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This book introduces the graduate mathematician and researcher to the effective use of nonstandard analysis (NSA). It provides a tutorial introduction to this modern theory of infinitesimals, followed by nine examples of applications, including complex analysis, stochastic differential equations, differential geometry, topology, probability, integration, and asymptotics. It ends with remarks on teaching with infinitesimals. Written by members of the Reeb-school, and favouring typical examples, the results are new or give at least new insights into classical problems. Even if its first purpose is to help the readers to use NSA, it gives a taste of the research in progress in this domain.
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