While mathematics students generally meet the Riemann integral early in their undergraduate studies, those whose interests lie more in the direction of applied mathematics will probably find themselves needing to use the Lebesgue or Lebesgue-Stieltjes Integral before they have acquired the necessary theoretical background. This book is aimed at exactly this group of readers. The authors introduce the Lebesgue-Stieltjes integral on the real line as a natural extension of the Riemann integral, making the treatment as practical as possible. They discuss the evaluation of Lebesgue-Stieltjes...
While mathematics students generally meet the Riemann integral early in their undergraduate studies, those whose interests lie more in the direction o...
The fundamental theorem of algebra states that any complex polynomial must have a complex root. This book examines three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis and topology. The first proof in each pair is fairly straightforward and depends only on what could be considered elementary mathematics. However, each of these first proofs leads to more general results from which the fundamental theorem can be deduced as a direct consequence. These general results constitute the second proof in each pair. To arrive at each of the...
The fundamental theorem of algebra states that any complex polynomial must have a complex root. This book examines three pairs of proofs of the theore...
The calculus has been one ofthe areas of mathematics with a large number of significant applications since its formal development in the seventeenth century. With the recent development of the digital computer, the range of applications of mathematics, including the calculus, has increased greatly and now includes many disciplines that were formerly thought to be non- quantitative. Some of the more traditional applications have been altered, by the presence of a computer, to an extent such that many problems hitherto felt to be intractable are now solvable. This book has been written as a...
The calculus has been one ofthe areas of mathematics with a large number of significant applications since its formal development in the seventeenth c...
Unlike other texts in differential geometry, this book develops the architecture necessary to introduce symplectic and contact geometry alongside its Riemannian cousin. Emphasizes the consequences of a definition and the use of examples and constructions.
Unlike other texts in differential geometry, this book develops the architecture necessary to introduce symplectic and contact geometry alongside its ...
For over three decades, this best-selling classic has been used by thousands of students in the United States and abroad as a must-have textbook for a transitional course from calculus to analysis. It has proven to be very useful for mathematics majors who have no previous experience with rigorous proofs. Its friendly style unlocks the mystery of writing proofs, while carefully examining the theoretical basis for calculus. Proofs are given in full, and the large number of well-chosen examples and exercises range from routine to challenging.
The second edition preserves the book's...
For over three decades, this best-selling classic has been used by thousands of students in the United States and abroad as a must-have textbook fo...
While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory-uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself.
By...
While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the rea...
Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface.In truth, the most profound application of differential geometry is to modern physics, which is beyond the scope of this book.
Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface.In truth, the most profound applicati...
This monograph is an introduction to optimal control theory for systems governed by vector ordinary differential equations. It is not intended as a state-of-the-art handbook for researchers. We have tried to keep two types of reader in mind: (1) mathematicians, graduate students, and advanced undergraduates in mathematics who want a concise introduction to a field which contains nontrivial interesting applications of mathematics (for example, weak convergence, convexity, and the theory of ordinary differential equations); (2) economists, applied scientists, and engineers who want to...
This monograph is an introduction to optimal control theory for systems governed by vector ordinary differential equations. It is not intended as a st...
Transformation geometry is a relatively recent expression of the successful venture of bringing together geometry and algebra. The name describes an approach as much as the content. Our subject is Euclidean geometry. Essential to the study of the plane or any mathematical system is an under- standing of the transformations on that system that preserve designated features of the system. Our study of the automorphisms of the plane and of space is based on only the most elementary high-school geometry. In particular, group theory is not a prerequisite here. On the contrary, this modern approach...
Transformation geometry is a relatively recent expression of the successful venture of bringing together geometry and algebra. The name describes an a...
In the past decade there has been a significant change in the freshman/ sophomore mathematics curriculum as taught at many, if not most, of our colleges. This has been brought about by the introduction of linear algebra into the curriculum at the sophomore level. The advantages of using linear algebra both in the teaching of differential equations and in the teaching of multivariate calculus are by now widely recognized. Several textbooks adopting this point of view are now available and have been widely adopted. Students completing the sophomore year now have a fair preliminary under...
In the past decade there has been a significant change in the freshman/ sophomore mathematics curriculum as taught at many, if not most, of our colleg...
