IEEE 260 3-1993 en American National Standard Mathematical Signs and Symbols for Use in Physical Sciences and Technology《物理科学和技术用数学符号》.pdf

1、1ANSI/IEEE Std 260.3-1993 (R2012)(Revision of ANSI Y10.20-1975)(Supersedes ANSI Y10.17-1961 (R 1988)American National Standard Mathematical Signs and Symbols for Use in Physical Sciences and TechnologySponsorIEEE Standards Coordinating Committee 14 onQuantities, Units, and Letter SymbolsApproved Mar

2、ch 18, 1993Reaffirmed August 30, 2012IEEE Standards BoardApproved August 30, 1993American National Standards InstituteAbstract: Signs and symbols used in writing mathematical text are defined. Special symbols peculiar to certainbranches of mathematics, such as non-Euclidean Geometries, Abstract Alge

3、bras, Topology, and Mathematics ofFinance, which are not ordinarily applied to the physical sciences and engineering, are omitted.Keywords: letter symbol, mathematical notation, mathematical sign, mathematical symbol, mathematics, operationsymbol, quantity symbol, unit symbolThe Institute of Electri

4、cal 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 1-55937-318-0No part of this publication may be reproduce

5、d in any form, in an electronic retrieval system or otherwise, without theprior written permission of the publisher.IEEE Standards documents are developed within the Technical Committees of the IEEE Societies and the StandardsCoordinating Committees of the IEEE Standards Board. Members of the commit

6、tees serve voluntarily and withoutcompensation. They are not necessarily members of the Institute. The standards developed within IEEE represent aconsensus of the broad expertise on the subject within the Institute as well as those activities outside of IEEE that haveexpressed an interest in partici

7、pating in the development of the standard.Use of an IEEE Standard is wholly voluntary. The existence of an IEEE Standard does not imply that there are no otherways to produce, test, measure, purchase, market, or provide other goods and services related to the scope of the IEEEStandard. Furthermore,

8、the viewpoint expressed at the time a standard is approved and issued is subject to changebrought about through developments in the state of the art and comments received from users of the standard. EveryIEEE Standard is subjected to review at least every five years for revision or reaffirmation. Wh

9、en a document is morethan five years old and has not been reaffirmed, it is reasonable to conclude that its contents, although still of somevalue, do not wholly reflect the present state of the art. Users are cautioned to check to determine that they have thelatest edition of any IEEE Standard.Comme

10、nts for revision of IEEE Standards are welcome from any interested party, regardless of membership affiliationwith IEEE. Suggestions for changes in documents should be in the form of a proposed change of text, together withappropriate supporting comments.Interpretations: Occasionally questions may a

11、rise regarding the meaning of portions of the standards as they relate tospecific applications. When the need for interpretations is brought to the attention of IEEE, the Institute will initiateaction to prepare appropriate responses. Since IEEE Standards represent a consensus of all concerned inter

12、ests, it isimportant to ensure that any interpretation has also received the concurrence of a balance of interests. For this reasonIEEE and the members of its technical committees are not able to provide an instant response to interpretation requestsexcept in those cases where the matter has previou

13、sly received formal consideration.Comments on standards and requests for interpretations should be addressed to:Secretary, IEEE Standards Board445 Hoes LaneP.O. Box 1331Piscataway, NJ 08855-1331USAIEEE Standards documents are adopted by the Institute of Electrical and Electronics Engineers without r

14、egard towhether their adoption may involve patents on articles, materials, or processes. Such adoption does not assumeany liability to any patent owner, nor does it assume any obligation whatever to parties adopting the standardsdocuments.iiiIntroduction(This Introduction is not part of ANSI/IEEE St

15、d 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 i

16、s 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

17、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 t

18、hat 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 Sp

19、ecial 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 Usedas Letter Symbols for Engineering Mathematics. Clause 10 of t

20、his Standard supersedes ANSI Y10.17-1961 entirely.The technical support provided by Delco Systems Operations of Delco Electronics Corporation, in the preparation ofthis revision, is gratefully acknowledged.At the time this Standard was completed, the membership of the Standards Coordinating Committe

21、e 14, Quantities,Units, and Letter Symbols, consisted of:Bruce B. Barrow, Chair Andrew F. DunnStanley L. EhrlichRobert V. EspertiJohn A. GoetzTruman S. GrayM. Harry HesseRon K. JurgenWilliam R. KruesiJack M. LoudonArthur O. McCoubreyConrad R. MullerChester H. PageRalph M. ShowersBarry N. TaylorAlan

22、S. WhelihanAt 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. NielsenMelvin D. SpringerThe following persons were on the balloting committee:Bruce B. BarrowAndrew F.

