ansi ieee std 260.3-1993 - american national standard mathematical signs and symbols for use in...

8/16/2019 ANSI IEEE Std 260.3-1993 - American National Standard Mathematical Signs and Symbols for Use in Physical Scien…

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ANSI/IEEE Std 260.3-1993(Revision of ANSI Y10.20-1975)

(Supersedes ANSI Y10.17-1961 (R 1988))

American National StandardMathematical Signs and Symbols for Usein Physical Sciences and Technology

Sponsor

IEEE Standards Coordinating Committee 14 onQuantities, Units, and Letter Symbols

Approved March 18, 1993

IEEE Standards Board

Approved August 30, 1993

American National Standards Institute

Abstract: Signs and symbols used in writing mathematical text are defined. Special symbols peculiar to

certain branches of mathematics, such as non-Euclidean Geometries, Abstract Algebras, Topology, andMathematics of Finance, which are not ordinarily applied to the physical sciences and engineering, are

omitted.Keywords: letter symbol, mathematical notation, mathematical sign, mathematical symbol, mathematics,operation symbol, quantity symbol, unit symbol

The Institute of Electrical and Electronics Engineers, Inc.

345 East 47th Street, New York, NY 10017-2394, USACopyright © 1993 by the Institute of Electrical and Electronics Engineers Inc.

All rights reserved. Published 1993. Printed in the United States of America.

ISBN 1-55937-318-0

No part of this publication may be reproduced in any form, in an electronic retrieval system or otherwise, without the prior written permission of the publisher.


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Introduction

(This Introduction is not part of ANSI/IEEE Std 260.3-1993, American National Standard Mathematical Signs and Symbols for Usein Physical Sciences and Technology.)

This Standard is a revision of ANSI Y10.20-1975, the original edition of Mathematical Signs and Symbols for Use inPhysical Sciences and Technology. The purpose of this second edition is twofold: to serve as an authoritative nationalstandard for mathematical notation and, as promised in the first edition, to include symbols of those lesser-knownbranches of mathematics that are increasingly being applied to the physical sciences. Added to this revision are signsand symbols used in Symbolic Logic, Set Theory, Arithmetic, Differential Geometry, Matrices, Probability andStatistics.

The Table of Signs and Symbols has been reorganized and the format modified by insertion of an additional column toexhibit, if appropriate, the application of each sign or symbol, with its meaning in that context, under the Descriptionheading. The former subclause on Trigonometry has been renamed Circular Functions and now includes principalvalues of the inverse functions. To encourage the use of roman type, rather than italic type, to symbolize specificmathematical functions, the subclause on Special Functions in the original edition now occupies five subclauses; thetotal number of items being increased from 8 to 54.

Incorporated in this Standard is a revision of ANSI Y10.17-1961 (R 1988), Guide for Selection of Greek Letters Used as Letter Symbols for Engineering Mathematics. Clause 10 of this Standard supersedes ANSI Y10.17-1961 entirely.

The technical support provided by Delco Systems Operations of Delco Electronics Corporation, in the preparation of this revision, is gratefully acknowledged.

At the time this Standard was completed, the membership of the Standards Coordinating Committee 14, Quantities,Units, and Letter Symbols, consisted of:

Bruce B. Barrow , Chair

Andrew F. DunnStanley L. Ehrlich

Robert V. EspertiJohn A. GoetzTruman S. Gray

M. Harry HesseRon K. Jurgen

William R. KruesiJack M. LoudonArthur O. McCoubrey

Conrad R. MullerChester H. Page

Ralph M. ShowersBarry N. TaylorAlan S. Whelihan

At the time this Standard was completed, the membership of Subcommittee 14.6, Mathematical Signs and Symbols,consisted of:

Robert V. Esperti , Chair

Ralph E. Ekstrom David D. LynchKaj L. Nielsen

Melvin D. Springer

The following persons were on the balloting committee:

Bruce B. BarrowAndrew F. DunnStanley L. EhrlichRobert V. EspertiJohn A. Goetz

Truman S. GrayM. Harry HesseRon K. JurgenWilliam R. KruesiJack M. LoudonArthur O. McCoubrey

Conrad R. MullerChester H. PageRalph M. ShowersBarry N. TaylorAlan S. Whelihan

http://260_3_bk.pdf/http://260_3_bk.pdf/


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When the IEEE Standards Board approved this Standard on March 18, 1993, it had the following membership:

Wallace S. Read , Chair

Donald C. Loughry , Vice Chair

Andrew G. Salem , Secretary

Gilles A. BarilClyde R. CampDonald C. FleckensteinJay Forster*David F. FranklinRamiro GarciaDonald N. HeirmanJim Isaak

Ben C. JohnsonWalter J. KarplusLorraine C. KevraE.G. “Al” KienerIvor N. KnightJoseph L. Koepfinger*D. N. “Jim” LogothetisDon T. Michael*

Marco W. MigliaroL. John RankineArthur K. ReillyRonald H. ReimerGary S. RobinsonLeonard L. TrippDonald W. Zipse

*Member Emeritus

Also included are the following nonvoting IEEE Standards Board liaisons:

Satish K. AggarwalJames Beall Richard B. EngelmanDavid E. Soffrin Stanley Warshaw

Rachel A. Meisel IEEE Standards Project Editor

This Standard has been coordinated with the following organizations:

American Mathematical Society

American Institute of Physics

National Institute of Standards and Technology


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v

CLAUSE PAGE

1. Scope...................................................................................................................................................................1

2. Letter Symbols ............. ............... .............. ............... ............... .............. ................ ............. .............. ............. ......1

3. Alphabets and Typography ............ ............... ............. ............... ............... ............. ............... ............... ............... .2

4. Quantity Symbols......... ............... .............. ................ ............... .............. ............... ............. ............... .............. ....3

5. Unit Symbols.............. .............. ............... ................ ............. ................ ............. ............... .............. ............... ......3

6. Operation Symbols........... ............. ............... .............. ............... ............... .............. ............... .............. ............... .5

7. Reference Documents ............. ............... ............. ................ ............. ............... .............. ............... .............. .........5

7.1 References.................................................................................................................................................. 6

8. Conventions ............... ............. ............... ............. ................ ............. ............... .............. ............... ............... ........7

9. Signs and Symbols ............ ................ ............. ............... .............. ............... ............... .............. ............... .............7

9.1 Miscellaneous Signs and Symbols............................................................................................................. 89.2 General Operations .................................................................................................................................... 99.3 Symbolic Logic and Set Theory............................................................................................................... 119.4 Arithmetic (Number Theory) ................................................................................................................... 129.5 Elementary Functions .............................................................................................................................. 149.6 Geometry.................................................................................................................................................. 199.7 Vectors ..................................................................................................................................................... 229.8 Matrices.................................................................................................................................................... 239.9 Real Variables (Calculus) ........................................................................................................................ 249.10 Complex Variables.......... ............... ............... .............. ............. ............... ............... ............... ............... ... 27

9.11 Special Functions.................................................................................................................................... 289.12 Probability and Statistics................ ............. ................ ............. ................ ............... .............. ............... ... 34

10. Greek Characters........ .............. ............... ............... .............. ............. ................ ............... .............. ............... ....35

11. Bibliography......................................................................................................................................................37


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vi Copyright © 1998 IEEE All Rights Reserved

Designation (Variable) HeaderTitleLeft (Variable)


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Copyright © 1993 IEEE All Rights Reserved 1

American National StandardMathematical Signs and Symbols for Usein Physical Sciences and Technology

1. Scope

Only signs and symbols used in writing mathematical text are contained in this Standard. Special symbols peculiar tocertain branches of mathematics, such as non-Euclidean Geometries, Abstract Algebras, Topology, and Mathematicsof Finance, which are not ordinarily applied to the physical sciences and engineering, have been omitted. Becausethere is no consensus in the literature for signs and symbols used in tensor analysis, the subject of tensors is relegatedto future editions when there is general agreement among authorities in the field.

2. Letter Symbols

Letter symbols1 include symbols for physical quantities (quantity symbols), symbols for units in which these quantitiesare measured (unit symbols), and symbols for operators on, and functions of, these quantities, as well as specialsymbols for frequently used words and phrases (operation symbols).

A quantity symbol is, in general, a single letter,2 e.g., I , to represent an electric current, modified, when appropriate, byone or more subscripts or superscripts, e.g., I i, to represent input current. A symbol assigned to denote a quantity in atreatise should be used consistently for that quantity throughout the work.

