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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.
See also multivariable calculus, list of multivariable calculus topics. Manifold. Differentiable manifold; Smooth manifold; Banach manifold; Fréchet manifold; Tensor analysis. Tangent vector
Calculus on Manifolds is a brief monograph on the theory of vector-valued functions of several real variables (f : R n →R m) and differentiable manifolds in Euclidean space. . In addition to extending the concepts of differentiation (including the inverse and implicit function theorems) and Riemann integration (including Fubini's theorem) to functions of several variables, the book treats ...
In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas [1]) are fundamental formulas that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold.
His five-volume A Comprehensive Introduction to Differential Geometry [11] is among his most influential and celebrated works. The distinctive pedagogical aim of the work, as stated in its preface, was to elucidate for graduate students the often obscure relationship between classical differential geometry—geometrically intuitive but imprecise—and its modern counterpart, replete with ...
In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces.
Some of his books are: Introduction to Topology and Modern Analysis (1963) [7] Differential Equations with Applications and Historical Notes (1972, 1991, 2016) [8] Precalculus Mathematics in a Nutshell (1981) [9] Calculus with Analytic Geometry (1985, 1996) [10] Calculus Gems: Brief Lives and Memorable Mathematics (1992) [11]
Hopf himself was motivated by problems from physics. When Hopf started to work in the mid 1920s, the theory of relativity was only 10 years old and it sparked a great deal of interest in differential geometry, especially in global structure of 4-manifolds, as such manifolds appear in cosmology as models of the universe.