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Let M be a smooth manifold. A (smooth) singular k-simplex in M is defined as a smooth map from the standard simplex in R k to M. The group C k (M, Z) of singular k-chains on M is defined to be the free abelian group on the set of singular k-simplices in M. These groups, together with the boundary map, ∂, define a chain complex.
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 ...
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle, length of curves, surface area and volume.
Download as PDF; Printable version; In other projects Wikidata item; ... Pages in category "Smooth manifolds" The following 19 pages are in this category, out of 19 ...
It follows that is a regular value of , so () and its quotient / are both smooth manifolds. The quotient inherits a symplectic form from M {\displaystyle M} ; that is, there is a unique symplectic form on the quotient whose pullback to μ − 1 ( 0 ) {\displaystyle \mu ^{-1}(0)} equals the restriction of ω {\displaystyle \omega } to μ − 1 ...
Important to applications in mathematics and physics [1] is the notion of a flow on a manifold. In particular, if is a smooth manifold and is a smooth vector field, one is interested in finding integral curves to .
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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 ...