1“Letter symbol” as a technical term does not have the same meaning as either “name” or “abbreviation”. An abbreviation is a letter or combinationof letters (sometimes with apostrophe(s) or period) that, by convention, represents a word or a name in a particular language; hence, an abbreviationmay be different in another language. A symbol represents a quantity, a unit , or an operation, and should be independent of language (except, bytradition, some unit symbols and their prefixes have Latin or Greek origins, and many operation symbols are Latin derivatives), e.g., the symbol forthe quantity: electromotive force is “ E ”, whereas the abbreviation is “emf” in English, “fem” in French, and “EMK” in German. The word for theunit of electric current “ampere” is often abbreviated “amp”, but the symbol for this unit is “A”. The international standard symbol for the circularfunction, “sine” is “sin”, although. for example, the word for “sine” in Spanish is “seno”.2Quantity symbols comprising two letters are sometimes used for dimensionless transport parameters, e.g., Prandtl Number, “Pr ”, and ReynoldsNumber, “ Re”, not to be confused with the complex variables function, “Re”, meaning “real part of” (qv 9.10.2, p 26).


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2 Copyright © 1993 IEEE All Rights Reserved

ANSI/IEEE Std 260.3-1993 AMERICAN NATIONAL STANDARD

A unit symbol3 is a letter or group of letters, e.g., “m” for meter(s) and “Hz” for hertz, or a special sign, such as ° fordegree(s), that may be used in place of the name of the unit.

An operation symbol is a letter, a group of letters, or special sign(s) that represents a mathematical operator, a specificmathematical function or relationship, a word, or a phrase.

3. Alphabets and Typography

Letter symbols are restricted primarily to the English and Greek alphabets.4 Script, black letter, or other special fontsto distinguish between possible conflicting uses of the same letter for different quantities should not be used.

Symbols for physical quantities, mathematical variables, indices, and general functions5 are printed in italic type, e.g.,

A areae eccentricity of a conic section x, y, z Cartesian coordinatesi, j, k indices

f ( x) f function of x

Symbols used for physical units, as well as mathematical constants, specific mathematical functions, operators, and allnumerals are printed in roman (upright) type, e.g.,

cm centimeter(s)i imaginary unit:sin 2θ sine of the angle: 2 × θ a × b + c a times b plus cd x differential of x

All punctuation,6 including grouping symbols, such as parentheses, brackets, and braces, are also printed in romantype, e.g.,

F(a, b; c; z) hypergeometric function[ x] integer function of x[ abc] triple scalar product of vectors{a, b} LCM of a and bn! factorial n

Subscripts and superscripts are governed by the above principles. Those that are letter symbols for physical quantities,mathematical variables, or for indices are printed in italic type, whereas others are printed in roman type, e.g.,

sin p x pth power of sin xaij, a45 matrix elements I i, I o input, output currentsB x(α, β) incomplete beta function

3It was once common to treat unit symbols in the same manner as general abbreviations, but the recommendations of the International Organizationfor Standardization (ISO) and many other international and national bodies concerned with standardization, emphasize the symbolic character of these designations and rigidly prescribe the manner in which they shall be treated. The concept of the unit symbol is therefore adopted in thisStandard.4Greek letters that are easily confused with English letters should be avoided. Clauses 8 and 10 provide a guide for selecting Greek letters to be usedas symbols.5The term “general functions” is used here to contrast with “specific mathematical functions”, discussed below.6It should be noted that the commas, semicolons, brackets, braces, and exclamation point in these examples are mathematical operators and,consistent with the previous paragraph, should be displayed in roman type.

1–


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MATHEMATICAL SIGNS AND SYMBOLS ANSI/IEEE Std 260.3-1993

πr 2 area enclosed by a circle

To indicate the vector character of a quantity, boldface type is used, italic for general vectors, roman for unit vectorsand symbols for special vector functions, e.g.,

Fi force on the ith element

grad f gradient of the scalar, f div F divergence of the vector, Fi, j, k orthogonal unit vectors kn normal curvature vector

The gradient symbol is boldface because its operation results in a vector, but “div” is not, as its operation results in ascalar. Ordinary italic type may be used to represent the magnitude of a general vector quantity.

4. Quantity Symbols

Quantity symbols may be used in mathematical expressions in any way consistent with good mathematical usage. The

product of two scalar quantities a and b is indicated by writing ab. The quotient may be indicated by writing any of thefollowing:

When more than one solidus (/) is used in an algebraic expression, grouping symbols shall be inserted to remove anyambiguity. Thus, one may write (a / b)/ c, or a /[b / c], but not a/b/c.

Subscripts and superscripts are commonly used with quantity symbols. Several subscripts or superscripts, sometimesseparated by commas, may be attached to a single letter; but, unless logical clarity dictates it, subscripts andsuperscriptsshould not be attached to other subscripts or superscripts.7 A symbol that has been modified by asuperscript shall be enclosed in grouping symbols before an exponent is appended.

Care should be taken not to assign the same symbol for different quantities in the same work. Use of different symbolsor appending subscripts to distinguish the symbols, is recommended.

5. Unit Symbols

Unit symbols are written in lowercase letters, except the initial letter is capitalized whenever the unit is derived froma proper name.8 The distinction between uppercase and lowercase letters shall be followed even when the symbolsappear in applications where the other lettering is in uppercase style. A unit symbol is printed in roman type, the formbeing the same for both singular and plural. A final period (.) shall not be part of a unit symbol.

7There are acceptable exceptions exhibited in Clause 9: Item Numbers 9.4.18, 9.4.19, 9.4.23, 9.4.24, 9.5.2.1.2, 9.8.10, 9.11.4.4.9, 9.11.4.4.10,9.11.5.4, 9.12.10, 9.12.11.8To prevent confusion with the numeral, 1, the uppercase letter, L, rather than the lowercase letter, 1, is the symbol for liter. Prefixes are consideredseparately. Some examples of frequently used unit symbols are listed below.picofarad pF decibel dBnanosecond ns volt Vmicroampere µA kilopascal kPamilliliter mL megawatt MWcentimeter cm gigahertz GHz

a

b--- a b a b or ab

1–,÷, ⁄ ,


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When a compound unit is formed by multiplication of two or more other units., the symbol consists of the symbols forthe separate units joined by a raised dot (preferred) or by a space, e.g., N ·m (or N m) for newton meter. The dot orspace may be omitted in the case of familiar compounds such as watthour (symbol: Wh) if no confusion would result.9

Hyphens shall not be used in symbols for compound units; however, exponents may be applied to unit symbols.

Should a compound unit be formed by division of one unit by another, its symbol consists of the symbols for theseparate units, either separated by a solidus or conjoined using negative powers, e.g., m/s or m

·s–1 for meter(s) per

second. In simple cases, use of the solidus is preferred, but in no case shall more than one solidus on the same line, ora solidus followed by a product, be included in such a combination unless grouping symbols, such as parentheses, areinserted to avoid ambiguity. In complicated cases, use of negative powers is recommended.10

The following prefixes from the International System of Units (SI) are used to indicate decimal multiples orsubmultiples of units:

9It may also be omitted where adjacent symbols are separated by an exponent, as in V ≡ kg·m2s–3A–1.10The notation for products and quotients of unit symbols is intentionally made more explicit than that given in Clause 4 for quantity symbolsbecause many unit symbols consist of more than one letter, qv Reference 3 in Subclause 7.1.

FACTOR SI PREFIX SYMBOL

1024 yotta Y

1021 zetta Z

1018 exa E

1015 peta P

1012 tera T

109 giga G

106 mega M

103 kilo k

102 hecto h

10 deka da

10–1 deci d

10–2 centi c

10–3 milli m

10–6 micro µ

10–9 nano n

10-12 pico P

10–15 femto f

10–18 atto a

10–21 zepto z

10–24 yocto y


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Values that are outside the range of the above prefixes should be expressed with powers of ten applied to the baseunit(s). In any case, compound prefixes shall not be used, e.g.,

6. Operation Symbols

In this Standard “operation symbols” include signs and symbols for mathematical operators, specific mathematicalfunctions, mathematical relations between quantities, and precisely defined symbols for words and phrases commonlyused in mathematical works; e.g., the operation symbol, ∇, represents

where the signs, ∂, +, and — (division sign) are also operation symbols, and i, j, k, as well as x, y, z, are quantitysymbols. Other examples of operation symbols are “exp” for the exponential function and the symbol “∋”, which iscommonly used in place of the phrase “such that”.

A compilation of operation symbols is listed in Clause 9, with their definitions and, where appropriate, typical use of the symbol and a description of its meaning in that application.

7. Reference Documents

The signs and symbols tabulated in Clause 9 are, for the most part, in accord with general usage. The original editionof this Standard and Reference 111 generally follow the rules stated in Clause 3 regarding the use of roman type forsymbols to designate mathematical constants, operators, and specific mathematical functions. However, the otherreferences do not generally conform to those precepts, often using e for the base of natural logarithms, i for theimaginary unit, , d for the differential operator, and, among other symbols for functions, J n( z) for the Besselfunction of the 1st kind of order n.

In addition to the references below and the handbooks listed in the bibliography, many modern textbooks wereconsulted to obtain a consensus for each sign and symbol; the font character being changed whenever necessary, tocomply with the aforementioned rules. In some cases subjective judgment was used, but in all cases the main concernwas to avoid ambiguity.

