ''dy'' are considered to be very small ([[Infinitesimal|infinitesimal]]), and this interpretation is made rigorous in [[Non-standard analysis|non-standard analysis]].}}
{{term|term=[[Differentiation rules|differentiation rules]]}}
{{defn|defn=.}}
{{term|term=[[Direct comparison test|direct comparison test]]}}
{{defn|defn=A convergence test in which an infinite series or an improper integral is compared to one with known convergence properties.}}
{{term|term=[[Dirichlet's test]]}}
{{defn|defn=Is a method of testing for the [[Convergent series|convergence]] of a [[Series (mathematics)|series]]. It is named after its author [[Biography:Peter Gustav Lejeune Dirichlet|Peter Gustav Lejeune Dirichlet]], and was published posthumously in the ''[[Journal de Mathématiques Pures et Appliquées]]'' in 1862.<ref>''Démonstration d’un théorème d’Abel.'' Journal de mathématiques pures et appliquées 2nd series, tome 7 (1862), [http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1862_2_7_A43_0 p. 253-255] {{Webarchive|url=https://web.archive.org/web/20110721011902/http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1862_2_7_A43_0 |date=2011-07-21 }}.</ref> The test states that if <math>\{a_n\}</math> is a [[Sequence|sequence]] of [[Real number|real number]]s and <math>\{b_n\}</math> a sequence of [[Complex number|complex number]]s satisfying

:* <math>a_{n+1} \le a_n</math>

:* <math>\lim_{n \rightarrow \infty}a_n = 0</math>

:* <math>\left|\sum^{N}_{n=1}b_n\right|\leq M</math> for every positive integer ''N''

where ''M'' is some constant, then the series

:<math>\sum^{\infty}_{n=1}a_n b_n</math>

converges.}}
{{term|term=[[Disc integration|disc integration]]}}
{{defn|defn=Also known in integral calculus as the '''disc method''', is a means of calculating the [[Physics:Volume|volume]] of a [[Solid of revolution|solid of revolution]] of a solid-state material when [[Integral|integrating]] along an axis "parallel" to the axis of revolution.}}
{{term|term=[[Divergent series|divergent series]]}}
{{defn|defn=Is an infinite series that is not [[Convergent series|convergent]], meaning that the infinite [[Sequence|sequence]] of the partial sums of the series does not have a finite [[Limit of a sequence|limit]].}}
{{term|term=discontinuity}}
{{defn|defn=[[Continuous function]]s are of utmost importance in [[Mathematics|mathematics]], functions and applications. However, not all [[Function (mathematics)|functions]] are continuous. If a function is not continuous at a point in its [[Domain of a function|domain]], one says that it has a '''discontinuity''' there. The set of all points of discontinuity of a function may be a discrete set, a [[Dense set|dense set]], or even the entire domain of the function.}}
{{term|term=[[Dot product|dot product]]}}
{{term|term=double integral}}
{{defn|defn= The '''multiple integral''' is a definite integral of a [[Function (mathematics)|function]] of more than one real [[Variable (mathematics)|variable]], for example, {{math|''f''(''x'', ''y'')}} or {{math|''f''(''x'', ''y'', ''z'')}}. Integrals of a function of two variables over a region in {{math|'''R'''<sup>2</sup>}} are called double integrals, and integrals of a function of three variables over a region of {{math|'''R'''<sup>3</sup>}} are called triple integrals.<ref name= "Stewart">{{cite book |last=Stewart |first=James |date=2008 |title=Calculus: Early Transcendentals |edition=6th |publisher=Brooks Cole Cengage Learning |isbn=978-0-495-01166-8 |url-access=registration |url=https://archive.org/details/calculusearlytra00stew_1 }}</ref>}}

{{glossary end}}

==E==
{{glossary}}

{{term|term=[[E (mathematical constant)|e (mathematical constant)]]}}
{{defn|defn=The number '''{{mvar|e}}''' is a [[Mathematical constant|mathematical constant]] that is the base of the [[Natural logarithm|natural logarithm]]: the unique number whose natural logarithm is equal to one. It is approximately equal to '''2.71828''',<ref>Oxford English Dictionary, 2nd ed.: [https://web.archive.org/web/20131006101640/http://oxforddictionaries.com/definition/english/natural-logarithm natural logarithm]</ref> and is the [[Limit of a sequence|limit]] of {{math|(1 + 1/''n'')<sup>''n''</sup>}} as {{mvar|n}} approaches [[Infinity|infinity]], an expression that arises in the study of [[Compound interest|compound interest]]. It can also be calculated as the sum of the infinite [[Series (mathematics)|series]]<ref>[[Encyclopedic Dictionary of Mathematics]] 142.D</ref>

:<math>e = \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} = \frac{1}{1} + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots</math>}}
{{term|term=[[Elliptic integral|elliptic integral]]}}
{{defn|defn=In integral calculus, '''elliptic integrals''' originally arose in connection with the problem of giving the [[Arc length|arc length]] of an [[Ellipse|ellipse]]. They were first studied by Giulio Fagnano and [[Biography:Leonhard Euler|Leonhard Euler]] ({{Circa|1750}}). Modern mathematics defines an "elliptic integral" as any [[Function (mathematics)|function]] {{math|''f''}} which can be expressed in the form

: <math> f(x) = \int_{c}^{x} R \left(t, \sqrt{P(t)} \right) \, dt, </math>

where {{math|''R''}} is a [[Rational function|rational function]] of its two arguments, {{math|''P''}} is a [[Polynomial|polynomial]] of degree 3 or 4 with no repeated roots, and {{math|''c''}} is a constant..}}
{{term|term=[[Classification of discontinuities#Essential discontinuity|essential discontinuity]]}}
{{defn|defn=For an essential discontinuity, only one of the two one-sided limits needs not exist or be infinite.
Consider the function 
:<math>f(x) = \begin{cases}
  \sin\frac{5}{x-1} & \mbox{ for } x < 1 \\
  0                 & \mbox{ for } x = 1 \\
  \frac{1}{x-1}   & \mbox{ for } x > 1
\end{cases}</math>

Then, the point <math>\scriptstyle x_0 \;=\; 1</math> is an ''essential discontinuity''.

In this case, <math>\scriptstyle L^{-}</math> doesn't exist and <math>\scriptstyle L^{+}</math> is infinite – thus satisfying twice the conditions of essential discontinuity. So ''x''<sub>0</sub> is an ''essential discontinuity'', ''infinite discontinuity'', or ''discontinuity of the second kind''. (This is distinct from the term ''[[Essential singularity|essential singularity]]'' which is often used when studying [[Complex analysis|functions of complex variables]].}}
{{term|term=[[Euler method]]}}
{{defn|defn=Euler's method is a numerical method to solve first order first degree differential equation with a given initial value. It is the most basic [[Explicit and implicit methods|explicit method]] for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after [[Biography:Leonhard Euler|Leonhard Euler]], who treated it in his book ''[[Institutionum calculi integralis]]'' (published 1768–1870).<ref>{{harvnb|Butcher|2003|p=45}}; {{harvnb|Hairer|Nørsett|Wanner|1993|p=35}}</ref>}}
{{term|term=[[Exponential function|exponential function]]}}
{{defn|defn=In [[Mathematics|mathematics]], an '''exponential function''' is a function of the form
{{block indent|<math> f(x) = ab^x, </math>}}
where {{mvar|b}} is a positive real number, and in which the argument {{math|''x''}} occurs as an exponent. For real numbers {{mvar|c}} and {{mvar|d,}} a function of the form <math>f(x)=ab^{cx+d}</math> is also an exponential function, as it can be rewritten as 
:<math>ab^{cx+d}=\left(ab^d\right) \left(b^c\right)^x.</math>}}
{{term|term=[[Extreme value theorem|extreme value theorem]]}}
{{defn|defn=States that if a real-valued [[Function (mathematics)|function]] ''f'' is [[Continuous function|continuous]] on the closed interval [''a'',''b''], then ''f'' must attain a maximum and a minimum, each at least once. That is, there exist numbers ''c'' and ''d'' in [''a'',''b''] such that:

:<math>f(c) \ge f(x) \ge f(d)\quad\text{for all }x\in [a,b].</math>

A related theorem is '''the boundedness theorem''' which states that a continuous function ''f'' in the closed interval [''a'',''b''] is [[Bounded function|bounded]] on that interval. That is, there exist real numbers ''m'' and ''M'' such that:

:<math>m < f(x) < M\quad\text{for all }x \in [a,b].</math>

The extreme value theorem enriches the boundedness theorem by saying that not only is the function bounded, but it also attains its least upper bound as its maximum and its greatest lower bound as its minimum.}}
{{term|term=[[Maxima and minima|extremum]]}}
{{defn|defn=In [[Mathematical analysis|mathematical analysis]], the '''maxima and minima''' (the respective plurals of '''maximum''' and '''minimum''') of a [[Function (mathematics)|function]], known collectively as '''extrema''' (the plural of '''extremum'''), are the largest and smallest value of the function, either within a given range (the '''local''' or '''relative''' extrema) or on the entire [[Domain of a function|domain of a function]] (the '''global''' or '''absolute''' extrema).<ref>{{cite book | last=Stewart | first=James |  title=Calculus: Early Transcendentals | publisher=Brooks/Cole | edition=6th | year=2008 | isbn=978-0-495-01166-8 | url=https://archive.org/details/calculusearlytra00stew_1 }}</ref><ref>{{cite book | last1=Larson | first1=Ron | author-link=Ron Larson (mathematician)| last2=Edwards | first2=Bruce H. | title=Calculus | publisher=Brooks/Cole | edition=9th | year=2009 | isbn=978-0-547-16702-2}}</ref><ref>{{cite book | last1 = Thomas | first1 = George B. | last2=Weir | first2= Maurice D. | last3=Hass | first3=Joel | author-link=George B. Thomas | title=Thomas' Calculus: Early Transcendentals | publisher=Addison-Wesley | year=2010 | edition=12th | isbn=978-0-321-58876-0}}</ref> [[Biography:Pierre de Fermat|Pierre de Fermat]] was one of the first mathematicians to propose a general technique, [[Adequality|adequality]], for finding the maxima and minima of functions.

As defined in [[Set theory|set theory]], the maximum and minimum of a [[Set (mathematics)|set]] are the [[Greatest and least elements|greatest and least elements]] in the set, respectively. Unbounded infinite sets, such as the set of [[Real number|real number]]s, have no minimum or maximum.}}

{{glossary end}}

==F==
{{glossary}}

{{term|[[Faà di Bruno's formula]]}}
{{defn|defn= Is an identity in [[Mathematics|mathematics]] generalizing the [[Chain rule|chain rule]] to higher derivatives, named after {{harvs|txt|first=Francesco|last= Faà di Bruno|year=1855|year2=1857}}, though he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French mathematician Louis François Antoine Arbogast stated the formula in a calculus textbook,<ref>{{harv|Arbogast|1800}}.</ref> considered the first published reference on the subject.<ref>According to {{harvtxt|Craik|2005|pp=120–122}}: see also the analysis of Arbogast's work by {{harvtxt|Johnson|2002|p=230}}.</ref>

Perhaps the most well-known form of Faà di Bruno's formula says that

:<math>{d^n \over dx^n} f(g(x))=\sum \frac{n!}{m_1!\,1!^{m_1}\,m_2!\,2!^{m_2}\,\cdots\,m_n!\,n!^{m_n}}\cdot f^{(m_1+\cdots+m_n)}(g(x))\cdot \prod_{j=1}^n\left(g^{(j)}(x)\right)^{m_j},</math>

where the sum is over all ''n''-[[Tuple|tuple]]s of nonnegative integers (''m''<sub>1</sub>, …, ''m''<sub>''n''</sub>) satisfying the constraint

:<math>1\cdot m_1+2\cdot m_2+3\cdot m_3+\cdots+n\cdot m_n=n.</math>

Sometimes, to give it a memorable pattern, it is written in a way in which the coefficients that have the combinatorial interpretation discussed below are less explicit:

:<math>{d^n \over dx^n} f(g(x))
=\sum \frac{n!}{m_1!\,m_2!\,\cdots\,m_n!}\cdot
f^{(m_1+\cdots+m_n)}(g(x))\cdot
\prod_{j=1}^n\left(\frac{g^{(j)}(x)}{j!}\right)^{m_j}.</math>

Combining the terms with the same value of ''m''<sub>1</sub>&nbsp;+&nbsp;''m''<sub>2</sub>&nbsp;+&nbsp;...&nbsp;+&nbsp;''m''<sub>''n''</sub>&nbsp;=&nbsp;''k'' and noticing that ''m''<sub>&nbsp;''j''</sub> has to be zero for ''j''&nbsp;>&nbsp;''n''&nbsp;&minus;&nbsp;''k''&nbsp;+&nbsp;1 leads to a somewhat simpler formula expressed in terms of Bell polynomials ''B''<sub>''n'',''k''</sub>(''x''<sub>1</sub>,...,''x''<sub>''n''&minus;''k''+1</sub>):

:<math>{d^n \over dx^n} f(g(x)) = \sum_{k=1}^n f^{(k)}(g(x))\cdot B_{n,k}\left(g'(x),g''(x),\dots,g^{(n-k+1)}(x)\right).</math>}}
{{term|[[Degree of a polynomial|first-degree polynomial]]}}
{{defn|defn=}}
{{term|[[Derivative test#First-derivative test|first derivative test]]}}
{{defn|defn=The first derivative test examines a function's [[Monotonic function|monotonic]] properties (where the function is increasing or decreasing) focusing on a particular point in its domain. If the function "switches" from increasing to decreasing at the point, then the function will achieve a highest value at that point. Similarly, if the function "switches" from decreasing to increasing at the point, then it will achieve a least value at that point. If the function fails to "switch", and remains increasing or remains decreasing, then no highest or least value is achieved.}}
{{term|[[Fractional calculus]]}}
{{defn|defn=Is a branch of [[Mathematical analysis|mathematical analysis]] that studies the several different possibilities of defining [[Real number|real number]] powers or [[Complex number|complex number]] powers of the [[Derivative|differentiation operator]] {{mvar|D}}

:<math>D f(x) = \dfrac{d}{dx} f(x)</math>,

and of the integration operator {{mvar|J}}

:<math>J f(x) = \int_0^x\!\!\!\! f(s) {ds}</math>,<ref group=Note>The symbol {{mvar|J}} is commonly used instead of the intuitive {{mvar|I}} in order to avoid confusion with other concepts identified by similar {{mvar|I}}–like [[Glyph|glyph]]s, e.g. [[Identity (mathematics)|identities]].</ref>

and developing a [[Calculus|calculus]] for such operators generalizing the classical one.

