By H. Keith Moffatt (auth.), Renzo L. Ricca (eds.)

ISBN-10: 1402002076

ISBN-13: 9781402002076

ISBN-10: 9401004463

ISBN-13: 9789401004466

Leading specialists current a different, necessary creation to the examine of the geometry and typology of fluid flows. From simple motions on curves and surfaces to the new advancements in knots and hyperlinks, the reader is progressively ended in discover the attention-grabbing international of geometric and topological fluid mechanics.

Geodesics and chaotic orbits, magnetic knots and vortex hyperlinks, continuous flows and singularities turn into alive with greater than one hundred sixty figures and examples.

within the establishing article, H. ok. Moffatt units the velocity, offering 8 awesome difficulties for the twenty first century. The e-book is going directly to supply innovations and strategies for tackling those and lots of different fascinating open problems.

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**Extra resources for An Introduction to the Geometry and Topology of Fluid Flows**

**Example text**

This map 'wraps' twice. ~ , deg = +2 deg = +1 I 6 , deg= 0 'Essentially' once. 2 Let f : M -. N be a map between closed oriented manifolds of equal dimension. e. at each of its inverse images det( of/ax) ::j:. O. The degree of f is degf = L xEf- 1 (yo) sgndet (~~) . 3 There is a theorem due to Sard guaranteeing that for every smooth f, vol {regular values of J} = volN; in other words, if we pick a value in N at random, it will be regular with probability 1. So there exist plenty of regular values to choose from.

K, self-crossing M", Tl ~ crossings. 5 If the deformation can be achieved without self-crossings, Po, PI are isotopic in M. 3, Ko and KI are not isotopic in IR3. 7 The concepts of homotopy and isotopy depend on the ambient space M. 3, if we replace M by IR4 , Ko and KI become isotopic. 4, if M = IR3 ~ ,G1 and G2 become homotopic (even isotopic). 8 In IR3 , these surfaces (E 2 ) are homotopic. Are they isotopic? 9 isotopic :::::} homotopic :::::} homeomorphic. The converses are false. 7), whereas 'homeomorphic' is intrinsic.

USA 88(4), 815-819. 8. B. (1978) Decomposition of the linking number of a closed ribbon: a problem from molecular biology. Proc Nat Acad. Sci. USA 75(8), 3557-3561. 9. F. (1833) Integral formula for linking number. In Zur Mathematischen Theorie de Electrodynamische Wirkungen. Collected Works 5, p. 605. Koniglichen Gesellschaft des Wissenschaften, Gottingen. 10. Milnor, J.. (1969) Topology from Differential Viewpoint. The University Press of Virginia, 2nd edition. 11. F. (1968) Some integral formulas for space curves and their generalization.

### An Introduction to the Geometry and Topology of Fluid Flows by H. Keith Moffatt (auth.), Renzo L. Ricca (eds.)

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