Although most of the signs and symbols depicted in Reference 1 agree in meaning with those of this Standard, thereare some differences dictated by tradition in the United States. For example, the International Organization forStandardization (ISO) prefers a comma (,) to denote the decimal sign; however, in the United States and some othercountries, a dot on the baseline (.) is used.

tera T not megamega MM,

giga G not kilomega kM,

nano n not millimicro mµ,

pico p not micromicro µµ.

11Information on references can be found in Subclause 7.1.

i ∂∂ x----- j

∂∂ y----- k

∂∂ z-----+ +

1–


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In general, this Standard favors the notation, definition, and meaning given in Reference 2 wherever disagreementexists with Reference 1. Since Reference 2 was published by the National Bureau of Standards 12 and is a standardreference, even for other references, it is appropriate that this Standard be consistent with Reference 2 except wheretype styles need to be modified to conform to the rules in Clause 3 of this Standard. Where an alternative sign orsymbol displayed in Reference I agrees with this Standard, the two Standards are assumed to be in accord.13 Thosesymbols in Reference 1 that belong to special fonts, violating the principles of Clause 3, are not included in thisStandard.

Some minor discrepancies between ISO and symbols of this Standard (ANS) exist for circular and hyperboliccotangents as well as their inverses (ISO 11–9; ANS 9.5.2.2, 9.5.2.3). In this Standard the cotangent functions aresymbolized by “ctn” to make them more distinguishable from “cos” than “cot”. Then each of the six basic three-lettersymbols has no more than one letter positionally in common with any of the other five. Also, the valid ranges in theprincipal values of the inverse secants and cosecants for negative arguments differ in the two Standards.

Two items in Reference 1, viz, ISO 11–11.3 and 11–13.4, use italic type, where, by the rules of Clause 3, roman typeis mandatory; qv ANS 9.8.4 and 9.7.1, respectively. The combinatorial symbol, ISO item 11–6.16, uses subscripts andsuperscripts for arguments, but to accommodate complicated literal values for them, an ordered pair is used in thisStandard, qv ANS 9.5.1.18.

The symbol, ∋, in Reference 1 (11–4.3) has the meaning “contains the element”, but means “such that” in ANS (9.3.7).Although zero belongs to the set, N, in Reference 1 (11–4.9), in ANS (9.3.15) the set, N, consists only of the naturalnumbers, i.e., the positive integers, which does not include zero. The ISO symbol for equality by definition (11–5.3) islanguage dependent (cf ANS 9.2.16). The order of the arguments in the symbols for elliptic integrals, as well as thesign of the characteristic, n, are different for the two Standards. Although the symbols used for polar, cylindrical, andspherical coordinates in Reference I are the same as those used in this Standard, the definitions of all of these symbols,except for z, disagree in the two documents (cf ISO 11–12, ANS 9.6.2).

The most serious variances are in the meanings attached to the associated Laguerre polynomial (ISO 11–14.13; ANS9.11.1.5) and the exponential integral (ISO 11–14.21; ANS 9.11.4.4.3). The coefficient of y in the differential equationsatisfied by the Laguerre polynomial in ISO is n – m, where in ANS, the (equivalent) coefficient of f(x) is n. Thedifference is reflected in the relationship between the associated Laguerre polynomial and the derivatives of the basicLaguerre polynomial, as shown by the formula in Item Number 11–14.13 of ISO 31-11:1992:

This equation differs from its counterpart in Item Number 9.11.1.5 of this Standard, which agrees with Reference 2,[B4],14 and [B9]. The exponential integral given in Reference 1, according to Reference 2, [B4], and [B5], isequivalent to –Ei(– x) in the notation of those references and is denoted by E 1( x) in Reference 2 and [B9]. Again, thisStandard subscribes to the meaning given in Reference 2.

7.1 References

[1] International Standard ISO 31-11:1992. Mathematical signs and symbols for use in the physical sciences and technology. 2nd ed. International Organization for Standardization.

[2] Abramowitz, Milton, and Stegun, Irene A. Handbook of Mathematical Functions, N. B. S. Applied MathematicsSeries • 55. Washington: National Bureau of Standards, 1972.

[3] ANSI / IEEE Std 260-1978 (Reaffirmed 1991). IEEE Standard Letter Symbols for Units of Measurement.

12The National Bureau of Standards is now known as the National Institute of Standards and Technology.13Reference 1, Introduction14Brackets identify treatises listed in bibliography, Clause 11.

Lnm x( ) d

m

d xm

---------Ln x( ) m n ε N; m n≤,( )=


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8. Conventions

Many conventions for quantity symbols have been adopted by authorities in various fields of the physical sciences andtechnology. Most authors adhere to these conventions even though they are not mandatory. Because they facilitatecomprehension by the reader who is familiar with the field, conventional symbols should be used whenever feasible.Some of the more common conventions are described here for reference.

Constants are generally denoted by the first few lowercase letters of the English alphabet: a, b, c, ⋅⋅⋅, while variablesare designated by the last few letters: ⋅⋅⋅, x, y, z. Integers and indices are usually indicated by the lowercase letters: i, j,k, l, m, and n. Lowercase Greek letters often symbolize dimensionless quantities, such as ratios or angles.15 Somephysical quantities are also represented by Greek letters, e.g., λ to denote wave length, ρ to indicate the density of asubstance, and ω to signify the magnitude of angular velocity.16

When describing the equations of motion of a physical body, such as a ship, an aircraft, a missile, etc, it is customaryto use reference triads, i.e., three mutually orthogonal axes oriented in inertial space, atmosphere, earth, body, etc. Agenerally accepted convention for orienting the body triad assigns the + x-axis to the forward direction in the plane of symmetry (if practicable) parallel to the waterline (or some arbitrary plane), the + y-axis toward starboard, normal tothe plane of symmetry, and the + z-axis pointed in a downward direction, perpendicular to the x-y plane. General

translational motion can be described along these x, y, z coordinate axes.17

Speeds are usually designated by u, v, andw along x, y, and z respectively. Full six-degree-of-freedom equations require rotations about each of the three primarybody axes. The rotations about the axes x, y, and z are commonly denoted by φ (roll), θ (pitch), and ψ (yaw),respectively, where positive rotations are in the right-handed sense, and time rates of change of φ , θ , and ψ arespecified by p, q, and r , respectively.

9. Signs and Symbols

The following table has been categorized into the various branches of mathematics18 to facilitate its use, primarily, butalso because some signs and symbols are common to two or more branches; e.g., ≡ generally means “is equivalent to”,as described in Item Number 9.2.13, but means “is congruent to” in Arithmetic (Item Number 9.4.3). Also, Item

Numbers 9.1.4, 9.4.24, and 9.11.2.2, each describe the letter symbol, “M”, but all have different meanings. However,confusion is not very likely to result from these ambiguous definitions; the meaning should be clear by the context inwhich it is used, if not by the subscript, superscript, or argument(s) attached to it. Some specific quantity symbols thatare associated with the various branches of mathematics are also included.19

15Angles are ratios when measured in radians or steradians.16When necessary to distinguish the parameters, subscripts are used, e.g., ρ0 to designate air density at sea level, and ωe for the earth's rotational ratewith respect to inertial space.17Correlatives of x, y, and z are sometimes specified by the lowercase Greek letters ξ, η, and ζ, respectively.18Not all branches are represented; qv Clause 1.19There are certain mathematical quantities that are represented by generally accepted symbols. One purpose of this Standard is to document suchsymbols. Some of the items in Subclauses 9.1, 9.3, 9.4, 9.5.1, 9.6.1, 9.7, 9.8, 9.10, 9.12, as well as all of Subclauses 9.6.2, 9.6.3, consist of quantitysymbols.


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9.1 Miscellaneous Signs and Symbols

ITEM

NUMBER

SIGN OR

SYMBOLAPPLICATION DEFINITION / MEANING

9.1.1 π A = πr 2 ratio of perimeter to diameter of a circle: 3.141 592 ···

9.1.2 e ln e = 1 natural logarithm base:

9.1.3 γ Γ′(1) = −γ Euler's constant:

9.1.4 M lg x ≡ M ln x lg e = 0.434294··· (qv 9.1.2, 9.5.2.1.4, 9.5.2.1.5)

9.1.5 ∞ infinity; unbounded number

9.1.6 B Bn Bernoulli number:

9.1.7 E En Euler number:

9.1.8 s s(n, m) Stirling number of the 1st kind:s(n + 1, m) s(n, m − 1) − ns(n, m), wheres(1, 1) 1 ; s(n, m) 0 for m 0, n 0, n < m

9.1.9 S S(n, m) Stirling number of the 2nd kind:S(n + 1, m) S(n, m − 1) + mS(n, m), whereS(1, 1) 1; S(n, m) 0 for m 0, n 0, n < m

9.1.10 δ Kronecker delta: 0 for i ≠ j, 1 for i = j9.1.11 . 3 210.123 456 78 radical point (termed decimal point when radix is ten)

Demarcates fractional part of the digital depiction of a realnumber. Spaces separate digits into groups of three.

