《结构工程中的数学方法》课程教学课件（讲稿）symbols

IntroductionListofSymbolsMeaning (if not self-explaining)Symbol and TermRreal numbersIRnn-dimensional Euclidean spaceIDuniversecomprehensivesetasthebasisforconsiderationsX, Avariable,term△difference8small numberOR"small change"variation8infinity[xabsolutevalueof x[a, b]closed intervalbounds a and b belong to the interval(a, b)open intervalaandbdo not belong to the interval[a, b)half-open intervala belongs to the interval but b does not9MichaelBeer,EngineeringMathematics

4 Symbol and Term 0 Introduction List of Symbols ∆  x Meaning (if not self-explaining) real numbers n n-dimensional Euclidean space universe x, A variable, term  comprehensive set as the basis for considerations difference δ small number OR variation [a, b] infinity bounds a and b belong to the interval (a, b) open interval a and b do not belong to the interval [a, b) half-open interval a belongs to the interval but b does not closed interval ∞ absolute value of x "small change" Michael Beer, Engineering Mathematics

oIntroductionListof SymbolsMeaning (if not self-explaining)Symbol and Termf(.)functionfunctionof thevariables inparenthesis0(hn)Landau notationthe variable h does not appear in O(hn)term of order hnwithan exponent smaller thannd, dnsingle, n-fold differentiationa, ansingle,n-fold partial differentiation,口first,second derivative",(4)third,fourth derivative2()sum over()fromi=a toi=n1j() dx integral over (.)dx from x=a to x=bn!factorialn!=1.2.....nlim()limesof()forn-mlimitof(.)whennapproachesmn-mexp() = e()exponential functionexp(0)with basee5Michael Beer,EngineeringMathematics

Michael Beer, Engineering Mathematics 5 Symbol and Term List of Symbols Meaning (if not self-explaining) ( ) sum over (.) from i=a to i=n = ∑ n i a . n! n! 1 2 n =⋅⋅ ⋅  ( ) n m→ lim . ( ) b a ∫ . dx integral over (.)dx from x=a to x=b factorial limes of (.) for n→m limit of (.) when n approaches m single, n-fold differentiation single, n-fold partial differentiation ￾', ￾'' first, second derivative d, dn ∂, ∂n ￾''' , ￾ third, fourth derivative (4) f(.) function function of the variables in parenthesis Landau notation, term of order hn O(hn) the variable h does not appear in O(hn) with an exponent smaller than n 0 Introduction exp .( ) exponential function with base e ( ) (.) exp . e =

IntroductionListof SymbolsMeaning (if not self-explaining)Symbol and Termgoesto,approaches ORterm and meaning→mappingORisgiven bythe contextreplacement1exchangeimplication,if....then...ORterm and meaningconclusion ORis given bythe contexttargetingat(optimisation)DequivalenceAmuchsmallerthanVuniversal quantifier,for all3existential guantifier,there existsEelementofnot element of史NAnotapplicable6Michael Beer, Engineering Mathematics

6 Symbol and Term List of Symbols Meaning (if not self-explaining) ⇒ ⇔ implication, if ., then . OR conclusion OR targeting at (optimisation) term and meaning is given by the context equivalence  ∀ much smaller than universal quantifier, for all $ ∉ existential quantifier, there exists not element of NA not applicable → goes to, approaches OR mapping OR replacement ↔ exchange term and meaning is given by the context ∈ element of 0 Introduction Michael Beer, Engineering Mathematics

IntroductionListof SymbolsMeaning (if not self-explaining)Symbol and TermX, Avector,matrixcolumn vector:Xx=1X.X = (Xi...,X.)row vector:ATtransposed matrix of AA-1inverse matrix of A1identitymatrixallelementsinthemaindiagonal are1all remaining elements are 0dot,inner,scalar productxcross,outer,Cartesian productrk(A)rank of matrix Adet(A)determinant of matrix AIA|determinant of matrix AMichaelBeer,EngineeringMathematics

7 Symbol and Term List of Symbols x A, Meaning (if not self-explaining) vector, matrix x = (x ,., x 1 n ) 1 n x x     =       x  T A −1 A transposed matrix of A column vector: row vector: inverse matrix of A rk(A) det(A) × I identity matrix all elements in the main diagonal are 1, all remaining elements are 0 · dot, inner, scalar product cross, outer, Cartesian product rank of matrix A determinant of matrix A |A| determinant of matrix A 0 Introduction Michael Beer, Engineering Mathematics

IntroductionListof SymbolsMeaning (if not self-explaining)Symbol and TermA = (..] setthe elements of A are given in the bracecsubsetBdomainformed bydefined setsas a subset of a universe"length"of a vector ormatrixIxll,l/A/|norm of vector x, of matrix A[AIcardinality of set A"number of elements"in ACcomplexnumbersRe(c)real part of ceCIm(c)imaginary part of c eCiimaginary unitwith i2=-1rpathhere:inthecomplexplanegeometrical summationconnectionof geometricalobjects$()contour integral8Michael Beer, EngineeringMathematics

8 Symbol and Term List of Symbols A = {} ⊆ Meaning (if not self-explaining) x,A set subset the elements of A are given in the brace B domain formed by defined sets as a subset of a universe norm of vector x, of matrix A "length" of a vector or matrix |A| cardinality of set A "number of elements" in A complex numbers real part of c ∈  imaginary part of c ∈  imaginary unit with i2 = −1 path  Re(c) Im(c) i Γ geometrical summation contour integral ⊕ connection of geometrical objects here: in the complex plane (.) Γ ∫ 0 Introduction Michael Beer, Engineering Mathematics