Search results
Results from the WOW.Com Content Network
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.
Diagram of an exhaust manifold from a Kia Rio. 1. manifold; 2. gasket; 3. nut; 4. heat shield; 5. heat shield bolt Ceramic-coated exhaust manifold on the side of a performance car. In automotive engineering, an exhaust manifold collects the exhaust gases from multiple cylinders into one pipe.
In mathematics, differential topology is the field dealing with the topological properties and smooth properties [a] of smooth manifolds.In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the geometric properties of smooth manifolds, including notions of size, distance, and rigid shape.
For manifolds of dimension at most 6, any piecewise linear (PL) structure can be smoothed in an essentially unique way, [2] so in particular the theory of 4 dimensional PL manifolds is much the same as the theory of 4 dimensional smooth manifolds. A major open problem in the theory of smooth 4-manifolds is to classify the simply connected ...
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
Important to applications in mathematics and physics [1] is the notion of a flow on a manifold. In particular, if M {\displaystyle M} is a smooth manifold and X {\displaystyle X} is a smooth vector field , one is interested in finding integral curves to X {\displaystyle X} .