Universität Karlsruhe (TH) Studienzentrum für Sehgeschädigte _________________________________________________________________________* *___________________ Sechste Auflage ~ August 2000 _________________________________________________________________________* *___________________ Universität Karlsruhe (TH) Studienzentrum für Sehgeschädigte Engesserstr. 4 76128 Karlsruhe Tel.: (0721) 608-2760 Fax.: (0721) 697377 E-Mail: szs@ira.uka.de Technische Universität Dresden Arbeitsgruppe Studium für Blinde und Sehbehinderte 01062 Dresden Tel.: (0351) 4575-467 Fax.: (0351) 4575-335 E-Mail: elvis@irz.inf.tu-dresden.de Introduction For performance of mathematical formulas in scientific texts you apply the followed described ASCII-mathematic-writing for blind persons (AMS). This notation was developed by the university of Karlsruhe in the experiment com- puterscience for blind and extremly visually impaired persons and was re- touched by the technical university of Dresden. At both universities study materials for visually impaired persons will create by student assistants (Tu- toren). Visually-impaired-persons-like means at this the creation of texts and graphics, which are readable for blind, visually impaired and perceive persons. The performance of mathematical symbols at the computer ensues as a rule graphicly. These performances aren t readable for blind and extremly visually impaired persons till now. In order to make mathematical symbols approacha- ble, they will convert it into a character presentation (AMS). The character set at this is limited of the ASCII-font, which is available on computer. With that the created texts are system-independent and could read and altered of every simple editor. With AMS the destination will persued to introduce beside the syntax-orientated notations for mathematical expressions also such ones, which emphazises the semantik of expressions better. Moreover AMS allows to aplly for expressions, which are often be found, short notations. Both support the improvement of the readability. The present collection of mathematical symbols and expressions should not only be an introduction of rendering order. It s thought as an instruction and referencebook at the practical rendering work. Therefore expressions would also received, which proper wouldn t have to be explained, because they immediate result from introduced notations. Mathematical texts live through own symbols of the authors. Should these symbols convinced into AMS, their definitions might be necessary. Extensions to AMS must find agreement with the responsable employees of the universities. INHALTSVERZEICHNIS 3 Inhaltsverzeichnis 1 Basic notation * * 5 1.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. 5 1.2 ASCII-font . . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. 5 1.2.1 ASCII-symbols with special meaning in AMS . . . . . . 6 1.3 In general guidelines of convertion . . . . . . . . . . . . . . . .* * 7 1.3.1 Structuring of AMS-expressions . . . . . . . . . . . . . . * * 7 1.3.2 Convertion of mathematical symbols, which aren t con- tained in AMS . . . . . . . . . . . . . . . . . . . . . . .* * 8 1.4 Strange alphabets . . . . . . . . . . . . . . . . . . . . . . . . .* * 9 1.4.1 Greek alphabet . . . . . . . . . . . . . . . . . . . . . . .* * 9 1.4.2 French alphabet . . . . . . . . . . . . . . . . . . . . . .* * 10 1.5 Special print-stylisation . . . . . . . . . . . . . . . . . . . . .* * . 10 1.6 Marking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. 10 1.7 Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. . 12 1.7.1 Postfixed indices . . . . . . . . . . . . . . . . . . . . . * *. 12 1.7.2 Prefixed indices . . . . . . . . . . . . . . . . . . . . . .* * . 14 1.8 Superpositioned signs . . . . . . . . . . . . . . . . . . . . . . .* * 14 1.9 Multiple-lined symbols . . . . . . . . . . . . . . . . . . . . . . .* * 14 1.9.1 Fractions and binominalcoefficiens . . . . . . . . . . . . * * 14 1.9.2 Big vertical curly brackets . . . . . . . . . . . . . . . . * *. 14 1.9.3 Pair of big vertical brackets or lines . . . . . . . . . . * *. 15 2 Mathematical symbols 17 2.1 Symbols, numbers, operation- and relationsigns . . . . . . . . . 17 2.1.1 Mathematical symbols . . . . . . . . . . . . . . . . . . . * * 17 INHALTSVERZEICHNIS 4 2.1.2 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . .* * 18 2.1.3 Operator signs . . . . . . . . . . . . . . . . . . . . . . * *. 19 2.1.4 Relation signs . . . . . . . . . . . . . . . . . . . . . . . * *. 