By H. Keith Moffatt (auth.), Renzo L. Ricca (eds.)
Leading specialists current a different, valuable creation to the research of the geometry and typology of fluid flows. From easy motions on curves and surfaces to the hot advancements in knots and hyperlinks, the reader is progressively 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 a hundred and sixty figures and examples.
within the commencing article, H. okay. Moffatt units the velocity, providing 8 impressive difficulties for the twenty first century. The ebook is going directly to supply ideas and methods for tackling those and lots of different fascinating open problems.
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Additional resources for An Introduction to the Geometry and Topology of Fluid Flows
12). g. seven cannot carry an everywhere non-zero vector field. 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.
9. Suggested Reading A beautiful book at a similar level as these lectures is .  is a systematic textbook.  is semi-popular and entertaining. For further topics in topology,  is recommendedj  is heftier and covers more. For fluid dynamicists, a natural sequel to these 4 days would be infinite-dimensional topological methods in partial differential equationsj they are nicely explained in . The practical calculation of topological invariants requires algebraic machinesj an efficient little book is .
12. 13 then implies that every closed hypersurface in IRn is 2-sided, hence orientable. 13 hold also on (d. 12). 9), cannot be realized as a surface in IR3 without self-intersections. It is tempting to picture manifolds as high-dimensional surfaces floating in some space. 15 shows that we must be careful. How much elbow room do we need to legitimize our picture? 16 (Whitney) Every closed manifold of dimension n can be realized as a submanifold of m2n. 5. Fixed Point Theorems Solving any species of equation (differential, integral, algebraic ...