9.1.12 (,) (a, b) ordered pair: a, b

9.1.13 (,,) ( x, y, z) ordered triplet: x, y, z

9.1.14 (, ,···,) ( x1, x2,···, xn) ordered n-tuplet: x1, x2, ···, xn

11n---+

n

n 8→lim 2.718 281 …=

1k ---

k 1=

n

∑ ln n–

n 8→lim 0.577 215 …=

2n( )!2

2n 1–1–( )π2n

-------------------------------------- 1–( )k 1– 1k

2n-------

k 1=

∞

∑

22n 2+

2n( )!π2n 1+

----------------------------- 1–( )k 1– 12k 1–( )2n 1+

--------------------------------

k 1=

∞

∑

∆=

∆=

∆=

<=

<=

∆= ∆

= ∆=

<=

<=

δ ji


15/43



9.2 General Operations

ITEM

NUMBER

SIGN OR


9.2.1 + p + q addition; a positive value; the sum: p plus q

9.2.2 – p − q subtraction; a negative value; the difference: p minus q9.2.3 ± p ± q plus or minus; the sum or difference of p and q

9.2.4 minus or plus; the difference or sum of the product pq,

e.g., ;ambiguous signs resolved by usingeither all upper signs or all lowersigns throughout the expression

9.2.5 pq multiplication; the product: p times q

9.2.6 · p·q multiplication; the product: p times q (cf 9.7.4)9.2.7 × p × q multiplication; the product: p times q (cf 9.7.5)

9.2.8 – division; the quotient: p divided by q

9.2.9 / p/q division; the quotient: p divided by q

9.2.10 ÷ p ÷ q division; the quotient: p divided by q

9.2.11 = p = q conditional equality; p is conditionally equal to q

9.2.12 ≠ p ≠ q conditional inequality; p is not equal to q

9.2.13 ≡ p2 ≡ pp unconditional equality; p2 is equivalent to pp (cf 9.4.3)

9.2.14 is not equivalent to p (cf 9.4.4)

(NB for real p: ≡ |p|, qv 9.2.31)

9.2.15 ≈ p ≈ q approximate equality; p is approximately equal to q

9.2.16 p q definition; p and q are defined to be equal

9.2.17 p q correspondence; p corresponds to q

+− pq+−

p3

q3± p q±( ) p2 pq q2++−( )≡

pq---

≡/ p2 ≡ p/ p2

p2

∆=

∆=

=̂ =̂


16/43



9.2 General Operations (continued)

ITEM

NUMBER

SIGN OR


9.2.18 ∝ p ∝ q proportionality; p is proportional to q

9.2.19 < p < q inequality; p is less than q

9.2.20 > p > q inequality; p is greater than q

9.2.21 p q p is less than or equal to q

9.2.22 p q p is greater than or equal to q

9.2.23 p q p is much less than q

9.2.24 p q p is much greater than q

9.2.25 ′ p′ prime; correlative of p (cf 9.9.16);also used with numerals to denote arc minute(s)

9.2.26 ″ p″ double prime; correlative of p′ (cf 9.9.17);also used with numerals to denote arc second(s)

9.2.27 () ( p + q)/ r parentheses; the sum, p plus q, divided by r

9.2.28 [] [ p + q]/ r brackets; the sum, p plus q, divided by r

9.2.29 {} { p + q}/ r braces; the sum, p plus q, divided by r

9.2.30 vinculum; the sum, p plus q, divided by r

9.2.31 | | |p| absolute value (magnitude) of p (cf 9.8.7, 9.10.6)

9.2.32 || || || p|| norm of p (exact meaning depends on context)

9.2.33 ⊕ p ⊕ q in a postulational system, the sum, p plus q

9.2.34 p q in a postulational system, the difference, p minus q

9.2.35 ⊗ p ⊗ q in a postulational system, the product, p times q9.2.36 p q in a postulational system, the quotient, p divided by q

9.2.37 ··· p, q, r, ··· , z ellipsis; et cetera (e.g., 9.1.1, 9.1.14, 9.5.1.20, 9.8.1)

<=

<=

>=

>=

>>

p q+ r ⁄

⊕ ⊕

∅ ∅


17/43



9.3 Symbolic Logic and Set Theory

ITEM

NUMBER

SIGN OR


9.3.1 ∧ p ∧ q conjunction; proposition p and proposition q

9.3.2 ∨ p ∨ q disjunction (inclusive); proposition p or proposition q(includes the case p ∧ q)

9.3.3 p q disjunction (exclusive); proposition p or proposition q(excludes the case p ∧ q)

9.3.4 ¬ ¬ p negation; the negative of proposition p

9.3.5 ⇒ p ⇒ q implication; the truth of proposition pimplies proposition q is also true

9.3.6 ⇔ p ⇔ q equivalence; propositions p and q are equivalent

9.3.7 ∋ such that

9.3.8 { , , ···, } {a1, a2,···, an} the set containing the elements ai, i = 1, 2, ···,n

9.3.9 ∈ a ∈ A a is an element (member) of set A

9.3.10 ∉ b ∉ A b is not an element (member) of set A

9.3.11 ∃ ∃ a there exists an element, a;∃! a: there exists a unique element, a

9.3.12 ∀ ∀ai for all elements, ai, i = 1, 2, 3,···, n

9.3.13 ∅ the null (empty) set: the set that contains no elements

9.3.14 P the set of prime numbers: {2, 3, 5, 7, 11, 13, ···}

9.3.15 N the set of natural numbers: {1, 2, 3, 4, 5, 6, ···}

9.3.16 Z the set of integers: {–n, 0, +n}, n ∈ N

9.3.17 Q the set of rational numbers: {m/n}, m ∈ Z, n ∈ N

9.3.18 R the set of real numbers: {r }, –∞ < r < +∞

9.3.19 C the set of complex numbers: { p + iq}, p ∈ R, q ∈ R

^

– ^–


18/43



9.3 Symbolic Logic and Set Theory (continued)

9.4 Arithmetic (Number Theory)

ITEM

NUMBER

SIGN OR


9.3.20 = A = B coincidence; the sets A and B coincide

9.3.21 ⊃ A ⊃ B the set A contains B as a proper subset of A:∃ a ∈ A a ∉ B

9.3.22 ⊇ A ⊇ B the set A contains B as a subset( A and B may coincide)

9.3.23 ⊂ A ⊂ B the set B contains A as a proper subset of B:∃ b ∈ B b ∉ A

9.3.24 ⊆ A ⊆ B the set B contains A as a subset( A and B may coincide)

9.3.25 ∪ A ∪ B the union of set A and set B:C ∀c ∈ C :c ∈ A ∨ c ∈ B

9.3.26 the union of all sets, Ai : A1 ∪ A2 ∪···∪ An

9.3.27 ∩ A ∩ B the intersection of set A and set B:C ∀c ∈ C :c ∈ A ∧ c ∈ B

9.3.28 Ai the intersection of all sets, Ai : A1 ∩ A2 ∩···∩ An

ITEMNUMBER

SIGN ORSYMBOL

APPLICATION DEFINITION / MEANING

9.4.1 | a | b a is a divisor of b, i.e., b / a is an integer

9.4.2 a b a is not a divisor of b, i.e., b / a is not an integer

9.4.3 ≡ (mod) a ≡ b (mod m) a is congruent to b, modulo m ; i.e., m | (a − b)

9.4.4 (mod) a b (mod m) a is not congruent to b, modulo m; i.e., m (a − b)

9.4.5 ( , ) (a, b) the greatest common divisor of a and b

∈

∈

∈

∪ Ai

∪ ∈

∩ Ai

i 1=

n

∩

| | | |

≡/ ≡/ | |


19/43



9.4 Arithmetic (continued)

ITEM

NUMBER

SIGN OR


9.4.6 { , } {a, b} the least common multiple of a and b

9.4.7 U U(n) the least common multiple of first n positive integers9.4.8 d d(n) the number of divisors of n : σ0(n) (qv 9.4.10)

9.4.9 σ σ(n) the sum of the divisors of n: σ1(n) (qv 9.4.10)

9.4.10 σm σm(n) the sum of the mth powers of the divisors of n

9.4.11 p pn p ∈ P; the nth prime number ( pi < pi + 1) (qv 9.3.14)

9.4.12 Ω Ω(n) the total number of prime factors of n

9.4.13 ω ω(n) the number of different prime factors of n

9.4.14 π π(n) pi function: the number of prime numbers n

9.4.15 ϑ ϑ(n) theta function: Σ ln p for all prime numbers, p n

9.4.16 Λ Λ(n) lambda function: ln p for n = pm, 0 for ≠ pm, m 1

9.4.17 ψ ψ (n) psi function:

9.4.18 λ λ(n) Liouville function:

9.4.19 µ µ(n) Möbius function:

9.4.20 ϕ ϕ(n) Euler totient (phi) function: for 0 < m n, the number of

integers, m (m, n) = 19.4.21 p p(n) the number of partitions of n

9.4.22 r r(n) the number of representations of n n = k 2 + m2

9.4.23 F Fn the nth Fermat number: 22n + 1

9.4.24 M the nth Mersenne number: – 1 (qv 9.4.11)

for any i >= 1, then µ n( ) ∆= 0

<=

∈

∈

M pn2 pn


20/43



9.5 Elementary Functions

9.5.1 Algebraic Functions

ITEM

NUMBER

SIGN OR


9.5.1.1 p x p x raised to the power, p: x x x··· x ( p factors for integral p > 0)

9.5.1.2 – p x– p 1/ x p ( x ≠ 0 if p 0)

9.5.1.3 p+q x p+q x p xq

9.5.1.4 pq x pq ( x p)q

9.5.1.5 square root of x :

9.5.1.6 nth root of x :

9.5.1.7 sgn sgn x signum of x :

9.5.1.8 [ ] [ x] largest integer not exceeding x :

[ x] is an integer x − 1 < [ x] x

9.5.1.9 〈 〉 〈 x〉 nearest integer to x :

9.5.1.10 sum: x1 + x2 + x3 +···+ xn (n terms)

9.5.1.11 sum: x1 + x2 + x3 +···+ xn (n terms)

9.5.1.12 Π product: x1 x2 x3 ··· xn (n factors)

9.5.1.13 product: x1 x2 x3 ··· xn (n factors)

9.5.1.14 ! n! factorial: n! n (n−1)!, where 0! 1;

9.5.1.15 cycle delineator for a repeating decimal number;

>=

x x

12---

n xn x

1n---

x x----- for x 0 0 for x,≠ 0 qv 9.2.31( )=

∈ = 1 (qv 9.5.1.13)

0.083, 0.027112------ 0.083 333…, 1

37------ 0.027 027 027…≡≡


21/43



9.5.1 Algebraic Functions (continued)

9.5.2 Elementary Transcendental Functions

9.5.2.1 Exponential and Logarithmic Functions

ITEM

NUMBER

SIGN OR


9.5.1.16 ( ) 2(3)17 common difference indicator for an arithmetic sequence;e.g., 2, 5, 8, 11, 14, 17

9.5.1.17 P P(n, m) permutation of n things, taken m at a time:

9.5.1.18 C C(n, m) combination of n things, taken m at a time:

9.5.1.19 binomial coefficient; generating function:

9.5.1.20 continued fraction:

ITEM

NUMBER

SIGN OR

SYMBOL APPLICATION DEFINITION / MEANING

9.5.2.1.1 exp exp x e x (qv 9.1.2, 9.5.1.1)

9.5.2.1.2 log logb x logarithm (with base b > 1) of x > 0; ≡ x

9.5.2.1.3 lb lb x binary logarithm of x: log2 x

9.5.2.1.4 lg lg x common (Briggsian) logarithm of x: log10 x

9.5.2.1.5 ln ln x natural (Napierian) logarithm of x: loge x;

lnn x ln(lnn−1 x, where ln0 x x

P n m,( ) n!n m–( )!

-------------------- (qv 9.1.12, 9.5.1.14)≡

P n m,( ) P n m,( )m!

------------------n!

n m–( )! m!---------------------------- (qv 9.5.1.17)≡ ≡

n

m

1 x+( )nn

m xm;

n

m C n m,( )≡

m 0=

n

∑≡

a0

b0

a1 +-------------

b1

a2+---------…+

a0

b0

a1

b1

a2 .+-------------

..+

-----------------------------+

bloge x

∆=

∆=


22/43



9.5.2.2 Circular Functions

Functions in Subclause 9.5.2.2 relate Cartesian ( x, y) coordinates (qv 9.6.2.1) with polar (r , θ ) coordinates (qv 9.6.2.6)whose origins coincide. The argument, θ , is the angle measured around the common origin from the + x-axis (towardthe + y-axis for the positive sense) to the radius vector, r, whose magnitude is r > 0. The Cartesian coordinates of theterminus of r are the abscissa, x, and the ordinate, y.

ITEM

NUMBER

SIGN OR


9.5.2.2.1 sin sin θ sine of the angle, θ (ratio: y / r )

9.5.2.2.2 cos cos θ cosine of the angle, θ (ratio: x / r )

9.5.2.2.3 tan tan θ tangent of the angle, θ (ratio: y / x);

9.5.2.2.4 ctn ctn θ cotangent of the angle, θ (ratio: x / y);θ ≠ k π, k = 0, ±1, ±2, ±3, ···

9.5.2.2.5 sec sec θ secant of the angle, θ (ratio: r / x);

9.5.2.2.6 csc csc θ cosecant of the angle, θ (ratio: r / y);θ ≠ k π, k = 0, ±1, ±2, ±3, ···

9.5.2.2.7 arcsin arcsin q angle whose sine is q

9.5.2.2.8 Arcsin Arcsin q principal value of arcsin q;

9.5.2.2.9 arccos arccos q angle whose cosine is q

9.5.2.2.10 Arccos Arccos q principal value of arccos q;

9.5.2.2.11 arctan arctan q angle whose tangent is q

9.5.2.2.12 Arctan Arctan q principal value of arctan q;

9.5.2.2.13 arcctn arcctn q angle whose cotangent is q

θ k 12---+

π, k ≠ 0 1 2, 3,…±±,±,=

θ k 12---+

π, k ≠ 0 1 2, 3, …±±,±,=

<q

21=( )

restricted toπ2---,–

π2---+ (qv 9.9.4)


23/43



9.5.2.2 Circular Functions (continued)

9.5.2.3 Hyperbolic Functions

ITEM

NUMBER

SIGN OR


9.5.2.2.14 Arcctn Arcctn q principal value of arcctn q;restricted to (0, +π) (qv 9.9.1)

9.5.2.2.15 arcsec arcsec q angle whose secant is q

9.5.2.2.16 Arcsec Arcsec q principal value of arcsec q;

9.5.2.2.17 arccsc arccsc q angle whose cosecant is q

9.5.2.2.18 Arccsc Arccsc q principal value of arccsc q;

9.5.2.2.19 vers vers θ versed sine of θ : 1 − cos θ

9.5.2.2.20 covrs covrs θ coversed sine of θ or versed cosine of θ : 1 − sin θ

9.5.2.2.21 exsec exsec θ exsecant of θ : sec θ − 1

9.5.2.2.22 hav hav θ haversine of θ :

ITEM

NUMBER

SIGN OR


9.5.2.3.1 sinh sinh θ hyperbolic sine of θ :

9.5.2.3.2 cosh cosh θ hyperbolic cosine of θ :

9.5.2.3.3 tanh tanh θ hyperbolic tangent of θ :

9.5.2.3.4 ctnh ctnh θ hyperbolic cotangent of θ :

9.5.2.3.5 sech sech θ hyperbolic secant of θ :

9.5.2.3.6 csch csch θ hyperbolic cosecant of θ :

>q2 1=( )

restricted to π,– π2---– , 0,

π2---+ (qv 9.9.3)

>q

21=( )

restricted to π,– π2---– , 0,

π2---+ (qv 9.9.2)

12--- vers θ 1

2--- 1 θcos–( ) sin2 θ

2---

≡≡

12--- e

θe

θ––( )

12--- e

θe

θ–+( )

sinh θcosh θ---------------

cosh θsinh θ--------------- θ 0≠( )

1cosh θ---------------

1hsin θ---------------- θ 0≠( )


24/43



9.5.2.3 Hyperbolic Functions (continued)

9.5.2.4 Miscellaneous Elementary Transcendental Functions

ITEM

NUMBER

SIGN OR


9.5.2.3.7 arsinh arsinh x reverse hyperbolic sine of x:

9.5.2.3.8 arcosh arcosh x inverse hyperbolic cosine of x:

9.5.2.3.9 Arcosh Arcosh x principal value of arcosh x:

9.5.2.3.10 artanh artanh x inverse hyperbolic tangent of x:

9.5.2.3.11 arctnh arctnh x inverse hyperbolic cotangent of x:

9.5.2.3.12 arsech arsech x inverse hyperbolic secant of x:

9.5.2.3.13 Arsech Arsech x principal value of arsech x:

9.5.2.3.14 arcsch arcsch x inverse hyperbolic cosecant of x:

ITEM

NUMBER

SIGN OR


9.5.2.4.1 gd gd x Gudermannian of x:

9.5.2.4.2 λ λ(θ ) lambda function: gd–1θ ≡ ln (sec θ + tan θ )

ln x x2

1++( )

>ln x x

21–±( ) x 1=( )

>ln x x2

1–+( ) x 1=( )

12--- ln

1 x+1 x–------------ x

21( )


25/43



9.6 Geometry

9.6.1 Elementary Geometry (Euclidean)

ITEM

NUMBER

SIGN OR


9.6.1.1 line segment or chord (joining points A and B)

9.6.1.2 directed line segment (from point C to point D)

9.6.1.3 circular arc (joining points E and F )