In this context, the term ''powers'' refers to iterative application of a linear operator to a function, in some analogy to [[Function composition|function composition]] acting on a variable, i.e. {{math|''f''&nbsp;{{ssup|∘2}}(''x'')&nbsp;{{=}}&nbsp;''f''&nbsp;∘&nbsp;''f''&nbsp;(''x'')&nbsp;{{=}}&nbsp;''f''&nbsp;(&nbsp;''f''&nbsp;(''x'')&nbsp;)}}.}}
{{term|[[Frustum|frustum]]}}
{{defn|defn=In [[Geometry|geometry]], a '''frustum''' (plural: ''frusta'' or ''frustums'') is the portion of a [[Polyhedron|solid]] (normally a cone or [[Pyramid (geometry)|pyramid]]) that lies between one or two parallel planes cutting it. A '''right frustum''' is a parallel [[Truncation (geometry)|truncation]] of a right pyramid or right cone.<ref>William F. Kern, James R. Bland, ''Solid Mensuration with proofs'', 1938, p.&nbsp;67</ref>}}
{{term|[[Function (mathematics)|function]]}}
{{defn|defn=Is a process or a relation<!-- Please, do not link to [[Binary relation]], this is not the technical meaning that is intended--> that associates each element {{mvar|x}} of a [[Set (mathematics)|set]] {{mvar|X}},  the ''[[Domain of a function|domain]]'' of the function, to a single element {{mvar|y}} of another set {{mvar|Y}} (possibly the same set), the ''codomain'' of the function. If the function is called {{mvar|f}}, this relation is denoted {{math|1=''y'' = ''f''{{space|hair}}(''x'')}} (read {{mvar|f}} of {{mvar|x}}), the element {{mvar|x}} is the ''[[Argument of a function|argument]]'' or ''input'' of the function, and {{mvar|y}} is the ''value of the function'', the ''output'', or the ''image'' of {{mvar|x}} by {{mvar|f}}.<ref name=MacLane>{{cite book | last1 = MacLane | first1 = Saunders |  last2 = Birkhoff | first2 = Garrett |  title = Algebra | url = https://archive.org/details/algebra00macl | url-access = registration | publisher = Macmillan | edition = First | year = 1967 | location = New York | pages = [https://archive.org/details/algebra00macl/page/1 1–13] }}</ref> The symbol that is used for representing the input is the [[Variable (mathematics)|variable]] of the function (one often says that {{mvar|f}} is a function of the variable {{mvar|x}}).}}
{{term|[[Function composition|function composition]]}}
{{defn|defn=Is an operation that takes two [[Function (mathematics)|functions]] {{math|''f''}} and {{math|''g''}} and produces a function {{math|''h''}} such that {{math|1=''h''(''x'') = ''g''(''f''(''x''))}}. In this operation, the function {{math|''g''}} is [[Function application|applied]] to the result of applying the function {{math|''f''}} to {{math|''x''}}. That is, the functions {{math|''f'' : ''X'' → ''Y''}} and {{math|''g'' : ''Y'' → ''Z''}} are '''composed''' to yield a function that maps {{math|''x''}} in {{math|''X''}} to {{math|''g''(''f''(''x''))}} in {{math|''Z''}}.}}
{{term|[[Fundamental theorem of calculus|fundamental theorem of calculus]]}}
{{defn|The '''fundamental theorem of calculus''' is a [[Theorem|theorem]] that links the concept of [[Derivative|differentiating]] a [[Function (mathematics)|function]] with the concept of [[Integral|integrating]] a function. The first part of the theorem, sometimes called the '''first fundamental theorem of calculus''', states that one of the [[Antiderivative|antiderivative]]s (also called ''indefinite integral''), say ''F'', of some function ''f'' may be obtained as the integral of ''f'' with a variable bound of integration. This implies the existence of [[Antiderivative|antiderivative]]s for [[Continuous function|continuous function]]s.<ref>{{Citation |last=Spivak|first=Michael|year=1980|title=Calculus|edition=2nd|location=Houston, Texas|publisher=Publish or Perish Inc.}}</ref>  Conversely, the second part of the theorem, sometimes called the '''second fundamental theorem of calculus''', states that the integral of a function ''f'' over some interval can be computed by using any one, say ''F'', of its infinitely many [[Antiderivative|antiderivative]]s. This part of the theorem has key practical applications, because explicitly finding the antiderivative of a function by [[Symbolic integration|symbolic integration]] avoids [[Numerical integration|numerical integration]] to compute integrals. This provides generally a better numerical accuracy.}}

{{glossary end}}

==G==
{{glossary}}

{{term|term=[[General Leibniz rule|general Leibniz rule]]}}
{{defn|defn= The '''general Leibniz rule''',<ref>{{cite book |last=Olver |first=Peter J. |year=2000 |title=Applications of Lie Groups to Differential Equations |publisher=Springer |pages=318–319 |url=https://books.google.com/books?id=sI2bAxgLMXYC&pg=PA318 |isbn=9780387950006 }}</ref> named after [[Biography:Gottfried Wilhelm Leibniz|Gottfried Wilhelm Leibniz]], generalizes the [[Product rule|product rule]] (which is also known as "Leibniz's rule").  It states that if <math>f</math> and <math>g</math> are <math>n</math>-times [[Differentiable function|differentiable function]]s, then the product <math>fg</math> is also <math>n</math>-times differentiable and its <math>n</math>th derivative is given by

:<math>(fg)^{(n)}=\sum_{k=0}^n {n \choose k} f^{(n-k)} g^{(k)},</math>

where <math>{n \choose k}={n!\over k! (n-k)!}</math> is the [[Binomial coefficient|binomial coefficient]] and <math>f^{(0)}\equiv f.</math>

This can be proved by using the product rule and [[Mathematical induction|mathematical induction]].}}
{{term|term=[[Maxima and minima|global maximum]]}}
{{defn|defn= In [[Mathematical analysis|mathematical analysis]], the '''maxima and minima''' (the respective plurals of '''maximum''' and '''minimum''') of a [[Function (mathematics)|function]], known collectively as '''extrema''' (the plural of '''extremum'''), are the largest and smallest value of the function, either within a given range (the '''local''' or '''relative''' extrema) or on the entire [[Domain of a function|domain of a function]] (the '''global''' or '''absolute''' extrema).<ref>{{cite book | last=Stewart | first=James |  title=Calculus: Early Transcendentals | publisher=Brooks/Cole | edition=6th | year=2008 | isbn=978-0-495-01166-8 | url=https://archive.org/details/calculusearlytra00stew_1 }}</ref><ref>{{cite book | last1=Larson | first1=Ron | author-link=Ron Larson (mathematician)| last2=Edwards | first2=Bruce H. | title=Calculus | publisher=Brooks/Cole | edition=9th | year=2009 | isbn=978-0-547-16702-2}}</ref><ref>{{cite book | last1 = Thomas | first1 = George B. | last2=Weir | first2= Maurice D. | last3=Hass | first3=Joel | author-link=George B. Thomas | title=Thomas' Calculus: Early Transcendentals | publisher=Addison-Wesley | year=2010 | edition=12th | isbn=978-0-321-58876-0}}</ref> [[Biography:Pierre de Fermat|Pierre de Fermat]] was one of the first mathematicians to propose a general technique, [[Adequality|adequality]], for finding the maxima and minima of functions.

As defined in [[Set theory|set theory]], the maximum and minimum of a [[Set (mathematics)|set]] are the [[Greatest and least elements|greatest and least elements]] in the set, respectively. Unbounded infinite sets, such as the set of [[Real number|real number]]s, have no minimum or maximum.}}
{{term|term=[[Maxima and minima|global minimum]]}}
{{defn|defn= In [[Mathematical analysis|mathematical analysis]], the '''maxima and minima''' (the respective plurals of '''maximum''' and '''minimum''') of a [[Function (mathematics)|function]], known collectively as '''extrema''' (the plural of '''extremum'''), are the largest and smallest value of the function, either within a given range (the '''local''' or '''relative''' extrema) or on the entire [[Domain of a function|domain of a function]] (the '''global''' or '''absolute''' extrema).<ref>{{cite book | last=Stewart | first=James |  title=Calculus: Early Transcendentals | publisher=Brooks/Cole | edition=6th | year=2008 | isbn=978-0-495-01166-8 | url=https://archive.org/details/calculusearlytra00stew_1 }}</ref><ref>{{cite book | last1=Larson | first1=Ron | author-link=Ron Larson (mathematician)| last2=Edwards | first2=Bruce H. | title=Calculus | publisher=Brooks/Cole | edition=9th | year=2009 | isbn=978-0-547-16702-2}}</ref><ref>{{cite book | last1 = Thomas | first1 = George B. | last2=Weir | first2= Maurice D. | last3=Hass | first3=Joel | author-link=George B. Thomas | title=Thomas' Calculus: Early Transcendentals | publisher=Addison-Wesley | year=2010 | edition=12th | isbn=978-0-321-58876-0}}</ref> [[Biography:Pierre de Fermat|Pierre de Fermat]] was one of the first mathematicians to propose a general technique, [[Adequality|adequality]], for finding the maxima and minima of functions.

