Geometry of Vector Fields
Presenting a classical approach to the foundations and development of the geometry of vector fields, this volume space, three orthogonal systems, and applications in mechanics. Other topics, including vector fields, Pfaff forms and systems in n-dimensional space, foliations and Godbillon-Vey invariant, are also considered. There is much interest in the study of geometrical objects in n-dimensional Euclidean space, and this volume provides a useful and comprehensive presentation.
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analogous arbitrary arc length arc length parameter asymptotic lines boundary Cartesian coordinates coefficients coincides components condition Consider const construct corresponding covariant derivatives curl curln curvature vector curvilinear coordinates defined Denote differential equations domain G equal Euclidean space expression exterior differential exterior product family of surfaces field n streamlines foliation formula fundamental form geodesic curvature geometrical given grad ip Hence holonomic homotopic Hopf invariant hypersurface integral intersection lemma linear manifold mapping ip matrix mean curvature mutually orthogonal n,curln neighborhood non-holonomicity value normal curvature obtain orientation parallel transport Pfaff equation plane orthogonal position vector principal curvatures principal directions proved respect Riemannian space satisfies second fundamental form second kind Section simplex singular points sphere S2 Suppose surface F symmetric polynomials system of coordinates tangent vector tensor THREE-DIMENSIONAL EUCLIDEAN SPACE total curvature triorthogonal unit sphere unit vector field zero