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Differential geometry finds applications throughout mathematics and the natural sciences. Most prominently the language of differential geometry was used by Albert Einstein in his theory of general relativity, and subsequently by physicists in the development of quantum field theory and the standard model of particle physics.
Download as PDF; Printable version; ... is a major result of differential geometry, ... Elementary Differential Geometry. New York: Academic Press. pp. 271–275.
In 1950 Struik published his Lectures on Classical Differential Geometry, [13] which gained praise from Ian R. Porteous: Of all the textbooks on elementary differential geometry published in the last fifty years the most readable is one of the earliest, namely that by D.J. Struik (1950). He is the only one to mention Allvar Gullstrand. [14]
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.
The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems and root data.
In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor.
In mathematics, differential refers to several related notions [1] derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions. [2] The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology.
The linear operator which assigns to each function its derivative is an example of a differential operator on a function space. By means of the Fourier transform, pseudo-differential operators can be defined which allow for fractional calculus. Some of these operators are so important that they have their own names:
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