By Prof. Dr. Konrad Königsberger (auth.)

ISBN-10: 3540203893

ISBN-13: 9783540203896

ISBN-10: 3540350772

ISBN-13: 9783540350774

Dieser Band behandelt die Differential- und Integralrechnung im R^{n} sowie Differentialgleichungen und Elemente der Funktionentheorie. Zu seinen Besonderheiten gehören eine neue, einfache Einführung des Lebesgueintegrals sowie der Gaußsche Integralsatz in großer, bedarfsgerechter Allgemeinheit. Ein umfangreiches Kapitel ist den Differentialformen gewidmet und als Einstieg in die Theorie der Mannigfaltigkeiten konzipiert. Historische und biographische Anmerkungen bereichern die Darstellung. Mit seinen zahlreichen Beispielen und interessanten Übungsaufgaben eignet sich dieses Lehrbuch auch sehr intestine zum Selbststudium.

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**Extra resources for Analysis 2**

**Example text**

The governing DE is therefore given by dP = kP(2000 - P), dt where k is an unknown proportionality constant. If 20 students are infected with the virus after 3 days, how many students were infected after 1 day and how many will be infected by the end of a week? Solution: Because only one student is infected with the virus at the beginning, we formulate the IVP we wish to solve by dP = kP(2000 - P), P(0) = 1. dt This equation is nonlinear but can be solved by separating the variables similar to part (d) of Example 7.

Indeed, the starting point is usually some real-world problem that must be mathematically formulated before it can be solved. The complete solution process, therefore, consists primarily of the following three steps (see also Fig. 1). 1. Construction of a mathematical model: The variables involved must be carefully defined and the governing physical (or biological) laws identified. The mathematical model is usually some differential equation(s) representing an idealization of the laws, taking into account some simplifying assumptions in order to make the model tractable.

Therefore, the governing IVP is m dv = mg - 5v, v(0) = 100. dt (a) The DE is linear and the mass of the skydiver is m = 150/32 = 75/16 slugs (a "slug" is a unit of mass). In normal form, the IVP becomes dv16 + — v=32, v(0) = 100. 178. org/terms ORDINARY DIFFERENTIAL EQUATIONS 15 and the corresponding particular solution (28) is V p (t) = 32e -16 t1t5 f t o 16 s /15 ds = 30 - 30e - 16 t/ 15 0 from which we deduce v(t) = v H(t) + v(t) = 30 + 70e -16t/15 (b) The limiting or terminal velocity of the skydiver is v_ = lim v(t) = 30ft/s.

### Analysis 2 by Prof. Dr. Konrad Königsberger (auth.)

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