9.6.1.4 || a || b parallel; line a is parallel to line b

9.6.1.5 ⊥ a ⊥ b perpendicular; line a is perpendicular to line b9.6.1.6 ∠ angle; angle A, formed by rays emanating from A

9.6.1.7 ∆ ∆ ABC triangle; triangle with vertices at A, B, and C

9.6.1.8 ABCD square; square with vertices at A, B, C , and D

9.6.1.9 A( B) circle; circle with center at A passing through B

9.6.1.10 ( ABC ) circle; circle passing through A, B, and C

9.6.1.11 ∼ ∆ ABC ∼ ∆ DEF similarity; ∆ ABC is similar to ∆ DEF

9.6.1.12 ≅ ∆ ABC ≅ ∆ DEF congruence; ∆ ABC is congruent to ∆ DEF

9.6.1.13 ∆ ABC ∆ DEF equiangularity; ∆ ABC and ∆ DEF are equiangular,

i.e.,

9.6.1.14 E E ∆ABC spherical excess (of ∆ ABC :

9.6.1.15 s semi-perimeter of a triangle (plane or spherical)

9.6.1.16 s ⊥s plural suffix; e.g., perpendiculars9.6.1.17 ∴ therefore; hence

9.6.1.18 because; since

9.6.1.19 QED quod erat demonstrandum: which was to he proved

9.6.1.20 QEF quod erat faciendum: which was to be constructed

AB

CD

)

EF )

A∠

^

– – ^

– – A∠ D, B E , C F ∠=∠∠=∠∠=

A B C 180° )–∠+∠+∠

∴


26/43



9.6.2 Analytic Geometry

ITEM

NUMBER

SIGN OR


9.6.2.1 x, y P( x, y) rectangular (Cartesian) coordinatesof a point (P) in a plane (qv 9.1.12)

9.6.2.2 h, k O′(h, k ) rectangular ( x, y) coordinates (respectively)of translated origin (O′) in a plane (qv 9.1.12)

9.6.2.3 x, y, z P( x, y, z) rectangular (Cartesian) coordinatesof a point (P) in space (qv 9.1.13 )

9.6.2.4 h, k , l O′(h, k , l) rectangular ( x, y, z) coordinates (respectively)of translated origin (O′) in space (qv 9.1.13)

9.6.2.5 a, b, c x, y, z intercepts; rectangular equation of a plane

9.6.2.6 α, β, γ direction angles (from x, y, z axes, respectively)

9.6.2.7 λ, µ, ν λ2 + µ2 + ν2 ≡ 1 direction cosines (of angles, α, β, γ , respectively)

9.6.2.8 r , θ P (r , θ ) polar coordinates of a point (P) in a plane;transformation to rectangular coordinates:

x r cos θ , y r sin θ (qv 9.1.12)

9.6.2.9 r , θ , z P(r , θ , z) cylindrical coordinates of a point (P) in space;transformation to rectangular coordinates:

x r cos θ , y r sin θ , z z (qv 9.1.13)

9.6.2.10 ρ, φ , θ P(ρ, φ, θ ) spherical coordinates of a point (P) in space;transformation to cylindrical coordinates: z ρ cos φ , r ρ sin φ , θ θ (qv 9.1.13)

9.6.2.11 m y = mx + b slope of a line in a plane; rectangular equation of a line

9.6.2.12 p y2 = 2 px semi-latus rectum; rectangular equation of a parabola

p is also used to designate the length of a normal(perpendicular) to its intersection with a line (or a plane)from the origin

9.6.2.13 e eccentricity of a conic section;polar equation of a conic section (focus at the origin)

x

a---

y

b---

z

c--+ + 1=

∆=

∆=

∆=

∆=

∆=

∆=

∆=

∆=

r p

1 e cos θ–--------------------------=


27/43



9.6.3 Differential Geometry

ITEM

NUMBER

SIGN OR


9.6.3.1 s arc length; independent intrinsic coordinate

9.6.3.2 κ κ (s) curvature; dependent intrinsic coordinate9.6.3.3 τ τ(s) torsion; dependent intrinsic coordinate

9.6.3.4 t unit tangent vector to a curve

9.6.3.5 n unit normal vector to a curve

9.6.3.6 b unit binormal vector to a curve:b t × n (qq v 9.6.3.4, 9.6.3.5, 9.7.5)

9.6.3.7 u, v, w curvilinear coordinates

9.6.3.8 N unit normal vector to a surface

9.6.3.9 E , 2F , G coefficients of the first fundamental form

9.6.3.10 e, 2 f , g coefficients of the second fundamental form9.6.3.11 κ 1, κ 2 principal curvatures of a surface:

κ 1 ≡ e / E , κ 2 ≡ g / G

9.6.3.12 M mean curvature of a surface:

9.6.3.13 K total (Gaussian) curvature of a surface: κ 1 κ 2

9.6.3.14 κ g surface geodesic (tangential) curvature

9.6.3.15 kg geodesic curvature vector: kg κ g (N × t)(component of k in plane tangent to surface)

9.6.3.16 κ n surface normal curvature

9.6.3.17 kn normal curvature vector: kn κ nN(component of k along normal to surface)

9.6.3.18 k curvature vector: k κ n ≡ kg + kn

∆=

12--- κ 1 κ 2+( )

∆=

∆=

∆=


28/43



9.7 Vectors

ITEM

NUMBER

SIGN OR


9.7.1 i, j, k ai + b j + ck i, j, k symbolize unit vectors parallel to, and directedalong, the positive x, y, z Cartesian axes, respectively;

ai + b j + ck represents the vector directed from the originof x, y, z coordinates to the diagonally opposite corner ofa rectangular parallelepiped with edges a, b, c,respectively, along x, y, z axes.

9.7.2 A vector A A x i + A y j + A z k

9.7.3 | | | A| magnitude of A :

9.7.4 · A · B scalar (dot) product of A and B: A x B x + A y B y + A z B z

9.7.5 × A × B vector (cross) product of A and B:( A y B z − Az B y) i + ( A z B x − A x B z) j + ( A x B y − A y B x) k

9.7.6 [ ] [ abc] triple scalar product: a · ( b × c) ≡ ( a × b) · c9.7.7 ∇ nabla or del :

9.7.8 grad grad f gradient of f :

9.7.9 div div F divergence of F : ∇· F ∇ · (F x i + F y j + F z k)

9.7.10 rot rot F curl of F : ∇ × F ∇ × (F x i + F y j + F z k)

9.7.11 ∇2 ∇2 f Laplace operator:

∆=

∆ A A x

2 A y2 A z

2+ + A A•≡=

i ∂∂ x----- j

∂∂ y----- k

∂∂ z-----+ + (qv 9.9.27)

∆∇ f = ∂ f ∂ x

-----i ∂ f

∂ y----- j

∂ f ∂ z-----k+ +

∆=

∂F x∂ x

---------≡ ∂F y

∂ y---------

∂F z∂ z

---------+ +

∆=

∂F z∂ y

--------- ∂F z

∂ z---------–

≡ i

+∂F x∂ z

--------- ∂F z

∂ x---------–

j ∂F y

∂ x---------

∂F x∂ y

---------– k+

∂2

∂ x2--------

∂2

∂ y2--------

∂2

∂2----- ∇ ∇ ;⋅≡+ +

∇2 f ∂2 f

∂ x2--------

∂2 f ∂ y2--------

∂2 f ∂ z2--------+ +≡


29/43



9.8 Matrices

ITEM

NUMBER

SIGN OR


9.8.1 matrix, A, delineated by its elements, aij, forming arectangular array (of numbers) having m rows and n

columns

9.8.2 [ ] [aij]mn matrix, A, having elements aij:i = 1, 2, 3,···, m (row indices) ,

j = 1, 2, 3,···, n (column indices)

9.8.3 O A + O ≡ A zero matrix : ∀(i, j): zij 0 in [ zij]mn

9.8.4 I AI ≡ I A ≡ A identity matrix: uii 1, uij 0 for i ≠ j, in [uij]nn(NB uij ≡ , qv 9.1.10)

9.8.5 ∼ Ã transpose of A : [a ji]nm, where A [aij]mn

9.8.6 conjugate of A : [ ij]mn, where A [aij]mn(qv 9.10.4)

9.8.7 | | | A| absolute value of A : [|aij|]mn , where A [aij]mn9.8.8 tr tr A trace of A : , where A [aij]nn

9.8.9 det det A determinant of A, where A [aij]nn

9.8.10 cof cof aij cofactor of aij (in det A) :(–1)i+ j det Aij, where Aij n

−1, n

−1,

obtained by deleting the ith row

and jth column from A [akl]nn

9.8.11 adj adj A adjoint of A : ,

where A [akl]nn, and cij cof aij in det A

9.8.12 –1 A–1 inverse of A :

a11

a21

a31

.

.

.am1

a12

a22

a32

.

.

. am2

………. .

.

…

a11

a21

a31

.

.