As defined in [[Set theory|set theory]], the maximum and minimum of a [[Set (mathematics)|set]] are the [[Greatest and least elements|greatest and least elements]] in the set, respectively. Unbounded infinite sets, such as the set of [[Real number|real number]]s, have no minimum or maximum.}}
{{term|term=[[Golden spiral|golden spiral]]}}
{{defn|defn=In [[Geometry|geometry]], a '''golden spiral''' is a [[Logarithmic spiral|logarithmic spiral]] whose growth factor is {{math|&phi;}}, the [[Golden ratio|golden ratio]].<ref>Chang, Yu-sung, "[http://demonstrations.wolfram.com/GoldenSpiral/ Golden Spiral] {{Webarchive|url=https://web.archive.org/web/20190728084311/http://demonstrations.wolfram.com/GoldenSpiral/ |date=2019-07-28 }}", The Wolfram Demonstrations Project.</ref> That is, a golden spiral gets wider (or further from its origin) by a factor of {{math|&phi;}} for every quarter turn it makes.}}
{{term|term=[[Gradient|gradient]]}}
{{defn|defn= Is a multi-variable generalization of the [[Derivative|derivative]].  While a derivative can be defined on functions of a single variable, for functions of several variables, the gradient takes its place.  The gradient is a [[Vector-valued function|vector-valued function]], as opposed to a derivative, which is scalar-valued.}}

{{glossary end}}

==H==
{{glossary}}

{{term|term=[[Harmonic progression (mathematics)|harmonic progression]]}}
{{defn|defn=In [[Mathematics|mathematics]], a '''harmonic progression''' (or '''harmonic sequence''') is a progression formed by taking the reciprocals of an [[Arithmetic progression|arithmetic progression]]. It is a [[Sequence|sequence]] of the form

:<math> \frac{1}{a} ,\ \frac{1}{a+d}\ , \frac{1}{a+2d}\ , \frac{1}{a+3d}\ , \cdots, \frac{1}{a+kd},</math>

where &minus;a/''d'' is not a [[Natural number|natural number]] and ''k'' '''is''' a natural number.

Equivalently, a sequence is a harmonic progression when each term is the [[Harmonic mean|harmonic mean]] of the neighboring terms.

It is not possible for a harmonic progression (other than the trivial case where ''a'' = 1 and ''k'' = 0)  to sum to an [[Integer|integer]]. The reason is that, necessarily, at least one denominator of the progression will be divisible by a [[Prime number|prime number]] that does not divide any other denominator.<ref>{{citation|first=P.|last= Erdős |title=Egy Kürschák-féle elemi számelméleti tétel általánosítása|trans-title=Generalization of an elementary number-theoretic theorem of Kürschák|language=hu|journal=Mat. Fiz. Lapok|volume=39|year=1932|pages=17–24|url=https://www.renyi.hu/~p_erdos/1932-02.pdf}}. As cited by {{citation
 | contribution = Paul Erdős and Egyptian fractions
 | doi = 10.1007/978-3-642-39286-3_9
 | mr = 3203600
 | pages = 289–309
 | publisher = János Bolyai Math. Soc., Budapest
 | series = Bolyai Soc. Math. Stud.
 | title = Erdős centennial
 | volume = 25
 | year = 2013}}.</ref>}}

{{term|term=[[Derivative#Higher derivatives|higher derivative]]}}
{{defn|defn= Let {{math|''f''}} be a differentiable function, and let {{math|''f'' ′}} be its derivative. The derivative of {{math|''f'' ′}} (if it has one) is written {{math|''f'' ′′}} and is called the ''[[Second derivative|second derivative]] of {{math|f}}''.  Similarly, the derivative of the second derivative, if it exists, is written {{math|''f'' ′′′}} and is called the ''[[Third derivative|third derivative]] of {{math|f}}''. Continuing this process, one can define, if it exists, the {{math|''n''}}th derivative as the derivative of the {{math|(''n''-1)}}th derivative. These repeated derivatives are called ''higher-order derivatives''. The {{math|''n''}}th derivative is also called the '''derivative of order {{math|''n''}}'''.}}
{{term|term=[[Homogeneous differential equation|homogeneous linear differential equation]]}}
{{defn|defn=A [[Differential equation|differential equation]] can be '''homogeneous''' in either of two respects.

A first order differential equation is said to be homogeneous if it may be written
:<math>f(x,y)dy = g(x,y)dx,</math>
where {{mvar|f}} and {{mvar|g}} are [[Homogeneous function|homogeneous function]]s of the same degree of {{mvar|x}} and {{mvar|y}}. In this case, the change of variable {{math|1=''y'' = ''ux''}} leads to an equation of the form 
:<math>\frac{dx}x = h(u)du,</math>
which is easy to solve by [[Integral|integration]] of the two members.

Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of [[Linear differential equation|linear differential equation]]s, this means that there are no constant terms. The solutions of any linear [[Ordinary differential equation|ordinary differential equation]] of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term.}}
{{term|term=[[Hyperbolic function|hyperbolic function]]}}
{{defn|defn= '''Hyperbolic functions''' are analogs of the ordinary trigonometric, or [[Unit circle|circular]], functions.}}

{{glossary end}}

==I==
{{glossary}}

{{term|term=[[Identity function|identity function]]}}
{{defn|defn= Also called an '''identity relation''' or '''identity map''' or '''identity transformation''', is a [[Function (mathematics)|function]] that always returns the same value that was used as its argument. In [[Equation|equation]]s, the function is given by {{math|1=''f''(''x'') = ''x''}}.}}
{{term|term=[[Imaginary number|imaginary number]]}}
{{defn|defn=Is a [[Complex number|complex number]] that can be written as a [[Real number|real number]] multiplied by the [[Imaginary unit|imaginary unit]] {{mvar|i}},<ref group=note>''j'' is usually used in Engineering contexts where ''i'' has other meanings (such as electrical current)</ref> which is defined by its property {{math|1=''i''<sup>2</sup> = −1}}.<ref>
{{cite book
|chapter-url=https://books.google.com/books?id=SGVfGIewvxkC&pg=PA38
|title=Fundamentals of Waves and Oscillations
|last=Uno Ingard
|first=K.
|publisher=Cambridge University Press
|year=1988
|isbn=0-521-33957-X
|page=38
|chapter=Chapter 2
}}
</ref> The [[Square (algebra)|square]] of an imaginary number {{mvar|bi}} is {{math|−''b''<sup>2</sup>}}. For example, {{math|5''i''}} is an imaginary number, and its square is {{math|−25}}. Zero is considered to be both real and imaginary.<ref>{{cite book|url=https://books.google.com/books?id=mqdzqbPYiAUC&pg=SA11-PA2|title=A Text Book of Mathematics Class XI|last=Sinha|first=K.C.|publisher=Rastogi Publications|year=2008|isbn=978-81-7133-912-9|edition=Second|page=11.2}}</ref>}}
{{term|term=[[Implicit function|implicit function]]}}
{{defn|defn=In [[Mathematics|mathematics]], an implicit equation is a [[Relation (mathematics)|relation]]  of the form <math>R(x_1,\ldots, x_n) = 0</math>, where <math>R</math> is a [[Function (mathematics)|function]] of several variables (often a [[Polynomial|polynomial]]). For example, the implicit equation of the [[Unit circle|unit circle]] is <math> x^2 +y^2-1 = 0</math>.

An '''implicit function''' is a [[Function (mathematics)|function]] that is defined implicitly by an implicit equation, by associating one of the variables (the [[Value (mathematics)|value]]) with the others (the [[Argument of a function|argument]]s).<ref name=Chiang>{{cite book |last=Chiang |first=Alpha C. |title=Fundamental Methods of Mathematical Economics |location=New York |publisher=McGraw-Hill |edition=Third |year=1984 |isbn=0-07-010813-7 |url=https://books.google.com/books?id=6gcoAQAAMAAJ }}</ref>{{rp|204–206}} Thus, an implicit function for <math>y</math> in the context of the [[Unit circle|unit circle]] is defined implicitly by <math> x^2 +f(x)^2-1 = 0</math>. This implicit equation defines <math>f</math> as a function of <math>x</math> only if <math>-1 \leq x \leq 1</math> and one considers only non-negative (or non-positive) values for the values of the function.

