Differential geometry

Synthetic differential geometry

In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory. There are several insights that allow for such a reformulation. The first is that most of the analytic data for describing the class of smooth manifolds can be encoded into certain fibre bundles on manifolds: namely bundles of jets (see also jet bundle). The second insight is that the operation of assigning a bundle of jets to a smooth manifold is functorial in nature. The third insight is that over a certain category, these are representable functors. Furthermore, their representatives are related to the algebras of dual numbers, so that smooth infinitesimal analysis may be used. Synthetic differential geometry can serve as a platform for formulating certain otherwise obscure or confusing notions from differential geometry. For example, the meaning of what it means to be natural (or invariant) has a particularly simple expression, even though the formulation in classical differential geometry may be quite difficult. (Wikipedia).

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From playlist Differential Geometry

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From playlist Differential Equations

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From playlist Popular Questions

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From playlist Differential Equations

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In this video I describe the logic of Synthetic Differential Geometry. This is a non-constructive theory collapsing in the presence of the law of excluded middle. As a logic al theory, it can be realized in a topos and it has sheave models giving a nice representation of tangent bundles.

From playlist Algebra

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This video defines a differential equation and then classifies differential equations by type, order, and linearity. Search Library at http://mathispower4u.wordpress.com

From playlist Introduction to Differential Equations

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From playlist Differential Geometry

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From playlist Solve Differential Equation (Particular Solution) #Integration

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From playlist Mathematics

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From playlist Differential Geometry

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Filiz Dogru: Outer Billiards: A Comparison Between Affine, Hyperbolic, and Symplectic Geometry

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From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022

Related pages

Functor | Smooth infinitesimal analysis | Mathematics | Differential geometry | Jet bundle | Jet (mathematics) | Representable functor | Category theory