China
Africa
Explore chapters and articles related to this topic
Differential geometry
Published in Louis Komzsik, Applied Calculus of Variations for Engineers, 2019
Differential geometry is a classical mathematical area that has become very important for engineering applications in the recent decades. This importance is based on the rise of computer-aided visualization and geometry generation technologies.
Differential geometry
Published in Louis Komzsik, Applied Calculus of Variations for Engineers, 2018
Differential geometry is a classical mathematical area that has become very important for engineering applications in the recent decades. This importance is based on the rise of computer-aided visualization and geometry generation technologies.
Orthogonal Expansions in Curvilinear Coordinates
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
In an effort to extend the concepts of length, volume, and curvature to the most general scenarios, mathematicians toward the end of the 19th century formulated the geometry of higher dimensional curved spaces without regard to whether or not these spaces fit in a higher dimensional Euclidean space.3 Roughly speaking, the only properties that were found to be important were: (1) that the curved space “locally looks like” ℝm; (2) that it is possible to construct a set of overlapping patches in the curved space in such a way that coordinates can be imposed on each patch; and (3) that the description of points in the overlapping patches are consistent with respect to the different parameterizations that are defined in the overlapping regions. In mathematics, the study of curved spaces is called differential geometry. Smooth curved spaces that for all points look locally like Euclidean space are called manifolds.4 Each manifold can be covered with a collection of overlapping patches, in which a local coordinate chart relates a point in the manifold to one in a copy of ℝM. The collection of all of the coordinate charts is called an atlas. The inverse of a coordinate chart gives the parameterization of a patch in the manifold. Precise definitions of these terms can be found in any of a number of excellent books on differential geometry, and a short review is provided in Appendix G. It is assumed from the beginning that all of the manifolds of interest in this book can be embedded in ℝNfor sufficiently large N.Hence, we need not use formal methods of coordinate-free differential geometry. Rather, it suffices to use the methods described previously for surfaces in higher dimensions, with the one modification described below.
Using design to develop an in-depth understanding of mathematical surfaces: The hairy Klein bottle
Published in International Journal of Mathematical Education in Science and Technology, 2021
Differential geometry is another discipline that uses calculus and linear algebra to explore curves and surfaces from a geometric point of view, i.e. quantitative properties such as curvature, fundamental forms, and others. It turns out that these two topics (topology and deferential geometry) that seem so different can ‘communicate’ with each other. One of the famous results is Gauss–Bonnet theorem, which determines the total of the Gauss curvature (a geometric property) using the Euler characteristic number of a surface (a topologic property) (for more details, see Lang, 2001).
The Dirac equation as a model of topological insulators
Published in Philosophical Magazine, 2020
Xiao Yuan, M. Bowen, P. S. Riseborough
The Gauss–Bonnet Theorem links differential geometry to the topology. It states that for a two-dimensional manifold with a boundary the Euler characteristic is given bywhere κ is the curvature and is the geodesic curvature. The Euler characteristic for a surface is a topological invariant given bywhere g, the genus, is equal to the number of holes. The Chern Theorem [26,27] is a generalisation of the Gauss–Bonnet Theorem to -dimensional manifolds. The Chern number for a band indexed by τ is related to the Berry phase. The Berry phase is an integral of the Berry curvature over an open surface, ie one that has a boundary. The Chern number (of the first kind) is defined as an integral over a closed, orientable, two-dimensional manifold and leads to integer topological invariants. For a crystalline lattice for which Bloch's Theorem applies, an integral over a two-dimensional Brillouin zone is equivalent to an integral over the closed surface of a torus (see Figure 8). For a solid which is time-reversal invariant the Chern number is zero. A non-zero Chern number indicates that the wave function does not have a smoothly varying and uniquely defined phase at each point of the closed surface. This may happen whenever the wave function goes to zero at a point, since the Berry phase of a closed contour encircling the point may change by multiples of . Hence, the Chern number describes the net number of vortices passing through the surface. For a manifold with boundaries, the integral of the Berry curvature is equal (modulo ) to the sum of the Berry phases around the boundaries. However, each Berry phase contribution from a boundary has an equivalent contribution from the Kramer's conjugate boundary. Hence, the index, through the Atiyah–Singer index Theorem [28], can be defined in terms of an integration over half the area of the Brillouin zone and the boundaries around the singularities. The volume of integration is chosen such that and its Kramer's conjugate point are never included in the area. The index is given byHence, has values of .