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For example, for differential geometry, the top-level code is 53, and the second-level codes are: A for classical differential geometry; B for local differential geometry; C for global differential geometry; D for symplectic geometry and contact geometry; In addition, the special second-level code "-" is used for specific kinds of materials.
Differential geometry is also indispensable in the study of gravitational lensing and black holes. Differential forms are used in the study of electromagnetism. Differential geometry has applications to both Lagrangian mechanics and Hamiltonian mechanics. Symplectic manifolds in particular can be used to study Hamiltonian systems.
See also multivariable calculus, list of multivariable calculus topics. Manifold. Differentiable manifold; Smooth manifold; Banach manifold; Fréchet manifold; Tensor analysis. Tangent vector
The differential-geometric properties of a parametric curve (such as its length, its Frenet frame, and its generalized curvature) are invariant under reparametrization and therefore properties of the equivalence class itself. The equivalence classes are called C r-curves and are central objects studied in the differential geometry of curves.
A Comprehensive Introduction to Differential Geometry, [16] [17] (1970, revised 3rd ed. 2005) The Joy of TeX: A Gourmet Guide to Typesetting with the AMS-TeX Macro package, (1990) A Hitchhiker's Guide to the Calculus, [18] (1995) Spivak, Michael (2010). Physics for mathematicians—Mechanics I. Houston, TX: Publish or Perish. ISBN 978-0-914098 ...
Calibrated geometry; Cartan connection; Cartan's equivalence method; Catalan's minimal surface; Caustic (mathematics) Cayley's ruled cubic surface; Center of curvature; Chentsov's theorem; Chern–Simons form; Chern–Weil homomorphism; Chern's conjecture (affine geometry) Chern's conjecture for hypersurfaces in spheres; Clairaut's relation ...
In the language of differential geometry, this derivative is a one-form on the punctured plane. It is closed (its exterior derivative is zero) but not exact , meaning that it is not the derivative of a 0-form (that is, a function): the angle θ {\\displaystyle \\theta } is not a globally defined smooth function on the entire punctured plane.
The differential geometry of surfaces revolves around the study of geodesics. It is still an open question whether every Riemannian metric on a 2-dimensional local chart arises from an embedding in 3-dimensional Euclidean space: the theory of geodesics has been used to show this is true in the important case when the components of the metric ...