Get Illustrierte Mathematik: Visualisierung von mathematischen PDF

By D.L. Johnson

ISBN-10: 1852332700

ISBN-13: 9781852332709

Die optische Darstellung von Objekten, die durch mathematische Sachverhalte beschrieben werden, sind für Schüler und Studenten während der gesamten Ausbildungszeit ein einprägsames Hilfsmittel. Die Visualisierung komplex beschriebener Gegenstände dient nicht nur dem Verständnis abstrakter Gebilde, sondern birgt in sich auch ein ästhetisches und künstlerisches Moment.
Die Autoren stellen auf ihrer CD-ROM eine umfangreiche Sammlung von Graphiken, die unter home windows und auf Macintosh betrachtet und weiter verarbeitet werden können, zur Verfügung. Die dafür notwendige software program MathReader ist auf der CD vorhandenen.
Als weitere Komponente enthält die CD eine Vielzahl von Programmen, mit denen der Benutzer durch edition von Parametern eigene Beispiele kreieren kann. Zur Arbeit mit diesen Paketen muß auf dem Rechner die software program Mathematica installiert sein.

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Additional resources for Illustrierte Mathematik: Visualisierung von mathematischen Gegenständen (German Edition)

Example text

We discard the trivial case I = V, when rr' — r 2 = 1, and assume at once that I jzV. We distinguish two cases according as / and /' meet (say in a point O) or not. These are illustrated in Fig. 1, where the angle from / to /' is a in case (i) and the perpendicular distance from / to /' is a in case (ii). 45 46 Symmetries Case (i): lf)V = {O}. In polar coordinates with centre O, base line / and a > 0, an arbitrary point P = (p, 6) is mapped by r to (p, —0), while r' sends F to the point P' = {p,), where ^ ^ = a, since /' bisects the angle POPr'.

24, and this gives the reason for its name. 5 Consider the subgroups T S E2, So = E/2TTZ of the Euclidean group E. It follows from the Normal Form Theorem that S 0 T = E+, S0HT={1}. 14)), we know that T is a normal subgroup of E + . This leads us to recognise E + as a semidirect product So Xa T, where, for s € So, sa is the automorphism of T sending £(a) to t(Rs). Now let r E E denote reflection in any axis through 0 and denote the group (r) = {l,r} by Z2. We have Z2E+ = E , Z 2 f l E + = {1}, and the normality of E + in E follows from considerations of orientation.

1, s = r(l')r(l), t= r(l)r(l"), whence u = st = r(l')r(l") 51 4. Products of Reflections as r(/) 2 = 1. 1 again. The reader is invited to supply an appropriate diagram. Moving on to the next case, u — ts, a similar argument can be applied. Or merely observe that u~l — s~xt~l is a rotation by the previous case, and therefore so is u. Finally, take the case of two rotations, s — s(0,a), s' = s(0',a'). To compute the product u = ss' when O ^ 0 ' , define three lines in E 2 as follows: • / is the line through O and 0 ' , • /' is the line through O such that the angle from /' to / is a / 2 , • I" is the line through O' such that the angle from I to I" is Q ' / 2 .

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Illustrierte Mathematik: Visualisierung von mathematischen Gegenständen (German Edition) by D.L. Johnson


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