![]() In the 1940s and beyond, she said consolidation saw schools in smaller towns lose out in favor of those in larger towns, like State Center, Le Grand and, in her case, Marshalltown. She said the first wave of consolidations in the late 1800s and early 1900s came as a result of country schools banding together. Lang said Van Cleve, just like some communities in the West Marshall, East Marshall and GMG school districts, could not keep up with the educational and facilities needs required by the state for student success. That small town was eventually added to the Marshalltown School District and its local building closed. Lang said she grew up in Van Cleve, east of Melbourne. “There are pros and cons like there is in everything.” It is not too far-fetched to argue that differential geometry should be in every mathematician's arsenal.“The small town people, like we were … thought consolidation was probably not good for us because we would lose our school,” said 34-year Marshalltown Schools teacher and current Green Mountain-Garwin substitute teacher Julie Lang. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. ![]() ![]() Additionally, in an attempt to make the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior power are included.ĭifferential geometry, as its name implies, is the study of geometry using differential calculus. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. A knowledge of de Rham cohomology is required for the last third of the text. After the first chapter, it becomes necessary to understand and manipulate differential forms. Initially, the prerequisites for the reader include a passing familiarity with manifolds. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. This text presents a graduate-level introduction to differential geometry for mathematics and physics students.
0 Comments
Leave a Reply. |