Analysis III [Lecture notes] by Dirk Ferus

By Dirk Ferus

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Extra resources for Analysis III [Lecture notes]

Example text

Wenn h kein Automorphismus, sondern ein Diffeomorphismus, also eine stetig differenzierbare Abbildung mit stetig differenzierbarem Inversen ist, liefert die allgemeine Philosophie der 57 Differentialrechnung sofort eine Vermutung, wie man diese Formel verallgemeinern sollte: Statt der Determinante von h sollte wohl die Determinante der Ableitung von h auftreten: f dµn = (f ◦ h)| det Dh | dµn . Der Beweis dieser Formel ist unser Ziel in den folgenden Abschnitten. Wenn wir f nur u ¨ber einen Bereich h(G) integrieren wollen, m¨ ussen wir f ersetzen durch f χh(G).

Dann ist h(W ) kompakt, also integrierbar, und es gilt µn (h(W )) = | det h | µn (W ). (21) Beweis. F¨ ur die beiden ersten Typen elementarer Isomorphismen ist die Behauptung klar, weil h(W ) ein Intervall ist, dessen Volumen sich elementar errechnet. Sei also h vom dritten Typ und W = W1 × . . × Wn . Dann gilt χh(W )(x1 , . . , xn ) = χ(x 2 + W1 )(x1 )χW2 × . . × Wn (x2 , . . , xn ). Dabei bezeichnet x2 + W1 das um x2 verschobene Intervall W1 , f¨ ur das offenbar µ1 (x2 + W1 ) = µ1 (W1 ). )χW2 × .

Wenn h kein Automorphismus, sondern ein Diffeomorphismus, also eine stetig differenzierbare Abbildung mit stetig differenzierbarem Inversen ist, liefert die allgemeine Philosophie der 57 Differentialrechnung sofort eine Vermutung, wie man diese Formel verallgemeinern sollte: Statt der Determinante von h sollte wohl die Determinante der Ableitung von h auftreten: f dµn = (f ◦ h)| det Dh | dµn . Der Beweis dieser Formel ist unser Ziel in den folgenden Abschnitten. Wenn wir f nur u ¨ber einen Bereich h(G) integrieren wollen, m¨ ussen wir f ersetzen durch f χh(G).

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