The [[Implicit function theorem|implicit function theorem]] provides conditions under which some kinds of relations define an implicit function, namely relations defined as the [[Indicator function|indicator function]] of the zero set of some continuously differentiable [[Multivariable calculus|multivariate]] function.}}
{{term|term=improper fraction}}
{{defn|defn=Common fractions can be classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise.<ref>{{cite web|url=http://www.worldwidewords.org/qa/qa-vul1.htm|title=World Wide Words: Vulgar fractions|work=World Wide Words|access-date=2014-10-30}}</ref><ref>{{MathWorld |title=Improper Fraction |id=ImproperFraction}}</ref> In general, a common fraction is said to be a proper fraction if the [[Absolute value|absolute value]] of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1.<ref>{{cite web|url=http://mathforum.org/library/drmath/view/65128.html|title=Math Forum – Ask Dr. Math:Can Negative Fractions Also Be Proper or Improper?|author=Laurel|date=31 March 2004|access-date=2014-10-30}}</ref><ref>{{cite web|url=http://www.necompact.org/ea/gle_support/Math/resources_number/prop_fraction.htm|title=New England Compact Math Resources|access-date=2019-06-16|archive-url=https://web.archive.org/web/20120415053421/http://www.necompact.org/ea/gle_support/Math/resources_number/prop_fraction.htm|archive-date=2012-04-15|url-status=dead}}</ref>
It is said to be an improper fraction, or sometimes top-heavy fraction,<ref>{{cite book|last1=Greer|first1=A.|title=New comprehensive mathematics for 'O' level|date=1986|publisher=Thornes|location=Cheltenham|isbn=978-0-85950-159-0|page=5|edition=2nd ed., reprinted.|url=https://books.google.com/books?id=wX2dxeDahAwC&pg=PA5|access-date=2014-07-29}}</ref> if the absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are 2/3, –3/4, and 4/9; examples of improper fractions are 9/4, –4/3, and 3/3.}}
{{term|term=[[Improper integral|improper integral]]}}
{{defn|defn=In [[Mathematical analysis|mathematical analysis]], an improper integral is the [[Limit (mathematics)|limit]] of a definite integral as an endpoint of the interval(s) of integration approaches either a specified [[Real number|real number]], <math>\infty</math>, <math>-\infty</math>, or in some instances as both endpoints approach limits. Such an integral is often written symbolically just like a standard definite integral, in some cases with ''infinity'' as a limit of integration.

Specifically, an improper integral is a limit of the form:
:<math>\lim_{b\to\infty} \int_a^bf(x)\, dx, \qquad \lim_{a\to -\infty} \int_a^bf(x)\, dx,</math>
or
:<math>\lim_{c\to b^-} \int_a^cf(x)\, dx,\quad
\lim_{c\to a^+} \int_c^bf(x)\, dx,</math>
in which one takes a limit in one or the other (or sometimes both) endpoints {{harv|Apostol|1967|loc=§10.23}}.}}
{{term|term=[[Inflection point|inflection point]]}} 
{{defn|defn= In [[Differential calculus|differential calculus]], an '''inflection point''', '''point of inflection''', '''flex''', or '''inflection''' (British English: '''inflexion''') is a point on a [[Continuous function|continuous]] [[Plane curve|plane curve]] at which the curve changes from being [[Concave function|concave]] (concave downward) to [[Convex function|convex]] (concave upward), or vice versa.}}
{{term|term=[[Derivative|instantaneous rate of change]]}}
{{defn|defn=The derivative of a function of a single variable at a chosen input value, when it exists, is the [[Slope|slope]] of the [[Tangent|tangent line]] to the [[Graph of a function|graph of the function]] at that point. The tangent line is the best [[Linear approximation|linear approximation]] of the function near that input value.  For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. .}}
{{term|term=[[Physics:Velocity#Instantaneous velocity|instantaneous velocity]]}}
{{defn|defn=If we consider {{math|'''''v'''''}} as velocity and {{math|'''''x'''''}} as the displacement (change in position) vector, then we can express the (instantaneous) velocity of a particle or object, at any particular time {{math|''t''}}, as the [[Derivative|derivative]] of the position with respect to time:

:<math>\boldsymbol{v} = \lim_{{\Delta t}\to 0} \frac{\Delta \boldsymbol{x}}{\Delta t} = \frac{d\boldsymbol{x}}{d\mathit{t}} .</math>

From this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time ({{math|'''''v'''''}} vs. {{math|''t''}} graph) is the displacement, {{math|'''''x'''''}}. In calculus terms, the [[Integral|integral]] of the velocity function {{math|'''''v'''''(''t'')}} is the displacement function {{math|'''''x'''''(''t'')}}. In the figure, this corresponds to the yellow area under the curve labeled {{math|'''''s'''''}} ({{math|'''''s'''''}} being an alternative notation for displacement).

:<math>\boldsymbol{x} = \int \boldsymbol{v} \ d\mathit{t} .</math>

Since the derivative of the position with respect to time gives the change in position (in [[Metre|metre]]s) divided by the change in time (in [[Second|second]]s), velocity is measured in [[Metre per second|metres per second]] (m/s). Although the concept of an instantaneous velocity might at first seem counter-intuitive, it may be thought of as the velocity that the object would continue to travel at if it stopped accelerating at that moment. .}}
{{term|term=[[Integral|integral]]}}
{{defn|defn=An integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining [[Infinitesimal|infinitesimal]] data. Integration is one of the two main operations of calculus, with its inverse operation, [[Derivative|differentiation]], being the other. .}}
{{term|term=[[Integral symbol|integral symbol]]}}
{{defn|defn=The integral symbol:
:<span style=font-size:200%>∫</span> ([[Unicode]]), <math>\displaystyle \int</math> ([[Software:LaTeX|LaTeX]])
is used to denote [[Integral|integral]]s and antiderivatives in [[Mathematics|mathematics]]. .}}
{{term|term=integrand}} 
{{defn|defn= The function to be integrated in an integral.}}
{{term|term=[[Integration by parts|integration by parts]]}}
{{defn|defn=In calculus, and more generally in [[Mathematical analysis|mathematical analysis]], '''integration by parts''' or '''partial integration''' is a process that finds the integral of a [[Product (mathematics)|product]] of functions in terms of the integral of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be readily derived by integrating the [[Product rule|product rule]] of [[Derivative|differentiation]].

If {{math|1=''u'' = ''u''(''x'')}} and {{math|1=''du'' = ''{{prime|u}}''(''x'') ''dx''}}, while {{math|1=''v'' = ''v''(''x'')}} and {{math|1=''dv'' = ''{{prime|v}}''(''x'') ''dx''}}, then integration by parts states that:

:<math>\begin{align}
   \int_a^b u(x) v'(x) \, dx 
   &= \Big[u(x) v(x)\Big]_a^b - \int_a^b u'(x) v(x) \, dx\\
   &= u(b) v(b) - u(a) v(a) - \int_a^b u'(x) v(x) \, dx 
 \end{align}</math>

or more compactly:

:<math>\int u \, dv = uv - \int v \, du.</math>

Mathematician [[Biography:Brook Taylor|Brook Taylor]] discovered integration by parts, first publishing the idea in 1715.<ref name="Brook Taylor biography, St. Andrews">{{cite web |url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.html |title=Brook Taylor |work=History.MCS.St-Andrews.ac.uk |access-date= May 25, 2018}}</ref><ref name="Brook Taylor biography, Stetson">{{cite web |url=https://www2.stetson.edu/~efriedma/periodictable/html/Tl.html |title=Brook Taylor |work=Stetson.edu |access-date=May 25, 2018 |archive-date=January 3, 2018 |archive-url=https://web.archive.org/web/20180103003304/http://www2.stetson.edu/~efriedma/periodictable/html/Tl.html |url-status=dead }}</ref> More general formulations of integration by parts exist for the [[Riemann–Stieltjes integral#Properties and relation to the Riemann integral|Riemann–Stieltjes]] and Lebesgue–Stieltjes integrals. The discrete analogue for sequences is called [[Summation by parts|summation by parts]].
 .}}
{{term|term=[[Integration by substitution|integration by substitution]]}}
{{defn|defn=Also known as ''u''-substitution, is a method for solving [[Integral|integral]]s. Using the [[Fundamental theorem of calculus|fundamental theorem of calculus]] often requires finding an [[Antiderivative|antiderivative]]. For this and other reasons, integration by substitution is an important tool in mathematics. It is the counterpart to the [[Chain rule|chain rule]] for [[Derivative|differentiation]]. .}}
{{term|term=[[Intermediate value theorem|intermediate value theorem]]}}
{{defn|defn=In [[Mathematical analysis|mathematical analysis]], the intermediate value theorem states that if a [[Continuous function|continuous function]], ''f'', with an [[Interval (mathematics)|interval]], [''a'', ''b''], as its [[Domain of a function|domain]], takes values ''f''(''a'') and ''f''(''b'') at each end of the interval, then it also takes any value between ''f''(''a'') and ''f''(''b'') at some point within the interval.

