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Home > Mathematics and Science Textbooks > Mathematics > Calculus and mathematical analysis > Analysis III: (153 Heidelberger Taschenbücher)
Analysis III: (153 Heidelberger Taschenbücher)

Analysis III: (153 Heidelberger Taschenbücher)


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3 Ftir die Funktion . 1(. \,"), =): = 1 zum Beispiel hat (1) den Wert (4n: /3)R, (2) aber 2 den Wert R '2n: 'n: =2n: R. Urn den wahren Sachverhalt zu ergrtinden, betrachten wir flir ein grofies, aber festes SElN die im Innern von Q enthaltenen s-Wtirfel 1 . . ., und bezeichnen sie mit Wj (1;. . j N). Die durch (251. 2) definierte Abbildung g: u: = (r, qJ, 9) . . . . . x =(x, y. z) flihrt jeden Wtirfel Wj bijektiv in ein krummlinig begrenztes "KlOtzchen" LI j C B 3. R tiber (siehe die Fig. 252. 1). Diese Klotzchen bilden zusammen ein die Kugel B 3 R von innen approximierendes Klotzchengebaude, somit gilt (wir verwenden wie de rum das Zeichen == flir "ungefahr gleich"): Es sei u das Zentrum des Wtirfels Wj und xj: =g(uj)ELl . Wir wollen annehmen, j j die Funktion f sei stetig; dann dtirfen wir weiter schreiben Nun ist g differenzierbar und Wj "klein," somit ist eine flir alle UE Wj brauchbare Approximation. Hiernach ist das Klotzchen LI j = g(W) in erster Naherung ein Parallelepiped, das durch 'Verzerrung des Wtirfels Wj mit der linearen Abbildung g*(u ) entstanden ist. Aufgrund von Satz j (23. 22) gilt daher Fig. 252. 1 78 25. Variablentransformation bei mehrfachen integralen so daB wir anstelle vori (4) erhalten: (5) J"J(x)d/lx == f(x) Idetg*(u)I/l(W) = l(u) IJ."

Table of Contents:
21. Hauptsätze der mehrdimensionalen Differentialrechnung.- 211. Stetige Differenzierbarkeit.- 212. Hilfssätze.- 213. Der Satz über die Umkehrabbildung.- 214. Die Funktionaldeterminante.- 215. Der Satz über implizite Funktionen.- 216. Der Immersionssatz.- 22. „Flächen” im IRn.- 221. Begriff der m-Fläche.- 222. Tangentialebene.- 223. Hyperflächen.- 224. Bedingt stationäre Punkte.- 225. Lagrangesche Multiplikatoren.- 226. Beispiele.- 227. Globale Extrema.- 23. Das Jordansche Maß im IRm.- 231. Vorbemerkungen.- 232. Äußeres und inneres Jordansches Maß.- 233. Grundeigenschaften des Maßes.- 234. Das Maß von Quadern. Translationsinvarianz.- 235. Verhalten des Maßes gegenüber C1-Abbildungen.- 236. Hilfssätze.- 237. Verhalten des Maßes gegenüber linearen Abbildungen.- 24. Mehrfache Integrale.- 241. Das Riemannsche Integral im IRm.- 242. Reduktionssatz („Satz von Fubini“).- 243. Integral über beliebige meßbare Mengen.- 244. Praktische Berechnung mehrfacher Integrale.- 245. Anwendung: Volumen der m-dimensionalen Kugel.- 246. Uneigentliche mehrfache Integrale.- 25. Variablentransformation bei mehrfachen Integralen.- 251. Zylinder- und Kugelkoordinaten.- 252. Problemstellung.- 253. Hilfssätze.- 254. Die Transformationsformel.- 26. Flächen im IR3.- 261. Das Vektorprodukt im IR3.- 262. Orientierung.- 263. Begriff des Flächeninhalts.- 264. Eigenschaften des Flächeninhalts.- 27. Vektorfelder.- 271. Vorbemerkungen. Begriff des Vektorfeldes.- 272. Linienintegrale.- 273. Konservative Felder.- 274. Infinitesimale Zirkulation.- 275. Rotation (zweidimensionaler Fall).- 276. Rotation (dreidimensionaler Fall).- 28. Die Greensche Formel für ebene Bereiche.- 281. Der Heine-Borelsche Überdeckungssatz.- 282. Zerlegung der Einheit.- 283. Die Greensche Formel fürglatt berandete Bereiche.- 284. Zulässige Bereiche.- 285. Anwendungen der Greenschen Formel.- 29. Der Satz von Stokes.- 291. Begriff des Flusses.- 292. Zulässige Flächen.- 293. Ein Übertragungsprinzip.- 294. Der Satz von Stokes.- 295. Einfach zusammenhängende Gebiete.- 296. Die Integrabilitätsbedingung.- 30. Der Satz von Gauß.- 301. Divergenz eines Vektorfeldes.- 302. Der Satz von Gauß für glatt berandete Bereiche.- 303. Zulässige Bereiche.- 304. Der Laplace-Operator.- 305. Ein Satz der Potentialtheorie.- Liste der Symbole und Abkürzungen.- Sachverzeichnis Analysis I bis III.


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Product Details
  • ISBN-13: 9783540069348
  • Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
  • Publisher Imprint: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Height: 203 mm
  • No of Pages: 186
  • Returnable: N
  • Width: 133 mm
  • ISBN-10: 3540069348
  • Publisher Date: 01 Sep 1981
  • Binding: Paperback
  • Language: German
  • Returnable: N
  • Series Title: 153 Heidelberger Taschenbücher


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