.am1

∆=

∆= ∆= δ j

i

∆=

A a ∆=

∆=

Σi 1=n aii∆=

∆=

∆=

ak i l j

[ ]

∆=

∆C ˜ c ji[ ]nn=

∆=

∆=

adj Adet A------------; AA

1– A

1– A≡ I≡ qv 9.8.4( )


30/43



9.9 Real Variables (Calculus)

ITEM

NUMBER

SIGN OR


9.9.1 ( , ) (a, b) open interval, a, b (of x: a


31/43



9.9 Real Variables (continued)

ITEM

NUMBER

SIGN OR


9.9.17 ″ f ″(a) 2nd derivative of f ( x) evaluated for x = a:derivative of f ′( x) evaluated for x = a

9.9.18 (n) f (n)(a) nth derivative of f ( x) evaluated for x = a

9.9.19 ∆ ∆ x increment of x; finite difference of x

9.9.20 derivative of y with respect to x :

9.9.21 nth derivative of y with respect to x :

9.9.22 · derivative of x with respect to t (time);d x /dt ; speed of x

9.9.23 ·· 2nd derivative of x with respect to t (time);

d2

x /dt 2

; acceleration of x9.9.24 (n) (n) nth derivative of x with respect to t (time);

dn x /dt n

9.9.25 ( , ,··· ) f x(a, b, ···) partial derivative of f ( x, y ···) with respect to xevaluated for x = a, y = b, ···

9.9.26 ( , ,··· ) f xy(a, b, ···) partial derivative of f x( x, y,···) with respect to yevaluated for x = a, y = b,···

9.9.27 partial derivative of u( x, y,···) with respect to x :

9.9.28 partial derivative of with respect to y

9.9.29 nth partial derivative of u( x, y,···) with respect to x

9.9.30 d du total differential of u( x,y,··· ):

9.9.31 det [ ] Jacobian of ui( x1, x2, x3, ···, xn ); i = 1, 2, 3,···, n(qq v 9.8.9, 9.9.27)

9.9.32 { } { xi} finite or infinite sequence of numbers, x1, x2, x3, ···

dd---

d yd x-------

∆ y∆ x------

∆ x 0→lim

dn

d n-------

dn y

d xn--------

dd x------

dd x------

dd x------

… d

d x------

d yd x------

…

x·

x··

x

∂∂--

∂u∂ x------ u x ∆ x y …, ,+( ) u x y …, ,( )–

∆ x----------------------------------------------------------------------∆ x 0→lim

∂2

∂ ∂--------- ∂2u∂ y∂ x------------ ∂

u

∂ x------

∂n

∂ n-------- ∂nu

∂ xn--------

u∂ x∂

-----d xu∂ y∂

-----d y …+ +

detu∂ i x∂ j

-------nn


32/43



9.9.33 inf inf{ xi} infimum (greatest lower bound) of sequence { xi}

9.9.34 sup sup{ xi} supremum (least upper bound) of sequence { xi}

9.9.35 indefinite integral; antiderivative of f ( x):

9.9.36 definite integral of f ( x):area defined by a x b and 0 y f ( x)

9.9.37 improper integral:

9.9.38 Cauchy principal value of

where a = –∞ and b = +∞; or for a < c < b:

9.9.39 F (b) − F (a); F ( x) is usually an antiderivative9.9.40 * f * g convolution of f and g:

9.9.41 infinite sum:

9.9.42 infinite product:

ITEM

NUMBER

SIGN OR


∫ f x( )d x

∫ dd x------ f x( )d x∫ f x( )≡

∫ f x( )d xa

b

∫ <=

<=

<=

<=

∞

∫ f x( )d xa

∞

∫ f x( )d xa

h

∫ h ∞→lim

f x( )d x ,a

b

∫

f x( ) x c –→

lim ∞ and f x( ) x c+→

lim ∞+−=±=

F x( )a

b

f y( )g x y–( )d y∞–

+∞∫ g* f ≡

∑ uii 1=

∑ uii 1=

∑n ∞→lim qv 9.5.1.11( )

∏ uii 1=

∏

ui

i 1=

∞

∏n ∞→lim qv 9.5.1.13( )


33/43



9.10 Complex Variables

ITEM

NUMBER

SIGN OR


9.10.1 i x + i y imaginary unit: ; j is used in electrical engineering

9.10.2 Re Re z real part of z :

9.10.3 Im Im z imaginary part of z:

9.10.4 conjugate of z :

9.10.5 cis cisθ cos θ + i sin θ ≡ eiθ

9.10.6 | | | z| absolute value of z :

9.10.7 mod mod z modulus of z : (r 0); | z| (cf 9.10.6)

9.10.8 arg arg z argument of z :

9.10.9 Arg Arg z principal value of arg z; restricted to (−π, +π]

9.10.10 exp exp z e z ≡ e x cis y, where (cf 9.5.2.1.1)

9.10.11 ln ln z natural logarithm of z : ln |z| + i arg z (qv 9.5.2.1.5)

9.10.12 Ln Ln z principal value of ln z: ln

|z

| + i Arg z (cf 9.10.11)

9.10.13 sgn sgn z signum of z :

9.10.14 ω k throot of the n roots of unity:cis(2k π / n), k = 0, 1, 2, …, n − 1

9.10.15 Res residue of f ( z) at z = a : coefficient of ( z − a)−1 in theLaurent series for f ( z) expanded about z = a

9.10.16 line integral of f ( z) along curve, C , in z-plane

9.10.17 line integral of f ( z) along closed path, C , in z-plane

1–

12--- z z+( ); x in z ∆= x i y+

12i----- z z–( ); y in z ∆= x i y+

z

Re z iIm z;– x i y– in z ∆= x i y+

zz ; x2 y2+ in z ∆= x i y+

r in z ∆= r eiθ >

=

θ in z ∆= r eiθ r 0>( ); arctanIm z

Re z---------

z ∆= x i y+

z

z----- cis arg z( ) for z 0, 0 for z≠≡ 0=

ωnk

Res z=a

f z( )

∫ f z( )d zC

∫

∫ ° f z( )d zC

∫ °


34/43



9.11 Special Functions

9.11.1 Orthogonal Polynomials

9.11.2 Hypergeometric Functions

ITEM

NUMBER

SIGN OR


9.11.1.1 T Tn( x) Chebychev polynomial (1st kind): solution of (1 − x2) f ″( x) − x f ′( x) + n2 f ( x) = 0

9.11.1.2 U Un( x) Chebychev polynomial (2nd kind): solution of

(1 − x2) f ″( x) − 3 x f ′( x) + n (n + 2) f ( x) = 0

9.11.1.3 H Hn( x) Hermite polynomial: solution of f ″( x) − 2 x f ′( x) + 2n f ( x) = 0

9.11.1.4 P Jacobi polynomial: solution of (1 − x2) f ″( x) − {α − β + (α + β + 2) x} f ′( x)+ n (n + α + β + 1) f ( x) = 0 (α, β > −1);

9.11.1.5 L Laguerre polynomial (associated): solution of x f ″( x) + (α + 1 − x) f ′( x) + n f ( x) = 0

m = 0, 1, 2, ···, n; Ln( x) ( x)

9.11.1.6 P Pn( x) Legendre polynomial: solution of (1 − x2) f ″( x) − 2 x f ′( x) + n (n + 1) f ( x) = 0

ITEM

NUMBER

SIGN OR


9.11.2.1 F F(a, b; c; z) hypergeometric function: solution of z (1 − z) f ″( z)

+ {c − (a + b + 1) z} f ′( z) − ab f ( z) = 0

9.11.2.2 M M(a, b, z) confluent hypergeometric function: solution of z f ″( z) + (b − z) f ′( z) − a f ( z) = 0

Pnα β,( )

x( )

Pn

12---–

12---–,

x( ) Tn x( ), Pn0 0,( )

x( ) Pn x( ) ,≡ ≡

Pn

12---

12---,

x( ) Ug x( ) ≡

Lnα( )

x( )

α 1–>( ); Lnm( ) x( ) 1–( )mdm

d xm---------Ln m+ x( )≡

∆= Ln

0( )


35/43



9.11.3 Bessel Functions

ITEM

NUMBER

SIGN OR


9.11.3.1 J Jn( z) Bessel function of the 1st kind of order n :

solution of

z2 f ″( z) + z f ′( z) + ( z2 − n2) f ( z) = 09.11.3.2 Y Yn( z) Bessel function of the 2

nd kind of order n:

9.11.3.3 I In( z) modified Bessel function of the 1st kind of order n:

solution of z2 f ″( z) + z f ′( z) − ( z2 + n2 f ( z) = 0

9.11.3.4 K Kn( z) modified Bessel function of the 2nd kind of order n:

9.11.3.5 H Bessel function of the 3rd kind of order n [often called Hankel function]:Jn( z) + i

2v − 1Yn( z); defined for v = 1, 2

9.11.3.6 ber bern x Kelvin function of the 1st kind of order n (real):

9.11.3.7 bei bein x Kelvin function of the 1st kind of order n (imaginary):

