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An in-depth survey of some of the most readily applicable essentials of modern mathematics, this concise volume is geared toward undergraduates of all backgrounds as well as future math majors. By focusing on relatively few fundamental concepts, the text delves deeply enough into each subject to challenge students and to offer practical applications. The opening chapter introduces the program of study and discusses how numbers developed. Subsequent...
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Induction in Geometry discusses the application of the method of mathematical induction to the solution of geometric problems, some of which are quite intricate. The book contains 37 examples with detailed solutions and 40 for which only brief hints are provided. Most of the material requires only a background in high school algebra and plane geometry; chapter six assumes some knowledge of solid geometry, and the text occasionally employs formulas...
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The first two chapters of this frequently cited reference provide background material in Riemannian geometry and the theory of submanifolds. Subsequent chapters explore minimal submanifolds, submanifolds with parallel mean curvature vector, conformally flat manifolds, and umbilical manifolds. The final chapter discusses geometric inequalities of submanifolds, results in Morse theory and their applications, and total mean curvature of a submanifold.
Suitable...
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Riemannian geometry is a fundamental area of modern mathematics and is important to the study of relativity. Within the larger context of Riemannian mathematics, the active subdiscipline of geodesics (shortest paths) in Riemannian spaces is of particular significance. This compact and self-contained text by a noted theorist presents the essentials of modern differential geometry as well as basic tools for the study of Morse theory. The advanced treatment...
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This highly regarded text is directed toward advanced undergraduates and graduate students in mathematics who are interested in developing a firm foundation in the theory of functions of a complex variable. The treatment departs from traditional presentations in its early development of a rigorous discussion of the theory of multiple-valued analytic functions on the basis of analytic continuation. Thus it offers an early introduction of Riemann surfaces,...
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Author Serge Lang defines algebraic geometry as the study of systems of algebraic equations in several variables and of the structure that one can give to the solutions of such equations. The study can be carried out in four ways: analytical, topological, algebraico-geometric, and arithmetic. This volume offers a rapid, concise, and self-contained introductory approach to the algebraic aspects of the third method, the algebraico-geometric. The treatment...
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The groundbreaking work of Russian mathematician A. M. Liapunov (1857–1918) on the stability of dynamical systems was overlooked for decades because of political turmoil. During the Cold War, when it was discovered that his method was applicable to the stability of aerospace guidance systems, interest in his research was rekindled. It has remained high ever since.
This monograph on both the theory and applications of Liapunov's direct method reflects...
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English
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This rich collection of fully worked problems in many areas of mathematics covers all the important subjects students are likely to encounter in their courses, from introductory to final-year undergraduate classes. Because lecture courses tend to focus on theory rather than examples, these exercises offer a valuable complement to classroom teachings, promoting the understanding of mathematical techniques and helping students prepare for exams. They...
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