20 2.2 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. 21 2.3 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. . 21 2.4 Vectors, matrices and determinants . . . . . . . . . . . . . . . . * * 22 2.5 Sequences and functions . . . . . . . . . . . . . . . . . . . . . . * * 24 2.6 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. . 25 2.6.1 Total derivative . . . . . . . . . . . . . . . . . . . . . .* * . 25 2.6.2 Partial derivative . . . . . . . . . . . . . . . . . . . . .* * . 25 2.6.3 Vectoranalysis . . . . . . . . . . . . . . . . . . . . . . * *. 26 2.7 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. . 27 1 BASIC NOTATION 5 1 Basic notation 1.1 Conventions In the definition the AMS-notation separated by means of := ,the ver* *bal explanation are on the left side and, separated by means of =: , a* *re the Schwarzschrift-notation. _________________________________________________________________________* *___________________|| ||||||| Examples:____ * * | ||||||| * * | ||||||| * * | ||||||| AMS-notation := verbal explanation =: Schwarzs* *chrift-notation | ||||||| (<=) := subset =: * * | ||||||| * * | ||||||| * * | |||||||__________________________________________________________________* *___________________ | 1.2 ASCII-font Printed symbols, which consists of a linear chane of ASCII-symbols, are discri- bed by means of such ones. The ASCII-font consists of 95 characters. 26 lower case character: a b c d e f g h i j k l m n o p q r s t u v w x y z 26 upper case character: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 10 figures: 0 1 2 3 4 5 6 7 8 9 32 special characters: | ( ) [ ] { } < > = \^~ ``'`; : , . ` + - = * # % & $ @ ! ? 1 BASIC NOTATION 6 1.2.1 ASCII-symbols with special meaning in AMS Space To avoid many-valuedness, spaces are absolut necessary at some places * * in ASCII-notation. That these spaces doesn t fail to notice at reading this in- struction, they are special marked by prefixed and postfixed quotation marks. are for a space to produce with the spacebar. Quotation marks aren t to write! Apostrophe The apostrophe is a special function indicator in AMS. With that the original meaning of the postfixed sign or also postfixed signs is canceled and replaced by a new meaning. _________________________________________________________________________* *___________________ ||||||| Example: * * | ||||||| ____________ * * | ||||||| * * | ||||||| 0 % := infinite (original meaning: per cent) * * =: 1 | ||||||| * * | ||||||| * * | ||||||| * * | |||||||__________________________________________________________________* *___________________ | Should the apostrophe be used in it s original meaning, you must place a space after it. Per-cent-sign, exclamation mark After the per-cent-sign and exclamation mark, if they should be understood in their original meaning, a space must be placed at each, that it couldn t come to a confusion with italic and bolt print-stylisations. Demarcator Demarcator separate keywords like function- and operationnames from prefi- xed or postfixed signs. Spaces, special characters and numbers could used as demarcator. 1 BASIC NOTATION 7 _________________________________________________________________________* *_____________________|| ||||||| Examples:____ * * | ||||||| * * | ||||||| * * O | ||||||| sin'' ''x; sin(x); sin10grd := sinussoidal function =: s* *inx; sin(x); sin10 | ||||||| sign'' ''b1 := signum of b index 1 =: s* *ignb1 | ||||||| * * | ||||||| An'' ''ver'' ''Bi := direct union of A in- =: A* *n [ Bi | ||||||| dex n and B index i * * | ||||||| * * | ||||||| * * | |||||||__________________________________________________________________* *_____________________ | 1.3 In general guidelines of convertion 1.3.1 Structuring of AMS-expressions Supplementary brackets and spaces In some cases the readability of difficult AMS-expressions could be improved by supplementary parenthesis.Otherwise supplementary signs take to a lengthe- ning of express. Spaces often show themselves as a better means of structuring. The problem can only be decided in individual case. If round or angular brackets are contained, you should pay attention, that it could come to confusions with marks or indices (section 1.5 and 1.6). Line break in long mathematical expressions Mathematical expressions, which reach over several lines, have to be changed so, that a operation- or relationsign stands at the end of the line. Reducted notation Reducted notations improve the readability of mathematical