9.11.3.8 ker kern x Kelvin function of the 2nd kind of order n (real):

9.11.3.9 kei kein x Kelvin function of the 2nd kind of order n (imaginary):

9.11.3.10 her hern x Kelvin function of the 3rd kind of order n (real):

9.11.3.11 hei hein x Kelvin function of the 3rd kind of order n (imaginary):

Jn ε+ z( ) cos n ε+( )π J n ε+( )– z( )–n ε+( )sin π

----------------------------------------------------------------------------------ε 0→lim

π2---

I n ε+( )– z( ) In ε+ z( )–n ε+( )sin π--------------------------------------------------ε 0→

lim

Hnv( ) z( )

Re Jn i

32---

x

for real n x, 0> qv 9.10.2( )

Im Jn i

32---

x

for real n x, 0> qv 9.10.3( )

Re i n– Kn i

12---

x

for real n x, 0> qv 9.10.2( )

Im i n– Kn i

12---

x

for real n x, 0> qv 9.10.3( )

Re Hn1( ) i

3

2

---

x

for real n x, 0> qv 9.10.2( )

Im Hn1( ) i

32---

x

for real n x, 0> qv 9.10.3( )


36/43


37/43



9.11.4.2 Jacobi Elliptic Functions (qv 9.11.4.1.1)

9.11.4.3 Gamma Functions

ITEM

NUMBER

SIGN OR


9.11.4.2.1 am am u amplitude of u: φ in u F(φ , k )

9.11.4.2.2 sn sn u sin φ in u F(φ , k )9.11.4.2.3 cn cn u cos φ in u F(φ , k )

9.11.4.2.4 dn dn u in u F(φ , k )

9.11.4.2.5 pq pq u p, q s, c, d ; p ≠ q

9.11.4.2.6 np np u p s, c, d

ITEM

NUMBER

SIGN OR


9.11.4.3.1 Γ Γ( x) gamma function (complete):

9.11.4.3.2 γ γ ( x, u) incomplete gamma function:

9.11.4.3.3 Γ Γ( x, u) incomplete gamma function: Γ( x) − γ ( x, u)

9.11.4.3.4 ψ ψ ( x) psi (digamma) function:

9.11.4.3.5 ψ ψ (n)( x) polygamma function:

∆=

∆= ∆=

1 k 2sin2φ– ∆= pn uqn u---------- ∆=

1pn u----------- ∆=

t x 1–

et –dt

0

∞

∫ x 0>( )

t x 1–

et –dt

0

u

∫ x 0>( )

t x 1–

et –dt

u

∞

∫ x 0>( )≡

Γ′ x( )Γ x( )-------------

d[ln Γ x( ) ]d x

-------------------------≡

e

t –

t ------

e xt –

1 et –

–---------------–

dt 0

∞

∫ x 0>( )≡

dnψ x( )

d xn

-----------------d

n 1+ln Γ x( )[ ]

d xn 1+

------------------------------------≡


38/43


39/43



9.11.5 Miscellaneous Special Functions

ITEM

NUMBER

SIGN OR


9.11.5.1 δ δ( x) Dirac delta function:

9.11.5.2 B Bn( x) Bernoulli polynomial of degree n; generating function:

nth Bernoulli number (n = 1, 2, 3, ··· ):

9.11.5.3 E En( x) Euler polynomial of degree n; generating function:

nth

Euler number (n = 0, 1, 2, ··· ):

9.11.5.4 ϑ ϑ n ( z, q) Jacobi theta functions: n = 1, 2, 3, 4; q nome:

9.11.5.5 p p( z) Weierstrass elliptic function ({ } is not recommended):

where mn 2mw1 + 2nw2

9.11.5.6 ζ ζ( z) Riemann zeta function:

9.11.5.7 Π Π(n, z) Gauss pi function:

f x( ) δ x a–( ) d x -∞

+∞

∫ ∆= f a( )

t e xt

et

1–------------- Bn x( )

t n

n!-----

n 0=

∑≡ t 2π


40/43



9.12 Probability and Statistics

ITEM

NUMBER

SIGN OR


9.12.1 E E ( x) expected value of x

9.12.2 P P( A) unconditional probability that an event, A, occur9.12.3 P P( B|A) conditional probability that an event, B, occur,

given that A has occurred

9.12.4 Q Q(A) unconditional probability that an event, A, not occur; Q( A) ≡ 1 − P( A)

9.12.5 pdf probability density function

9.12.6 cdf cumulative distribution function

9.12.7 irv independent random variable

9.12.8 d.f. degrees of freedom

9.12.9 ^ unbiased estimate of the value of

a statistical parameter, θ , from a population

9.12.10 ϕ ϕ( x) normal (Gaussian) pdf;with mean, µ, and standard deviation, σ:

9.12.11 Φ Φ( x) normal (Gaussian) cdf; standard form,i.e., zero mean, unit standard deviation:

9.12.12 χ2 χ2( x) chi-square pdf:

where n number of degrees of freedom

9.12.13 µ µm mth moment about the mean:

where n number of samples of x

9.12.14 α αm standardized µm: ;

α1 = 0, α2 = 1, α3 skewness, α4 kurtosis9.12.15 − sample mean value of x: ,


9.12.16 ~ population mean value of x: E ( )

θ̂

1

σ 2π--------------e

x µ–( )2 /2σ2–

∆ϕ t ( )dt

∞–

x

∫ 12π

---------- et 2 /2–

dt ∞–

x

∫ =

xn /2 1–

e x /2–

2n /2Γ n /2( )

---------------------------- x 0>( ) qv 9.11.4.3.1( )

∆=

1n--- xi x–( )

m,

i 1=

n

∑∆=

µmµ2m

-----------

∆= ∆= x

1n--- xi,

i 1=

n

∑∆=

x̃ x


41/43



10. Greek Characters

Because Greek letters traditionally have been used extensively in mathematical works, it is appropriate to provide aguide for their use in mathematical writing. With the advent of electronic computers and their associated software,publications that contain mathematical symbols no longer need to be typeset manually or even with automaticmechanical processes. Computer software for composing mathematical text, including Greek characters, is widelyavailable.20 Most all computer generated Greek letter fonts contain, at least, both lowercase and uppercase letters thatdiffer from roman fonts. Such a font, in 10 point type, is displayed in the table below. Those letter symbols enclosedin brackets were produced from a roman font.

9.12.17 s2 sample variance of x:


9.12.18 σ2 population variance of x:


9.12.19 s s x sample standard deviation of x:

9.12.20 σ σ x population standard deviation of x:


9.12.21 S S y standard error of estimate of y:

where n number of x− y sample pairs

9.12.22 r 2 r 2( y) coefficient of determination:

9.12.23 r r xy correlation coefficient of x with y:

where n number of x− y sample pairs

20This Standard was composed using a document preparation system, freely available, known as LATEX, qv [B8], Clause 11.

ITEM

NUMBER

SIGN OR


s x2 1

n--- xi x–( )

2 µ2 ,≡

i 1=

∑∆=

σ x2 nn 1–------------ E s x

2( ),

∆=

s x2

σ x2

n

2---

Γ n 1/2–( ) Γ n /2( )

--------------------------- E s x( ) qv 9.11.4.4.3.1( ),≡

∆=

1n--- y xi( ) yi–{ }

2

i 1=

n

∑

2---

,

∆=

1 S y2 / s y

2–

1n---

xi x–

s x------------

yi y–

s y------------

,

i 1=

n

∑∆=


42/43



LETTER

NAME

LETTER SYMBOL

UPPERCASE LOWERCASE VARIANT

alpha [ A ] α

beta [ B ] β

gamma Γ γ

delta ∆ δ

epsilon [ E ] ∈ ε

zeta [ Z ] ζ

eta [ H ] η

theta Θ θ ϑ

iota [ I ] ι

kappa [ K ] κ κ lambda Λ λ

mu [ M ] µ

nu [ N ] ν

xi Ξ ξ

omicron [ O ] [ o ]

pi Π π ϖ

rho [ P ] ρ

sigma Σ σ ς

tau [ T ] τ

upsilon ϒ υ

phi Φ φ ϕ

chi [ X ] χ

psi Ψ ψ

omega Ω ω

ρ ∼


43/43


11. Bibliography

[B1] Adams, Edwin P., Ph.D. Smithsonian Mathematical Formulae and Tables of Elliptic Functions. Washington:Smithsonian Institution, 1957

[B2] Burington, Richard Stevens, Ph.D. Handbook of Mathematical Tables and Formulas, 5th ed. New York;McGraw-Hill Book Company, 1973

[B3] Dwight, Herbert Bristol, D.Sc. Tables of Integrals and Other Mathematical Data, 4th ed. New York: TheMacmillan Company, 1961

[B4] Gröbner, Wolfgang, and Hofreiter, Nikolaus. Integraltafel Zweiter Teil Bestimmte Integrale. Vienna: Springer-Verlag, 1973

[B5] Jahnke, Dr. Eugene, and Emde, Fritz. Table