This has two important [[Corollary|corollaries]]:

# If a continuous function has values of opposite sign inside an interval, then it has a root in that interval ('''Bolzano's theorem''').<ref>{{MathWorld |title=Bolzano's Theorem |urlname=BolzanosTheorem}}</ref> 
# The [[Image (mathematics)|image]] of a continuous function over an interval is itself an interval. .}}
{{term|term=[[Inverse trigonometric functions|inverse trigonometric functions]]}}
{{defn|defn=(Also called arcus functions,<ref>Taczanowski, Stefan (1978-10-01). "On the optimization of some geometric parameters in 14 MeV neutron activation analysis". Nuclear Instruments and Methods. ScienceDirect. 155(3): 543–546. doi:10.1016/0029-554X(78)90541-4.</ref><ref>Hazewinkel, Michiel (1994) [1987]. Encyclopaedia of Mathematics (unabridged reprint ed.). Kluwer Academic Publishers / Springer Science & Business Media. {{ISBN|978-155608010-4}}.</ref><ref>Ebner, Dieter (2005-07-25). Preparatory Course in Mathematics (PDF) (6 ed.). Department of Physics, University of Konstanz. Archived (PDF) from the original on 2017-07-26. Retrieved 2017-07-26.</ref><ref>Mejlbro, Leif (2010-11-11). Stability, Riemann Surfaces, Conformal Mappings - Complex Functions Theory (PDF) (1 ed.). Ventus Publishing ApS / Bookboon. {{ISBN|978-87-7681-702-2}}. Archived (PDF) from the original on 2017-07-26. Retrieved 2017-07-26.</ref><ref>Durán, Mario (2012). Mathematical methods for wave propagation in science and engineering. 1: Fundamentals (1 ed.). Ediciones UC. p. 88. {{ISBN|978-956141314-6}}.</ref> antitrigonometric functions<ref>Hall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Chapter II. The Acute Angle [14] Inverse trigonometric functions". Written at Ann Arbor, Michigan, USA. Trigonometry. Part I: Plane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. p. 15. Retrieved 2017-08-12. […] α = arcsin m: It is frequently read "arc-sinem" or "anti-sine m," since two mutually inverse functions are said each to be the anti-function of the other. […] A similar symbolic relation holds for the other trigonometric functions. […] This notation is universally used in Europe and is fast gaining ground in this country. A less desirable symbol, α = sin-1m, is still found in English and American texts. The notation α = inv sin m is perhaps better still on account of its general applicability. […]</ref> or cyclometric functions<ref>Klein, Christian Felix (1924) [1902]. Elementarmathematik vom höheren Standpunkt aus: Arithmetik, Algebra, Analysis (in German). 1 (3rd ed.). Berlin: J. Springer.</ref><ref>Klein, Christian Felix (2004) [1932]. Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis. Translated by Hedrick, E. R.; Noble, C. A. (Translation of 3rd German ed.). Dover Publications, Inc. / The Macmillan Company. {{ISBN|978-0-48643480-3}}. Retrieved 2017-08-13.</ref><ref>Dörrie, Heinrich (1965). Triumph der Mathematik. Translated by Antin, David. Dover Publications. p. 69. {{ISBN|978-0-486-61348-2}}.</ref>) are the [[Inverse function|inverse function]]s of the [[Trigonometric functions|trigonometric functions]] (with suitably restricted [[Domain of a function|domain]]s). Specifically, they are the inverses of the [[Sine|sine]], [[Cosine|cosine]], [[Tangent (trigonometry)|tangent]], [[Cotangent|cotangent]], [[Secant (trigonometry)|secant]], and [[Cosecant|cosecant]] functions, and are used to obtain an angle from any of the angle's trigonometric ratios.}}

{{glossary end}}

==J==
{{glossary}}

{{term|term=[[Classification of discontinuities#Jump discontinuity|jump discontinuity]]}}
{{defn|defn= Consider the function  
:<math>f(x) = \begin{cases}
  x^2         & \mbox{ for } x < 1 \\
  0           & \mbox{ for } x = 1 \\
  2 - (x-1)^2 & \mbox{ for } x > 1
\end{cases}</math>

Then, the point {{math|''x''<sub>0</sub>}} = 1 is a ''jump discontinuity''.

In this case, a single limit does not exist because the one-sided limits, {{math|''L''<sup>−</sup>}} and {{math|''L''<sup>+</sup>}}, exist and are finite, but are not equal: since, {{math|''L''<sup>−</sup>}} ≠ {{math|''L''<sup>+</sup>}}, the limit {{math|''L''}} does not exist. Then, {{math|''x''<sub>0</sub>}} is called a ''jump discontinuity'', ''step discontinuity'', or ''discontinuity of the first kind''. For this type of discontinuity, the function {{math|''f''}} may have any value at {{math|''x''<sub>0</sub>}}.}}

{{glossary end}}

==L==

{{glossary}}

{{term|term=[[Lebesgue integration]]}}
{{defn|defn= In mathematics, the [[Integral|integral]] of a non-negative [[Function (mathematics)|function]] of a single variable can be regarded, in the simplest case, as the [[Area|area]] between the [[Graph of a function|graph]] of that function and the {{math|''x''}}-axis. The '''Lebesgue integral''' extends the integral to a larger class of functions. It also extends the [[Domain of a function|domain]]s on which these functions can be defined.}}
{{term|term=[[L'Hôpital's rule]]}}
{{defn|defn= '''L'Hôpital's rule''' or '''L'Hospital's rule'''  uses [[Derivative|derivative]]s to help evaluate [[Limit of a function|limits]] involving [[Indeterminate form|indeterminate form]]s.  Application (or repeated application) of the rule often converts an indeterminate form to an expression that can be evaluated by substitution, allowing easier evaluation of the limit. The rule is named after the 17th-century {{wipe|France}} [[Biography:Mathematician|mathematician]] [[Biography:Guillaume de l'Hôpital|Guillaume de l'Hôpital]]. Although the contribution of the rule is often attributed to L'Hôpital, the theorem was first introduced to L'Hôpital in 1694 by the Swiss mathematician [[Biography:Johann Bernoulli|Johann Bernoulli]].

L'Hôpital's rule states that for functions {{math|{{var|f}}}} and {{math|{{var|g}}}} which are [[Differentiable function|differentiable]] on an open [[Interval (mathematics)|interval]] {{math|{{var|I}}}} except possibly at a point {{math|{{var|c}}}} contained in {{math|{{var|I}}}}, if
<math>\lim_{x\to c}f(x)=\lim_{x\to c}g(x)=0 \text{ or } \pm\infty,</math> <math>g'(x)\ne 0</math> for all {{math|{{var|x}}}} in {{math|{{var|I}}}} with {{math|1={{var|x}} ≠ {{var|c}}}}, and <math>\lim_{x\to c}\frac{f'(x)}{g'(x)}</math> exists, then
:<math>\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}.</math>

The differentiation of the numerator and denominator often simplifies the quotient or converts it to a limit that can be evaluated directly.}}
{{term|term=[[Limit comparison test|limit comparison test]]}}
{{defn|defn= The limit comparison test allows one to determine the convergence of one series based on the convergence of another.}}
{{term|term=[[Limit of a function|limit of a function]]}}
{{defn|defn= .}}
{{term|term=[[Limits of integration|limits of integration]]}}
{{defn|defn= .}}
{{term|term=[[Linear combination|linear combination]]}}
{{defn|defn=In [[Mathematics|mathematics]], a linear combination is an [[Expression (mathematics)|expression]] constructed from a [[Set (mathematics)|set]] of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be any expression of the form ''ax'' + ''by'', where ''a'' and ''b'' are constants).<ref>{{cite book | last=Lay | first=David C. | title=Linear Algebra and Its Applications | url=https://archive.org/details/studyguidetoline0000layd | url-access=registration | publisher=Addison–Wesley | year=2006 | edition = 3rd | isbn=0-321-28713-4}}</ref><ref>{{cite book | last=Strang | first=Gilbert |  title=Linear Algebra and Its Applications | publisher=Brooks Cole | year=2006 | edition = 4th | isbn=0-03-010567-6}}</ref><ref>{{cite book | last = Axler | first = Sheldon | title = Linear Algebra Done Right | publisher = [[Company:Springer Science+Business Media|Springer]] | year = 2002 | edition = 2nd | isbn = 0-387-98258-2}}</ref> The concept of linear combinations is central to [[Linear algebra|linear algebra]] and related fields of mathematics.}}
{{term|term=[[Linear equation|linear equation]]}}
{{defn|defn=A linear equation is an equation relating two or more variables to each other in the form of <math>a_1x_1+\cdots +a_nx_n+b=0,</math> with the highest power of each variable being 1.}}
{{term|term=[[Physics:Linear system|linear system]]}}
{{defn|defn= .}}
{{term|term=list of integrals}}
{{defn|defn= .}}
{{term|term=[[Logarithm|logarithm]]}}
{{defn|defn= .}}
{{term|term=[[Logarithmic differentiation|logarithmic differentiation]]}}
{{defn|defn= .}}
{{term|term=[[Upper and lower bounds#Bounds of functions|lower bound]]}}
{{defn|defn= .}}