expressions. Ho- wever they may be used, if many-valuedness doesn t arise by that. Before their first usage, reductions must be explained by the short notation and the number of the page, at there it is used first time, should be filed in an info-file, w* *hich 1 BASIC NOTATION 8 belong to the text. _________________________________________________________________________* *___________________|||| ||||||| Example:____ * * | ||||||| * * | ||||||| * * | ||||||| in current text: * * | ||||||| x(i;)=xi * * | ||||||| * * | ||||||| * * | ||||||| in info-file: * * | ||||||| * * | ||||||| x(i;)=xi := redundant of the under index * * | ||||||| * * | |||||||__________________________________________________________________* *___________________ | 1.3.2 Convertion of mathematical symbols, which aren t con- tained in AMS In the list of mathematical symbols in section 2, ASCII-performances are defi- ned for following groups of symbols: group 1: indexde or with accents, ornaments or similar ones pro- vided ASCII-signs (i.e. x with tilde) and multiple-lined expressions from ASCII-signs (i.e. matrices) group 2: graphic-symbols (i.e. integral signs) Strictly speaking, the symbols of group 1 shouldn t be included to A* *MS, because their performance result from the introduced notations (section 1.3 to 1.8). These symbols are contained in AMS, because they often appears. The practical transformation-work should be made easier by being provided in ASCII-notation. Should a symbol be transformed, that isn t contained in this instruction, then first it must be checked, to which group the symbol belongs to. If it is a symbol of group 1, then -according to the instructions- it s convert* *ed in the sections 1.3 to 1.8) This is also valid, if an expression with exact the same meaning but with other notation is already contained. The transformation should be occured in 1 BASIC NOTATION 9 possible tight according to the model. Only if it shows, that it comes to heavy limitation of readability, because of the difficult structure of the expression, you can skip to the already availabl* *e, shorter, synonymous notation.The reduction have to be explained by the de- tailed notation, how the other reductions too (section 1.2.1.3). Than, in the following text, you only use the reducted notation. In case of symbols of group 2 the AMS have to be enlarged by definitions of a new ASCII-notation. For that you have to consult the responsible employees. 1.4 Strange alphabets 1.4.1 Greek alphabet ?A ?a := Alpha =: A ff ?B ?b := Beta =: B fi ?G ?g := Gamma =: fl ?D ?d := Delta =: ffi ?E ?e := Epsilon =: E ffl ?Z ?z := Zeta =: Z i ?H ?h := Eta =: H j ?Q ?q := Theta =: ` ode* *r # ?I ?i := Iota =: I ' ?K ?k := Kappa =: K ~ ?L ?l := Lambda =: ~ ?M ?m := My =: M ~ ?N ?n := Ny =: N ?X ?x := Xi =: , ?O ?o := Omikron =: O o ?P ?p := Pi =: ß ode* *r $ ?R ?r := Rho =: P æ ?S ?s := Sigma =: oe od* *er & ?T ?t := Tau =: T ø ?Y ?y := Ypsilon =: Y Æ ?F ?f := Phi =: OE od* *er ' ?C ?c := Chi =: X Ø ?V ?v := Psi =: _ ?W ?w := Omega =: ! 1 BASIC NOTATION 10 1.4.2 French alphabet Accents described by two signs. The first sign is the special-function-pointer e" := accent aigu * * =: 'e apostrophe. e'`e'^ :=:accent=graveaccent circonflex * * =:=`e: ^e e'.. := e with two points * * =: ë These accents can also stand above other vowels. c'> := cedille =: ç o'e := o with closely following e =: oe 1.5 Special print-stylisation `a`A`b`B := script attributes bolt, only for =: a* *AbA characters and numbers %a%A%b%B := script attributes italic, only =: a* *AbB for characters and numbers At the transformation of bolt resp. italic written words and shortcuts it suffi* *ces the marking of the first emphasized sign. !a!A!b!B := fractional scripts (i.e. Gothic, =: A* *aBb Gothic type) and alphabets, that aren t used in europe 1.6 Marking Markings are a sign directly above or under of a sign. _________________________________________________________________________* *__________________ ||||||| * * | ||||||| | x[` ;~] := x with lower marking ` and upper marking~ * * =: ~x_|| |||||||_|________________________________________________________________* *__________________|| If a sign contains both markings and indices, the markings specified first. 