![]() Volkswagen first started using this type of design in 2013, as did Ford with the EcoBoost three- and four-cylinder engines. Lowering exhaust temps helps extend the life of the catalytic converter, too, and the short distance between exhaust valves and turbocharger improves throttle response and power output. It also reduces exhaust temperatures, a task that's usually accomplished by running a richer (and thus less economical) air/fuel ratio. Plunging the hot exhaust passages through the cylinder head's coolant paths allows a cold engine to get up to operating temperature much more quickly, reducing wear and improving fuel economy. Sounds unusual-but it's a nifty solution with real performance and efficiency benefits, as Jason Fenske of YouTube's Engineering Explained will now elucidate. Structure is called a symplectic manifold.When Volkswagen's third-generation EA888 1.8-liter turbo four-cylinder engine came out, we noted a particularly unique feature that's somewhat uncommon in passenger car engines: The exhaust manifold is integrated into the cylinder head, with coolant passages routing water around the exhaust runners. Riemannian manifold, and one with a symplectic Or even algebraic (in order of specificity).Ī smooth manifold with a metric is called a For example, it could be smooth, complex, In fact, Whitney showed in the 1930s that any manifold can be embeddedĪ manifold may be endowed with more structure than a locally Euclidean topology. Many common examples of manifolds are submanifolds of Euclidean space. For example, the equator of a sphere is a ManifoldsĪre therefore of interest in the study of geometry,Ī submanifold is a subset of a manifold that is itself a manifold, but has smaller dimension. Of a subset of Euclidean space, like the circle or the sphere, is a manifold. The basic example of a manifold is Euclidean space, and many of its properties carry over to manifolds. The objects that crop up are manifolds.įrom the geometric perspective, manifolds represent the profound idea having to do Of a robot arm or all the possible positions and momenta of a rocket, an object is Objects." For example, in order to precisely describe all the configurations ManifoldsĪrise naturally in a variety of mathematical and physical applications as "global Meaning that the inverse of one followed by the other is an infinitely differentiable Smooth manifolds (also called differentiable manifolds) are manifolds for which overlapping charts "relate smoothly" to each other, Unless otherwise indicated, a manifold is assumed to have In addition, a manifold must have a secondĬountable topology. Of that neighborhood with an open ball in. The concept can be generalized to manifolds with corners.īy definition, every point on a manifold has a neighborhood together with a homeomorphism If a manifold contains its own boundary, it is called, not surprisingly, a " manifold with boundary." The closed unitīall in is a manifold with boundary, and itsīoundary is the unit sphere. Manifold without boundary or closed manifold for However, an author will sometimes be more preciseĪnd use the term open manifold for a noncompact Commonly, the unqualified term "manifold"is used to mean Topologically the same as the surface of the donut, and this type of surface is calledĪs a topological space, a manifold can be compact or noncompact, and connected Similarly, the surface of a coffee mug with a handle is For instance, a circle is topologically the same as any closed loop, no matter how different these One of the goals of topology is to find ways of distinguishing manifolds. More concisely, any object that can be "charted" is a manifold. We would encounter the round/flat Earth problem, as first codified by Poincaré. General, any object that is nearly "flat" on small scales is a manifold,Īnd so manifolds constitute a generalization of objects we could live on in which The fact that on the small scales that we see, the Earth does indeed look flat. With the modern evidence that it is round. To illustrate this idea, consider the ancient belief that the Earth was flat as contrasted Is topologically the same as the open unit A manifold is a topological space that is locally Euclidean (i.e.,Īround every point, there is a neighborhood that
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