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A plane containing a cross-section of the solid may be referred to as a cutting plane. The shape of the cross-section of a solid may depend upon the orientation of the cutting plane to the solid. For instance, while all the cross-sections of a ball are disks, [2] the cross-sections of a cube depend on how the cutting plane is related to the ...
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In the mathematical field of topology, a section (or cross section) [1] of a fiber bundle is a continuous right inverse of the projection function. In other words, if E {\displaystyle E} is a fiber bundle over a base space , B {\displaystyle B} :
If we take the other projection pr 2 : M × N → N : (x, y) → y to define the fiber bundle B 2 := (M × N, pr 2) then the vertical bundle will be VB 2 = TM × N. In both cases, the product structure gives a natural choice of horizontal bundle, and hence an Ehresmann connection: the horizontal bundle of B 1 is the vertical bundle of B 2 and ...
A principal -bundle, where denotes any topological group, is a fiber bundle: together with a continuous right action such that preserves the fibers of (i.e. if then for all ) and acts freely and transitively (meaning each fiber is a G-torsor) on them in such a way that for each and , the map sending to is a homeomorphism.
As he observed, for most such sections the cross section consists of either one or two ovals; however, when the plane is tangent to the inner surface of the torus, the cross-section takes on a figure-eight shape, which Proclus called a horse fetter (a device for holding two feet of a horse together), or "hippopede" in Greek. [8]
A two-dimensional Poincaré section of the forced Duffing equation. In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensional subspace, called the Poincaré section, transversal to the flow of the system.
Similarly, the natural monomorphism Z/2Z → Z/4Z doesn't split even though there is a non-trivial morphism Z/4Z → Z/2Z. The categorical concept of a section is important in homological algebra, and is also closely related to the notion of a section of a fiber bundle in topology: in the latter case, a section of a fiber bundle is a section of ...