1 BASIC NOTATION 11 If a semicolon is contained in a marking, the special-function-pointer is pre- fixed of him. _________________________________________________________________________* *___________________|||| ||||||| Examples:____ * * | ||||||| * * | ||||||| * * | ||||||| * * kP | ||||||| Sum[k=1;n] := sum of k=1 to n * * =: | ||||||| * * n=1 | ||||||| * * Rb | ||||||| Int[a;b] := integral of a to b * * =: | ||||||| * * | ||||||| * * a | ||||||| x[;~] := x-snake * * =: ~x | ||||||| * * | ||||||| h[;^] := h-roof * * =: ^x | ||||||| M[; ()] := M annulation, the inside of set M * * =: O | ||||||| * * M | ||||||| lim [x->?x;] := limit x against Greek xi * * =: lim | ||||||| * * x!, | ||||||| * * | |||||||__________________________________________________________________* *___________________ | Following exceptions applies to introduce shorter notations for often found mar- kings. Exceptions: x~ := x top cross =: _* *_x x` := x down cross =: x_ a^ := vector a =: ~a x.; x.. := temporal derivatives, x point; x two =: x* *`x" point _________________________________________________________________________* *___________________ ||||||| Examples: * * | ||||||| _____________ * * | ||||||| * * __ | ||||||| a~ := towards a conjugated complex number * * =: a | ||||||| * * ____ | ||||||| (PQ)~ := line of P to Q * * =: P Q | ||||||| (lim)` := limit inferior * * =: lim | ||||||| * * ____ | ||||||| * * | |||||||__________________________________________________________________* *___________________ | 1 BASIC NOTATION 12 1.7 Indices Indices are superior or inferior signs, which stand before signs or follow sign* *s. 1.7.1 Postfixed indices _________________________________________________________________________* *______________________ ||||||| * * j | ||||||| | x(i;j) := x with postfixed lower index i and postfixed* * upper =: xi || ||||||| | * * || ||||||| | index j * * || |||||||__|_______________________________________________________________* *_____________________|| x(i;) := x with postfixed lower index i =* *: xi xi := x with postfixed lower index i, shortend notation; =* *: xi only if the index contains of a number of character The reducted notation have to be explained by the supplementary notation before. x(;j) := x with postfixed upper index j =* *: xj If a semicolon is contained in a postfixed index, the special-function-pointer is to prefix to him. _________________________________________________________________________* *_____________________|| ||||||| Examples: * * | ||||||| _____________ * * | ||||||| * * b | ||||||| [F(x)](a;b) := difference of the primitive of the ba* *- =: [F (x)]a | ||||||| se function at the digits x=b a* *nd | ||||||| * * | ||||||| x=a * * | ||||||| * * | ||||||| f(n;) ; fn := n-th function of a function sequence * * =: fn | ||||||| [f(x)](x=g(t);) := notation for substitutions * * =: [f (x)]x=g(t)| ||||||| * * | ||||||| f(x(k;);) := f with lower index - x index k - * * =: fxk | ||||||| * * ` | ||||||| U[;.](?d;) ; U[;.]?d := doted delta-surroundings * * =: Uffi | ||||||| _R(;+) := set of positive real numbers * * =: R+ | |||||||__________________________________________________________________* *_____________________| The index-notation aren t used for: o exponents o operator symbols 1 BASIC NOTATION 13 Exponents The AMS-notation differences exponents of postfixed upper indices. _________________________________________________________________________* *_____________________ ||||||| * * n | ||||||| | x**n := n-th power of x, x top n * * =: x || |||||||__|_______________________________________________________________* *____________________|| x**(-1) := x invers =: x-1 x**-1 := x invers, reducted =: x-1 _________________________________________________________________________* *____________________ ||||||| Examples:____ * * | ||||||| * * | ||||||| * * | ||||||| x**2 := 2. power of x, x square * * =: x2 | |||||||__________________________________________________________________* *____________________| Operator symbols The AMS-notation differences operatorsymbols (even more exact: superior ele- ments of these symbols), both of postfixed upper indices and of exponents. _________________________________________________________________________* *_____________________ ||||||| 0 * * n | ||||||| | O n := O...O operator chane of n operators * * =: O || |||||||__|_______________________________________________________________* *____________________|| _________________________________________________________________________* *____________________ ||||||| Examples: * * | ||||||| _____________ * * | ||||||| * * n | ||||||| f0(n) ; (d0n)f/dx**n := n-th derivative of f to x * * =: f (n); d_f_n| ||||||| * * 2 dx | ||||||| (?d02)f/?dx**2 := 2. partial derivation of f to x * * =: ?d_f_2 | |||||||__________________________________________________________________* *_________?dx________| 1 BASIC NOTATION 14 1.7.2 Prefixed indices _________________________________________________________________________* *____________________ ||||||| * * j | ||||||| | (i;j)x := x with prefixed lower index i and prefi* *- =: ix || ||||||| | * * || ||||||| | xed upper index j * * || |||||||__|_______________________________________________________________* *___________________|| If a semicolon is contained in a prefixed index, put the special-function-point* *er before it. 1.8 Superpositioned signs _________________________________________________________________________* *____________________ ||||||| * * | ||||||| | X\\y := sign y, that lays above sign x * * =: Xy || |||||||__|_______________________________________________________________* *___________________|| _________________________________________________________________________* *___________________ ||||||| Example:____ * * | ||||||| * * | ||||||| * * | ||||||| h\\- := h-cross, constant of Plank * * =: h~ | |||||||__________________________________________________________________* *___________________ | 1.9 Multiple-lined symbols 1.9.1 Fractions and binominalcoefficients Fraction and binominalcoefficients will be linearized. _________________________________________________________________________* *___________________|| ||||||| Examples:____ * * | ||||||| * * | ||||||| (a+b)/(a-b) := fraction, a+b devided by a-b * * =: a+b_ | ||||||| * * a-b | ||||||| (k\n) := binominalcoefficient, k above n * * =: k | |||||||__________________________________________________________________* *___________n_______ | 1.9.2 Big vertical curly brackets The bracket is written only in the first line. The complete expression can also be linearized. 1 BASIC NOTATION 15 ||||||| * * | _________________________________________________________________________* *___________________|| ||||||| Examples: * * | ||||||| _____________ * * | ||||||| * * | ||||||| f(x) = { 1- for x < 0 * * | ||||||| * * | ||||||| 0 for x = 0 * * | ||||||| * * | ||||||| +1 for x > 0 ; f(x) = { -1 for x < 0, 0 for x = * *0, +1 for x > 0 | ||||||| * * | ||||||| * * | ||||||| := function discript* *ion | ||||||| 8 * * | ||||||| * * | ||||||| < -1 * *f ür x < 0 | ||||||| =: f (x) = 0 * *f ür x = 0 | ||||||| : * * | ||||||| +1 * *f ür x > 0 | ||||||| * * | ||||||| f: -A~>B * * | ||||||| a_->f(a) ; f: -A~>B, a_->f(a) * * | ||||||| * * æ | ||||||| * * A ! B | ||||||| := notation of functions =: f * *: a 7! f (a) | ||||||| * * | ||||||| * * | |||||||__________________________________________________________________* *___________________ | 1.9.3 Pair of big vertical brackets or lines At vectors, matrices and determinants the left bracket resp. the left line is written before the first element line and the right bracket resp. the right li* *ne after the last element line. Operation- and relationsigns stand in the first line. Vectors, matrices and de- terminants can also be linearized. 1 BASIC|NOTATION|| 16 * * || ||||||| * * | _________________________________________________________________________* *___________________|| ||||||| Examples: * * | ||||||| _____________ * * | ||||||| * * 0 1 | ||||||| * * x1 | ||||||| ` x = (x1 * * B C | ||||||| x2 * * B x2 C | ||||||| := vector x * * =: x = B . C | ||||||| : * * @ .. A | ||||||| xn) * * | ||||||| * * xn | ||||||| * * | ||||||| * * 0 1 | ||||||| * * x1 | ||||||| * * B x C | ||||||| * * B 2 C | ||||||| x^=(x1, x2, ..., xn) := vector x * * =: x = B .. C | ||||||| * * @ . A | ||||||| * * | ||||||| * * xn | ||||||| ` A = (a11a12...a1q * * | ||||||| * * | ||||||| a12a22...a2q * * | ||||||| : := (p,q)-matrix A * * | ||||||| * * | ||||||| ap1ap2...apq) * * | ||||||| 0 * * 1 | ||||||| a11 a1* *2 . . . a1q | ||||||| B a a * * . . . a C | ||||||| B 21 2* *2 2q C | ||||||| =: ` A = B@ .. * * CA | ||||||| . * * | ||||||| a a * * . . . a | ||||||| p1 p* *2 pq | ||||||| * * | ||||||| * * | ||||||| ` A= a11 a12 ... a1q, a21 a22 ... := (p,q)-matrix A * * | ||||||| a2q, ..., ap1 ap2 ... apq) * * | ||||||| 0 * * 1 | ||||||| a a * * . . . a | ||||||| 11 1* *2 1q | ||||||| B a21 a2* *2 . . . a2q C | ||||||| =: ` A = BB . * * CC | ||||||| @ . * * A | ||||||| . * * | ||||||| ap1 ap* *2 . . . apq | |||||||__________________________________________________________________* *___________________ | 2 MATHEMATICAL SYMBOLS 17 2 Mathematical symbols 2.1 Symbols, numbers, operation- and relationsigns 2.1.1 Mathematical symbols __ := parallel =: k -__ := not parallel =: 6k __= := parallel und equal =: __= `_` := orthogonal =: ?; ?