By H. Keith Moffatt (auth.), Renzo L. Ricca (eds.)
Leading specialists current a different, necessary advent to the examine of the geometry and typology of fluid flows. From simple motions on curves and surfaces to the hot advancements in knots and hyperlinks, the reader is steadily resulted 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 beginning article, H. ok. Moffatt units the speed, providing 8 remarkable difficulties for the twenty first century. The ebook is going directly to offer suggestions and strategies for tackling those and plenty of different attention-grabbing open problems.
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Extra info for An Introduction to the Geometry and Topology of Fluid Flows
Aref, D. Mackay, R. Ricca, and E. Spiegel for their kind hospitality in Cambridge and generous feedback. 55 References Bott, R. & Tu, 1. (1982) Differential Forms in Algebraic Topology. Springer. T. P. (1984-1990) Modern Geometry: Methods and Applications. Volumes I-III, Springer. 3. Guillemin, V. & Pollack, A. (1974) Differential Topology. Prentice-Hall. 4. Milnor, J. (1965) Topology from the Differentiable Viewpoint. University Press Virginia. 5. V. (1994) Intuitive Topology. Amererican Mathematical Society.
4). 5 The standard volume form on sn-l can be written n n = 2) -l)kxkdxl A··· A ~ A··· A dx n . 6 If v is the outward unit normal vector field on a hypersurface H of IIr', then degH v = vOl(~n-l) k J( dO' , where J( = Gaussian curvature (product of principal curvatures) of H, and dO' = (n - 1 )-dimensional volume element on H induced by the ambient euclidean metric on IRn. Proof. de gH v = r deg G n vol(sn-l) lsn-l = K large: } normals dispersed ~ ! , __ : -# l K small: I normals aligned r r 1 G*n = 1 J( dO' .
16 Construct an everywhere nonzero vector field on each sodd. 17 Can S4 carry a Minkowski metric without any singularities? 18 For odd-dimensional, closed, orientable M, X(M) = O. Proof. 13. QED 7. Degree Theory Once we discover that solutions or equilibria exist (sections 5 and 6), the next issue is to count their number. 1 @ ~ ~. ; . i i! i ...... ,,~- = -3 Pi , \ I' . I . c::: )""): t deg 'Wraps against' thrice. , , , . , :~ This map 'wraps' twice. ~ , deg = +2 deg = +1 I 6 , deg= 0 'Essentially' once.