{{glossary end}}

==M==

{{glossary}}

{{term|term=[[Mean value theorem|mean value theorem]]}}
{{defn|defn= .}}
{{term|term=[[Monotonic function#Monotonicity in calculus and analysis|monotonic function]]}}
{{defn|defn= .}}
{{term|term=[[Multiple integral|multiple integral]]}}
{{defn|defn= .}}
{{term|term=Multiplicative calculus}}
{{defn|defn= .}}
{{term|term=[[Multivariable calculus|multivariable calculus]]}}
{{defn|defn= .}}

{{glossary end}}

== N ==
{{glossary}}

{{term|term=[[Natural logarithm|natural logarithm]]}}
{{defn|defn= The '''natural logarithm''' of a number is its [[Logarithm|logarithm]] to the [[Base (exponentiation)|base]] of the [[Mathematical constant|mathematical constant]] [[E (mathematical constant)|''e'']], where ''e'' is an [[Irrational number|irrational]] and [[Transcendental number|transcendental]] number approximately equal to {{val|2.718281828459}}. The natural logarithm of ''x'' is generally written as {{nowrap|ln ''x''}}, {{nowrap|log<sub>''e''</sub> ''x''}},  or sometimes, if the base ''e'' is implicit, simply {{nowrap|log ''x''}}.<ref>{{cite book |title=Mathematics for physical chemistry |edition=3rd |author-first=Robert G. |author-last=Mortimer |publisher=[[Company:Academic Press|Academic Press]] |date=2005 |isbn=0-12-508347-5 |page=9 |url=https://books.google.com/books?id=nGoSv5tmATsC}} [https://books.google.com/books?id=nGoSv5tmATsC&pg=PA9 Extract of page 9]</ref> Parentheses are sometimes added for clarity, giving ln(''x''), log<sub>''e''</sub>(''x'') or log(''x''). This is done in particular when the argument to the logarithm is not a single symbol, to prevent ambiguity.}}
{{term|term=non-Newtonian calculus}}
{{defn|defn= .}}
{{term|term=[[Nonstandard calculus|nonstandard calculus]]}}
{{defn|defn= .}}
{{term|term=[[Notation for differentiation|notation for differentiation]]}}
{{defn|defn= .}}
{{term|term=[[Numerical integration|numerical integration]]}}
{{defn|defn= .}}

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==O==
{{glossary}}

{{term|term=[[One-sided limit|one-sided limit]]}}
{{defn|defn= .}}
{{term|term=[[Ordinary differential equation|ordinary differential equation]]}}
{{defn|defn= .}}

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==P==
{{glossary}}

{{term|term=[[Pappus's centroid theorem]]}}
{{defn|defn= (Also known as  the '''Guldinus theorem''', '''Pappus–Guldinus theorem''' or '''Pappus's theorem''') is either of two related [[Theorem|theorem]]s dealing with the [[Surface area|surface area]]s and [[Physics:Volume|volume]]s of [[Surface of revolution|surface]]s and [[Solid of revolution|solid]]s of revolution.}}
{{term|term=[[Parabola|parabola]]}}
{{defn|defn= Is a [[Plane curve|plane curve]] that is [[Reflection symmetry|mirror-symmetrical]] and is approximately U-[[Shape|shape]]d. It fits several superficially different other [[Mathematics|mathematical]] descriptions, which can all be proved to define exactly the same curves.}}
{{term|term=[[Paraboloid|paraboloid]]}}
{{defn|defn= .}}
{{term|term=[[Partial derivative|partial derivative]]}}
{{defn|defn= .}}
{{term|term=[[Partial differential equation|partial differential equation]]}}
{{defn|defn= .}}
{{term|term=[[Partial fraction decomposition|partial fraction decomposition]]}}
{{defn|defn= .}}
{{term|term=particular solution}}
{{defn|defn= .}}
{{term|term=[[Piecewise|piecewise-defined function]]}}
{{defn|defn= A function defined by multiple sub-functions that apply to certain intervals of the function's domain.}}
{{term|term=[[Position (vector)|position vector]]}}
{{defn|defn= .}}
{{term|term=[[Power rule|power rule]]}}
{{defn|defn= .}}
{{term|term=[[Product integral|product integral]]}}
{{defn|defn= .}}
{{term|term=[[Product rule|product rule]]}}
{{defn|defn= .}}
{{term|term=proper fraction}}
{{defn|defn= .}}
{{term|term=proper rational function}}
{{defn|defn= .}}
{{term|term=[[Pythagorean theorem]]}}
{{defn|defn= .}}
{{term|term=[[Pythagorean trigonometric identity]]}}
{{defn|defn= .}}

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==Q==
{{glossary}}

{{term|term=[[Quadratic function|quadratic function]]}}
{{defn|defn= In [[Algebra|algebra]], a '''quadratic function''', a '''quadratic polynomial''', a '''polynomial of degree 2''', or simply a '''quadratic''', is a [[Polynomial function|polynomial function]] with one or more variables in which the highest-degree term is of the second degree. For example, a quadratic function in three variables ''x'', ''y,'' and ''z'' contains exclusively terms ''x''<sup>2</sup>, ''y''<sup>2</sup>, ''z''<sup>2</sup>, ''xy'', ''xz'', ''yz'', ''x'', ''y'', ''z'', and a constant:

:<math>f(x,y,z)=ax^2+by^2+cz^2+dxy+exz+fyz+gx+hy+iz +j,</math>

with at least one of the [[Coefficient|coefficient]]s ''a, b, c, d, e,'' or ''f'' of the second-degree terms being non-zero.

A ''univariate'' (single-variable) quadratic function has the form<ref name="wolfram">{{cite web | url=http://mathworld.wolfram.com/QuadraticEquation.html | title=Quadratic Equation -- from Wolfram MathWorld | access-date=January 6, 2013}}</ref>

:<math>f(x)=ax^2+bx+c,\quad a \ne 0</math>
in the single variable ''x''. The [[Graph of a function|graph]] of a univariate quadratic function is a [[Parabola|parabola]] whose axis of symmetry is parallel to the {{math|''y''}}-axis, as shown at right.

If the quadratic function is set equal to zero, then the result is a [[Quadratic equation|quadratic equation]]. The solutions to the univariate equation are called the [[Root of a function|root]]s of the univariate function.

The bivariate case in terms of variables ''x'' and ''y'' has the form

:<math> f(x,y) = a x^2 + by^2 + cx y+ d x+ ey + f \,\!</math>

with at least one of ''a, b, c'' not equal to zero, and an equation setting this function equal to zero gives rise to a [[Conic section|conic section]] (a [[Circle|circle]] or other [[Ellipse|ellipse]], a [[Parabola|parabola]], or a [[Hyperbola|hyperbola]]).