23、DunnStanley L. EhrlichRobert V. EspertiJohn A. GoetzTruman S. GrayM. Harry HesseRon K. JurgenWilliam R. KruesiJack M. LoudonArthur O. McCoubreyConrad R. MullerChester H. PageRalph M. ShowersBarry N. TaylorAlan S. WhelihanivWhen the IEEE Standards Board approved this Standard on March 18, 1993, it ha

24、d 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 IsaakBen C. JohnsonWalter J. KarplusLorraine C. KevraE.G. “Al” KienerIvor N. Kni

25、ghtJoseph L. Koepfinger*D. N. “Jim” LogothetisDon T. Michael*Marco W. MigliaroL. John RankineArthur K. ReillyRonald H. ReimerGary S. RobinsonLeonard L. TrippDonald W. Zipse*Member EmeritusAlso included are the following nonvoting IEEE Standards Board liaisons:Satish K. AggarwalJames BeallRichard B.

26、EngelmanDavid E. SoffrinStanley WarshawRachel A. MeiselIEEE Standards Project EditorThis Standard has been coordinated with the following organizations:American Mathematical SocietyAmerican Institute of PhysicsNational Institute of Standards and TechnologyvCLAUSE PAGE1. Scope.12. Letter Symbols 13.

27、Alphabets and Typography .24. Quantity Symbols35. Unit Symbols.36. Operation Symbols57. Reference Documents .57.1 References 68. Conventions 79. Signs and Symbols 79.1 Miscellaneous Signs and Symbols. 89.2 General Operations 99.3 Symbolic Logic and Set Theory. 119.4 Arithmetic (Number Theory). 129.5

28、 Elementary Functions 149.6 Geometry 199.7 Vectors . 229.8 Matrices 239.9 Real Variables (Calculus) 249.10 Complex Variables 279.11 Special Functions 289.12 Probability and Statistics. 3410. Greek Characters.3511. Bibliography37vi Copyright 1998 IEEE All Rights ReservedDesignation (Variable) HeaderT

29、itleLeft (Variable)Copyright 1993 IEEE All Rights Reserved 1American National Standard Mathematical Signs and Symbols for Use in Physical Sciences and Technology1. ScopeOnly signs and symbols used in writing mathematical text are contained in this Standard. Special symbols peculiar tocertain branche

30、s 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

31、subject of tensors is relegatedto future editions when there is general agreement among authorities in the field.2. Letter SymbolsLetter symbols1include symbols for physical quantities (quantity symbols), symbols for units in which these quantitiesare measured (unit symbols), and symbols for operato

32、rs 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,2e.g., I, to represent an electric current, modified, when appropriate, byone or more subscripts or superscripts, e.g., Ii,

33、 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 lett

34、ers (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, som

35、e 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

36、 “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 transp

37、ort 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). 2 Copyright 1993 IEEE All Rights ReservedANSI/IEEE Std 260.3-1993 AMERICAN NATIONAL STANDARDA unit symbol3is a letter or group

38、 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 relat

39、ionship, a word, or a phrase.3. Alphabets and TypographyLetter symbols are restricted primarily to the English and Greek alphabets.4Script, black letter, or other special fontsto distinguish between possible conflicting uses of the same letter for different quantities should not be used.Symbols for

40、physical quantities, mathematical variables, indices, and general functions5are printed in italic type, e.g.,A areae eccentricity of a conic sectionx, y, z Cartesian coordinatesi, j, k indicesf(x) f function of xSymbols used for physical units, as well as mathematical constants, specific mathematica

41、l functions, operators, and allnumerals are printed in roman (upright) type, e.g.,cm centimeter(s)i imaginary unit: sin 2 sine of the angle: 2 a b + c a times b plus cdx differential of xAll punctuation,6including grouping symbols, such as parentheses, brackets, and braces, are also printed in roman

42、type, e.g.,F(a, b; c; z) hypergeometric functionx integer function of xabc triple scalar product of vectorsa, bLCM of a and bn! factorial nSubscripts and superscripts are governed by the above principles. Those that are letter symbols for physical quantities,mathematical variables, or for indices ar

43、e printed in italic type, whereas others are printed in roman type, e.g.,sinpxpthpower of sin xaij, a45matrix elementsIi, Ioinput, output currentsBx(, ) incomplete beta function3It was once common to treat unit symbols in the same manner as general abbreviations, but the recommendations of the Inter

44、national Organizationfor Standardization (ISO) and many other international and national bodies concerned with standardization, emphasize the symbolic character ofthese designations and rigidly prescribe the manner in which they shall be treated. The concept of the unit symbol is therefore adopted i

45、n 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 b

46、e 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.1Copyright 1993 IEEE All Rights Reserved 3MATHEMATICAL SIGNS AND SYMBOLS ANSI/IEEE Std 260.3-1993r2a

47、rea enclosed by a circleTo 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.,Fiforce on the ithelementgrad f gradient of the scalar, fdivF divergence of the vector, Fi, j, k orthogonal

48、unit vectorsknnormal curvature vectorThe 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 SymbolsQuantity symbols may be u

49、sed in mathematical expressions in any way consistent with good mathematical usage. Theproduct 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 lette