By Wolfgang Nolting
Der Grundkurs Theoretische Physik deckt in sieben Bänden alle für Diplom- und Bachelor/Master-Studiengänge maßgeblichen Gebiete ab. Jeder Band vermittelt das im jeweiligen Semester nötige theoretisch-physikalische Rüstzeug. Übungsaufgaben mit ausführlichen Lösungen dienen der Vertiefung des Stoffs. Band 1 behandelt die klassische Mechanik. Vorausgesetzt wird nur die übliche Schulmathematik, andere mathematische Hilfsmittel werden zu Beginn ausführlich erläutert. Die zweifarbig gestaltete Neuauflage wurde grundlegend überarbeitet und ergänzt.
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72) as ‘difference quotient’. If we now shift the point Q along the curve towards the point P then the increase of the secant becomes the increase of the tangent on the Fig. 1 Elements of Differential Calculus 19 Fig. x/ at P (broken line in Fig. 11), tan ˛ D lim tan ˛ 0 D lim 0 x ! ˛ y x and one arrives at the ‘differential quotient’ lim x ! x/ D 2x : All the differential quotients do not exhibit a unique limit everywhere! The curve in Fig. 12 is continuous at P, but has there different slopes if we come, respectively, 20 1 Mathematical Preparations Fig.
16 1 Mathematical Preparations Fig. 8 Schematic behavior of the exponential function In Sect. 65) Thus, if a is raised to the power of loga y one gets y. Rather often one uses a D 10 and calls it then ‘common (decimal) logarithm’: log10 100 D 2 I log10 1000 D 3 I : : : However, in physics we use most frequently the ‘natural logarithm’ with base a D e denoted by the symbol loge Á ln. 17) which we had to postpone because it exploits properties of the logarithm. 17) is concerned with the following statement about the limit of the sequence fan g D fqn g !
U/ du : In case of a definite integral we have to notice that the limits of integration are also usually changed with the substitution (xi ! xi /). • It is integrated now with respect to u. x/ ! u/ ! u/ D eu . 123) c is a real constant. u/ D u5 . 3 C 4x/6 C c : 24 c is again an arbitrary real constant. 3. x/. 5 Multiple Integrals Multiple integrals as volume or surface integrals are reduced for their calculation to a set of simple one-dimensional integrals of the kind we have inspected up to now in the preceding sections (Fig.