In general there can be an arbitrarily large number of variables, in which case the resulting surface is called a [[Quadric|quadric]], but the highest degree term must be of degree 2, such as ''x''<sup>2</sup>, ''xy'', ''yz'', etc.}}
{{term|term=quadratic polynomial}}
{{defn|defn= .}}
{{term|term=[[Quotient rule|quotient rule]]}}
{{defn|defn= A formula for finding the derivative of a function that is the ratio of two functions.}}

{{glossary end}}

==R==
{{glossary}}

{{term|term=[[Radian|radian]]}}
{{defn|defn=Is the [[International System of Units|SI unit]] for measuring [[Angle|angle]]s, and is the standard unit of angular measure used in many areas of [[Mathematics|mathematics]].  The length of an arc of a [[Unit circle|unit circle]] is numerically equal to the measurement in radians of the [[Angle|angle]] that it subtends; one radian is just under 57.3 [[Degree (angle)|degrees]] (expansion at {{OEIS2C|A072097}}). The unit was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an [[SI derived unit]].<ref>{{cite web |title=Resolution 8 of the CGPM at its 20th Meeting (1995) |url=http://www.bipm.org/en/CGPM/db/20/8/ |access-date=2014-09-23 |publisher=Bureau International des Poids et Mesures}}</ref>  Separately, the SI unit of [[Solid angle|solid angle]] measurement is the [[Steradian|steradian]] .}}
{{term|term=[[Ratio test|ratio test]]}}
{{defn|defn= .}}
{{term|term=reciprocal function}}
{{defn|defn= .}}
{{term|term=[[Reciprocal rule|reciprocal rule]]}}
{{defn|defn= .}}
{{term|term=[[Riemann integral]]}}
{{defn|defn= .}}
{{term|term=[[Related rates|related rates]]}}
{{defn|defn= .}}
{{term|term=[[Classification of discontinuities#Removable discontinuity|removable discontinuity]]}}
{{defn|defn= .}}
{{term|term=[[Rolle's theorem]]}}
{{defn|defn= .}}
{{term|term=[[Root test|root test]]}}
{{defn|defn= .}}

{{glossary end}}

==S==
{{glossary}}

{{term|term=[[Scalar (mathematics)|scalar]]}}
{{defn|defn= .}}
{{term|term=[[Secant line|secant line]]}}
{{defn|defn= .}}
{{term|term=second-degree polynomial}}
{{defn|defn= .}}
{{term|term=[[Second derivative|second derivative]]}}
{{defn|defn= .}}
{{term|term=[[Second derivative#Second derivative test|second derivative test]]}}
{{defn|defn= .}}
{{term|term=[[Differential equation#Equation order|second-order differential equation]]}}
{{defn|defn= .}}
{{term|term=[[Series (mathematics)|series]]}}
{{defn|defn= .}}
{{term|term=[[Shell integration|shell integration]]}}
{{defn|defn= .}}
{{term|term=[[Simpson's rule]]}}
{{defn|defn= .}}
{{term|term=[[Sine|sine]]}}
{{defn|defn= .}}
{{term|term=[[Physics:Sine wave|sine wave]]}}
{{defn|defn= .}}
{{term|term=[[Slope field|slope field]]}}
{{defn|defn= .}}
{{term|term=[[Squeeze theorem|squeeze theorem]]}}
{{defn|defn= .}}
{{term|term=[[Sum rule in differentiation|sum rule in differentiation]]}}
{{defn|defn= .}}
{{term|term=[[Sum rule in integration|sum rule in integration]]}}
{{defn|defn= .}}
{{term|term=[[Summation|summation]]}}
{{defn|defn= .}}
{{term|term=supplementary angle}}
{{defn|defn= .}}
{{term|term=[[Surface area|surface area]]}}
{{defn|defn= .}}
{{term|term=[[System of linear equations|system of linear equations]]}}
{{defn|defn= .}}

{{glossary end}}

==T==
{{glossary}}

{{term|term=table of integrals}}
{{defn|defn= .}}
{{term|term=[[Taylor series]]}}
{{defn|defn= .}}
{{term|term=[[Taylor's theorem]]}}
{{defn|defn= .}}
{{term|term=[[Tangent|tangent]]}}
{{defn|defn= .}}
{{term|term=[[Degree of a polynomial|third-degree polynomial]]}}
{{defn|defn= .}}
{{term|term=[[Third derivative|third derivative]]}}
{{defn|defn= .}}
{{term|term=[[Toroid|toroid]]}}
{{defn|defn= .}}
{{term|term=total differential}}
{{defn|defn= .}}
{{term|term=[[Trigonometric functions|trigonometric functions]]}}
{{defn|defn= .}}
{{term|term=[[List of trigonometric identities|trigonometric identities]]}}
{{defn|defn= .}}
{{term|term=[[Trigonometric integral|trigonometric integral]]}}
{{defn|defn= .}}
{{term|term=[[Trigonometric substitution|trigonometric substitution]]}}
{{defn|defn= .}}
{{term|term=[[Trigonometry|trigonometry]]}}
{{defn|defn= .}}
{{term|term=triple integral}}
{{defn|defn= .}}

{{glossary end}}

==U==
{{glossary}}

{{term|term=[[Upper and lower bounds#Bounds of functions|upper bound]]}}
{{defn|defn= .}}

{{glossary end}}

==V==
{{glossary}}

{{term|term=[[Variable (mathematics)|variable]]}}
{{defn|defn= .}}
{{term|term=[[Vector (mathematics and physics)|vector]]}}
{{defn|defn= .}}
{{term|term=[[Vector calculus|vector calculus]]}}
{{defn|defn= .}}

{{glossary end}}

==W==
{{glossary}}

{{term|term=[[Disc integration#Washer method|washer]]}}
{{defn|defn= .}}
{{term|term=[[Disc integration#Washer method|washer method]]}}
{{defn|defn= .}}

{{glossary end}}

== See also ==
* [[Outline of calculus]]
* [[Glossary of areas of mathematics]]
* [[Astronomy:Glossary of astronomy|Glossary of astronomy]]
* Glossary of biology
* [[Biology:Glossary of botanical terms|Glossary of botany]]
* [[Chemistry:Glossary of chemistry terms|Glossary of chemistry]]
* [[Earth:Glossary of ecology|Glossary of ecology]]
* [[Engineering:Glossary of engineering|Glossary of engineering]]
* Glossary of physics
* [[Glossary of probability and statistics]]

==References==
{{Reflist}}

===Works cited===
* {{citation |last=Apostol |first=T |title=Calculus, Vol. 1 |publisher=Jon Wiley & Sons |edition=2nd |year=1967}}.
* {{Citation |first=L. F. A. |last=Arbogast |title=Du calcul des derivations |trans-title=On the calculus of derivatives |year=1800 |language=fr |publisher=Levrault |place=Strasbourg |url=https://books.google.com/books?id=YoPq8uCy5Y8C |pages=xxiii+404}}.
* {{cite book |last1=Butcher |first1=John C. |title=Numerical Methods for Ordinary Differential Equations |publisher={{wipe|John Wiley & Sons}} |location=New York |isbn=978-0-471-96758-3 |year=2003}}
* {{Citation |first=Alex D. D. |last=Craik |title=Prehistory of Faà di Bruno's Formula |journal=[[American Mathematical Monthly]] |volume=112 |date=February 2005 |pages=217–234 |issue=2 |jstor=30037410 |doi=10.2307/30037410 |mr=2121322 |zbl=1088.01008}}.
* {{cite book |last1=Hairer |first1=Ernst |last2=Nørsett |first2=Syvert Paul |last3=Wanner |first3=Gerhard |title=Solving ordinary differential equations I: Nonstiff problems |publisher=[[Physics:Springer-Verlag|Springer-Verlag]] |location=Berlin, New York |isbn=978-3-540-56670-0 |year=1993}}
* {{Citation |last=Johnson |first=Warren P. |title=The Curious History of Faà di Bruno's Formula |journal=[[American Mathematical Monthly]] |volume=109 |issue=3 |pages=217–234 |date=March 2002 |url=https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Johnson217-234.pdf |jstor=2695352 |doi=10.2307/2695352 |mr=1903577 |zbl=1024.01010 |citeseerx=10.1.1.109.4135}}.

==Notes==

{{reflist|group=note}}
{{reflist|group=Note}}
{{reflist|group=lower-alpha}}

{{Glossaries of science and engineering}}
{{Integral}}
{{Infinitesimals}}
{{Calculus topics}}

[[Category:Fields of mathematics|*]]
[[Category:Glossaries of mathematics|C]]

{{Sourceattribution|Glossary of calculus}}