_ _` := right angle =: b <) := angle =: ^ '" " := minute =: 0 "" " := second; Attention: The measu- =: 00 re of angle second is described by the quotation mark, not by 2 apostrophs! -: := line-colon =: ._. () := little circle, annulation =: O grd := degree =: O ' := curve =: ` _> := equilateral triangle, point right =: . <_ := equilateral triangle, point left =: / -> := arrow of limit =: ! - := poor convergent =: * -> := not convergent =: 6! => := implication, results =: ) <=> := douple-end, equality =: , ~>; _->; -_>; := horizontal arrows to right =: - !; 7!; 6!; =) ==> <~; <-_; <_-; := horizontal arrows to left =: - ; 6 ; (= <== 2 MATHEMATICAL SYMBOLS 18 <-> := horizontal double-arrow =: $ <==> := horizontal double-line double- =: () arrow _^ := perpendicular arrow up =: " __^ := perpendicular double-line- =: * arrow up '_v := perpendicular arrow down =: # '__v := perpendicular double-line- =: + arrow down '__^v := perpendicular double-line =: m double-arrow /> := oblique arrow right-up =: % < := oblique arrow left-up =: - \> := oblique arrow right-down =: & := greater =: > > := strict preference-regularity, after =: ~ := much greater =: >= := greater than or equal =: >~ := greater than or approximately =: * *>s '>~ := preference-regularity, not less than similar =: s " "~= := asymptotical =: * *~= 2 MATHEMATICAL SYMBOLS 21 2.2 Statements Al" "x := universal quantifier, for all x =: * *8x Ex" "x := existential quantifier, there are an x =: * *9x '- := negation, not =: * *:;~ '& := conjugation, and =: ^ 'v := disjunction, or =: _ >-< := non equality, either or =: * *>-< '-&" " := NAND =: * *:^; ~^ '-v" " := NOR =: * *:_; ~_ * * nV '&[k=1;n]Ak := A1 and A2 and ... and A(n) =: * * Ak * *k=1n * * W 'v[k=1;n]Ak := A1 or A2 or ... or A(n) =: * * Ak * *k=1 ggT := highest common factor =: u kgV := group generated by a given set of subgroups =: t 2.3 Sets Number sets are described accordingly the model. _________________________________________________________________________* *___________________|| ||||||| Examples:____ * * | ||||||| * * | ||||||| * * | ||||||| |N := set of natural numbers * * =: N | ||||||| Q := set of relational numbers * * =: Q | |||||||__________________________________________________________________* *___________________ | 2 MATHEMATICAL SYMBOLS 22 - " := void set =: ; * *{} ; @ := is element =: 2 -@ := isn t element =: 62 (<=) := subset =: (<) := real subset =: -(<) := isn t real subset =: 6 ;* * : (>=) := subset =: (>) := real subset =: -(>) := isn t real subset =: 6 ;* * : dif := difference, without =: \; - sd := symmetrical difference =: ver := union =: [S VerM := union of sets M =: M nS Ver[k=1;n]Mk := M1 union M2 union ... union Mn =: * * Mk k=1S Ver[M@!S;]M := direct union of sets M of an setsy- =: * * M stem M2S dur := intersection =: \T DurM := intersection of sets M =: M car := cartesian product =: x Nn Car[i=1;n]Ai := cartesian product of n sets Ai =: * * Ai i=1 _\; _ := restriction =: ~; | (A_B) := Dedekind cut, A cut B =: (A|* *B) 2.4 Vectors, matrices and determinants Vestors are created accordingly the model. _________________________________________________________________________* *___________________|| ||||||| Examples: * * | ||||||| _____________ * * | ||||||| * * | ||||||| a^ := vector a * * =: ~a | ||||||| * * | ||||||| `b := vector b * * =: b | ||||||| c` := vector c * * =: c_ | |||||||__________________________________________________________________* *___________________ | 2 MATHEMATICAL SYMBOLS 23 x^(;T) := transpose of vestor x =: ~eT x^*y := dot product =: ~x.* * ~y (x^_y^); [x^_y^] := inner product of two vec- =: (~x* *|~y); [~x|~y] tors x^'xy^ := vector product, cross =: ~xx* * ~y product _x^_ := absoluted value of vector =: |~x| x __x^__ := norm of vector x =: ||~* *x|| 0 1 a11 a12 ... a1q B a21 a22 ... a2q C `A = B@ : CA := (p,q)-matrix ap1 ap2 ... apq 0 * * 1 a11 a12 . . . a1q B a21 a22 . . . a2* *qC =: A = BB . . .* * CC @ .. .. . . . .* *.A ap1 ap2 . . . apq 0 1 0 * * 1 1 0 0 1 0 * * 0 `I; `E := unit matrix =: I = @ 0 1 0A ; E = @ 0 1 * * 0A 0 0 1 0 0 * * 1 ` ' `0 := zero matrix =: 0 = 00 00 fifi fi fifia11a12 ... a1q fifi fia21 a22 ... a2q fifi det A=fifi: fifi := determinant of square matrix A fifi fi fiap1 ap2 ... apq fifi fifi * * fi fifia11a12 . . * *. a1qfifi a21 a22 . . * *. a2qfifi =: det A=fififi.... * * ..fi fifi. . . . * *. . fifi ap1 ap2 . . * *. apqfi Attention: The apostrophe for lines are the colon! 