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Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. [1] In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the symplectic form should allow one to obtain a vector field describing the flow of the system from the differential of a ...
In automotive engineering, an exhaust manifold collects the exhaust gases from multiple cylinders into one pipe. The word manifold comes from the Old English word manigfeald (from the Anglo-Saxon manig [many] and feald [fold]) [ 1 ] and refers to the folding together of multiple inputs and outputs (in contrast, an inlet or intake manifold ...
The homotopy type of a simply connected compact 4-manifold only depends on the intersection form on the middle dimensional homology. A famous theorem of Michael Freedman () implies that the homeomorphism type of the manifold only depends on this intersection form, and on a / invariant called the Kirby–Siebenmann invariant, and moreover that every combination of unimodular form and Kirby ...
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Let : be a smooth map of smooth manifolds. Given , the differential of at is a linear map : from the tangent space of at to the tangent space of at (). The image of a tangent vector under is sometimes called the pushforward of by .
Internal pressure acts upon the faces of the V, forcing the gasket to seal against the flange faces. Most spiral wound gasket applications will use two standard gasket thicknesses: 1/8 inch and 3/16 inch. With 1/8 inch thick gaskets, compression to a 0.100 inch thickness is recommended. For 3/16 inches, compress to a 0.13 inch thickness.
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Theorem: Every smooth manifold admits a (non-canonical) Riemannian metric. [13] This is a fundamental result. Although much of the basic theory of Riemannian metrics can be developed using only that a smooth manifold is a locally Euclidean topological space, for this result it is necessary to use that smooth manifolds are Hausdorff and paracompact.