Applied Exterior Calculus [antikvár]

PREFACE The purpose of this book is to provide upper division undergraduate and beginning graduate students with access to the exterior calculus. Such access is essential simply because, much of the modern literature both in mathematics and in the quantified sciences has come to use the exterior calculus as an expository vehicle. Whole segments of the literature and, of greater importance, essential and often simplifying concepts are thus unintelligible to the student who is unfamiliar with the exterior calculus. Most texts that treat the exterior calculus are oriented toward global results and the needs of the research worker. The reader thus is confronted at the very beginning, and rightly so, with fundamental aspects of topology and the theory of differentiable manifolds. The demands on the student are great, a quantum jump in both sophistication and maturity often being self-evident from the start. A significant portion of the exterior calculus and its attendant concepts can be mastered, however, without recourse to global concepts. In fact, most of the theory can be developed using only local notions, and it can be done in such a way that it places only moderate demands on a student with previous exposure to upper division algebra and analysis courses. These priorities are the basis on which this book has been written. The analysis and discussions are confined from the outset to local questions. We start with standard «-dimensional number space and restrict attention to what happens in a single neighborhood of a point. It is even possible to restrict attention to what happens in a single neighborhood of a point that carries a single fixed coordinate cover. This, however, is adequate for discussion of tangents to curves and for the attachment of an «-dimensional vector space to each point of the neighborhood in a natural way. Tangent spaces and vector fields then follow as direct consequences, and a change of viewpoint leads to