2 MATHEMATICAL SYMBOLS 24 2.5 Sequences and functions n min[k=1;n]ak := minimum of a1, a2, ..., an =: min* *k=1ak max[k=1;n]ak := maximum of a1, a2, ..., an =: man* *xk=1ak sup[n=1;'%]an := supremum of sequence an =: s1u* *pan n=11 inf[n=1;'%]an := infimum of sequence an =: inf* *n=1an lim[n->'%;]an := limit of sequence an =: n-l* *im>1an liminf ; (lim)` := limit inferior =: lim* * inf ;_lim__ limsup ; (lim~) := limit superior =: lim* * sup; lim " " " " := nest of intervall =: < a* *n |bn > " " " " := compact intervall =: < x* *0, x > f(x) := function f of x =: f (* *x) f(;-1) := reciprocal map, inverse function to f =: f -1 _f_ := absolute value of f =: |f | f+; f+(x) := positive part of function f =: f +* *; f +(x) f-; f-(x) := negative part of function f =: f -* *; f -(x) __f__ := norm of f =: ||f* * || __f__'% := supremum norm of f := ||f* * ||1 f()g := composition, compositum, f to g =: f O* * g f'*g := convolution =: f ** * g " "(;g)log" ä := logarithm of a to basis g =: g l* *og a log(g;)a := logarithm of a to basis g =: log* *g a V(a;b)(g) := total variation of g to [a;b] =: Vab* *(g) f_A := restriction of f to A =: f |A (g_f) := inner product of two functions =: (g|* *f ) (A'n)f := operator linear fig. A, used n times to f _________________________________________________________________________* *___________________|| ||||||| Examples: * * | ||||||| _____________ * * | ||||||| * * k | ||||||| (%D'k)`y := k-th difference sequence of sequence y y* * =: D y | ||||||| * * k | ||||||| (?D'k)yj := j-th component of k-th difference s* *e- =: yj | ||||||| quence of sequence y * * | |||||||__________________________________________________________________* *___________________ | 2 MATHEMATICAL SYMBOLS 25 2.6 Differentials 2.6.1 Total derivative f` := f0, f line =: * *f 0 f`` := f00, f two line =: * *f 00 f`` ` := f000, f three line =: * *f 000 f`(n) := n-th derivative of f =: * *f (n)n (d'n)/dx**n := n-th derivative to x =: * *_d__dxn (D'n) := differentation operator D, used n times =: * *Dn _________________________________________________________________________* *___________________|||| ||||||| Examples: * * | ||||||| _____________ * * | ||||||| * * | ||||||| df/dx; df(x)/dx; d/dxf(x) := 1. derivative of the functi* *on f(x) to x | ||||||| * * | ||||||| * * | ||||||| =: _df* *_; df(x)_; _d_f (x) | ||||||| dx * * dx dx | ||||||| * * | ||||||| * * | ||||||| f'(x0); Df(x0) := 1. derivative of f at x0 * * | ||||||| * * | ||||||| 0* * | ||||||| =: f (* *x0); Df (x0) | ||||||| * * | ||||||| * * | ||||||| (f'(k+1)); (D`k+1)f := (k+1)-th derivative of f * * | ||||||| * * | ||||||| * * | ||||||| =: f (* *k+1); Dk+1 f | |||||||__________________________________________________________________* *___________________ | df/dx_((x=?x);); := 1. derivative of f to x at x=xi; detai- =: _df* *_dx|x=, df/dx_x=?x led and reducted notation. After the first statement by the detailed nota- tion the reducted notation should be used! f. := 1. derivative of f after time t =: * *f. f.. := 2. derivative of f after time t =: * *..f 2.6.2 Partial derivative ?d/?dxk; Dk := 1. partial derivative after the k-th =: __* *ffi_ffixk; Dk variable xk 2 MATHEMATICAL SYMBOLS 26 ||||||| * * | _________________________________________________________________________* *___________________|| ||||||| Examples: * * | ||||||| _____________ * * | ||||||| * * | ||||||| * * | ||||||| * * | ||||||| D1f := 1. partial derivative of f after * *the first | ||||||| variable * * | ||||||| * * | ||||||| * * | ||||||| =: D1f* * | ||||||| * * | ||||||| * * | ||||||| ?df/?dx := 1. partial derivative of f to x * * | ||||||| * * | ||||||| * * | ||||||| =: ffi* *f_ | ||||||| ffi* *x | ||||||| * * | ||||||| ?df(x0,y0)/?dy; := 1. partial derivative of f to y at* * (x0, y0) | ||||||| * * | ||||||| ?df/?dy(x0,y0) * * | ||||||| ffi* *f(x0,y0) ffif | ||||||| =: ___* *______ffiy; ___ffiy(x0,|y0) |||||||__________________________________________________________________* *___________________ | * * 2 DjDk; := 2. partial derivative after the j-th =: DjDk; ___* *ffi_ffixjffixk (?d`2)/?dxj?dxk| and the k-th variable xj and xk * * | _________________________________________________________________________* *___________________|||| ||||||| Examples:____ * * | ||||||| * * | ||||||| * * | ||||||| * * | ||||||| D1D2f := 2. partial derivative after the * *first and the | ||||||| second variable * * | ||||||| * * | ||||||| * * | ||||||| =:* * D1D2f | ||||||| * * | ||||||| * * | ||||||| (?d`2)f/?dx?dy; := 2. partial derivative of f to x an* *d y | ||||||| ?d/?dx(?df/?dy) * * | ||||||| * * | ||||||| * * _ffi2f_ _ffi_ ffif_| ||||||| =:* * ffixffiy; ffix( ffiy)| ||||||| * * | ||||||| * * | ||||||| (?d`2)f/?dx**2 := 2. partial derivative of f to x * * | ||||||| * * | ||||||| * * 2 | ||||||| =:* * ffi_f2 | |||||||__________________________________________________________________* *_ffix______________ | 2.6.3 Vectoranalysis ?d/?d`v; D(`v;) := directional derivative in direction v =: _f* *fi_ffiv; Dv 2 MATHEMATICAL SYMBOLS 27 _________________________________________________________________________* *___________________|| ||||||| Example:____ * * | ||||||| * * | ||||||| * * | ||||||| ?df(`?x)/?d`v; := direction derivative if the functi* *on f at | ||||||| ?df/?d`v(`?x); greek xi in direction v * * | ||||||| * * | ||||||| D(`v;)f(`?x) * * | ||||||| ffif(,)* * | ||||||| =: _____; * *ffif ffiv(,); Dvf (,)| |||||||_____________________________________________________________ffiv_* *___________________ | Nab := nabla operator =: r Lap := laplacian operator =: 4 2.7 Integrals R Int := integral =: _________________________________________________________________________* *___________________|||| ||||||| Examples:____ * * | ||||||| * * | ||||||| * * b | ||||||| * * R | ||||||| Int[a;b]f(x)dx := integral of f(x) at the peripherie* * =: f (x)dx | ||||||| of a to b * * a | ||||||| * * +1 | ||||||| * * R | ||||||| Int[-'%;+'%] := improper integral * * =: | ||||||| * * R1 | ||||||| Int[I;] := integral above I * * =: | ||||||| * * I | ||||||| * * R | ||||||| Int[?g;]`f(`x)d`x := way integral of f along gamma * * =: f (x)dx | ||||||| * * flR | ||||||| Int[`?F;]fd?s := surface integral of the scalar fie* *ld =: f doe | ||||||| * * | ||||||| f above the surface Phi * * R | ||||||| * * | ||||||| Int[`\F;]`F*`nd?s := surface integral of the vector fie* *ld =: F . ndoe | ||||||| f above the surface Phi * * | |||||||__________________________________________________________________* *___________________ | R Int` := lower Darboux integral =: - -R Int~ := upper Darboux integral =: 2 MATHEMATICAL SYMBOLS 28 _________________________________________________________________________* *___________________|| ||||||| Examples:____ * * | ||||||| * * | ||||||| * * int | ||||||| Int`[I;] := lower integral of Darboux to I * * =: I | |||||||__________________________________________________________________* *___________________ | H Int\\() := integral above a closed path, a clo- =: sed surface _________________________________________________________________________* *___________________|| ||||||| Examples:____ * * | ||||||| * * H | ||||||| * * ~ | ||||||| Int\\()H^dr^ := integral of vector H above a close* *d =: H d~r | ||||||| path * * | |||||||__________________________________________________________________* *___________________ | R Int\\ (^) := integral above a closed way in =: O mathematical postitive circulation meaning /\ := wedge product, outer product =: ^ _________________________________________________________________________* *___________________|| ||||||| Examples: * * | ||||||| _____________ * * | ||||||| * * | ||||||| ?F/\?V := wedge product of the alternating l* *i- =: ^ | ||||||| * * | ||||||| near forms Phi and Psi * * | ||||||| * * | ||||||| Int[?F;]Pdy/\dz+Qdz/\dx+ Rdx/\dy * * | ||||||| * * | ||||||| * * | ||||||| := surface integral of a vector field* * above the sur- | ||||||| face Phi * * | ||||||| R * * | ||||||| =: P dy ^ dz + Qdz ^ dx + Rdx ^ dy * * | |||||||__________________________________________________________________* *___________________ | R R Int2 := double integral =: R * *R R Int3 := triply integral =: 2 MATHEMATICAL SYMBOLS 29 _________________________________________________________________________* *___________________|||| ||||||| Examples:____ * * | ||||||| * * | ||||||| * * | ||||||| * * | ||||||| Int2[(`B);]f(x,y)d`B := surface integral of f a* *bove the | ||||||| * * | ||||||| sphere B at the x,y-level * * | ||||||| R R * * | ||||||| =: B f* * (x, y)dB | ||||||| * * | ||||||| * * | ||||||| Int2\\()B^dA^ := integral of the vector fiel* *d B abo- | ||||||| ve the closed surface A * * | ||||||| * * | ||||||| =: ~BdA~ * * | ||||||| * * | ||||||| * * | ||||||| Int3[(`K);]f(x,y,z)dxdydz := space integral of f above t* *he spa- | ||||||| cial sphere K * * | ||||||| R R R* * | ||||||| =: * * f (x, y, z)dxdydz | |||||||______________________________________________________________K___* *___________________ |