THE MAIN INEQUALITY OF 3D VECTOR ANALYSIS

2004 ◽  
Vol 14 (01) ◽  
pp. 79-103 ◽  
Author(s):  
GILES AUCHMUTY
Keyword(s):  
Boundary Data ◽  
Vector Fields ◽  
Eigenvalue Problems ◽  
Three Dimensional ◽  
Orthogonal Decomposition ◽  
Vector Analysis ◽  
Mixed Case ◽  
Optimal Constants ◽  
Main Inequality

This paper proves some simple inequalities for Sobolev vector fields on nice bounded three-dimensional regions, subject to homogeneous mixed normal and tangential boundary data. The fields just have divergence and curl in L2. For the limit cases of prescribed zero normal, respectively zero tangential, data on the whole boundary, the inequalities were proved by Friedrichs who called the result the main inequality of vector analysis. For this mixed case, the optimal constants in the inequality are described, together with the fields for which equality holds. The detailed results depend on a special orthogonal decomposition and the analysis of associated eigenvalue problems.

Flow field around a vibrating cantilever: coherent structure eduction by continuous wavelet transform and proper orthogonal decomposition

2011 ◽  
Vol 669 ◽  
pp. 584-606 ◽  
Author(s):  
Y.-H. KIM ◽  
C. CIERPKA ◽  
S. T. WERELEY
Keyword(s):  
Wavelet Transform ◽  
Proper Orthogonal Decomposition ◽  
Continuous Wavelet Transform ◽  
Vector Fields ◽  
Three Dimensional ◽  
Orthogonal Decomposition ◽  
Continuous Wavelet ◽  
Vortical Structures ◽  
Flow Features ◽  
Proper Orthogonal

The velocity field around a vibrating cantilever plate was experimentally investigated using phase-locked particle image velocimetry. Experiments were performed at Reynolds numbers of Reh = 101, 126 and 146 based on the tip amplitude and the speed of the cantilever. The averaged vector fields indicate a pseudo-jet flow, which is dominated by vortical structures. These vortical structures are identified and characterized using the continuous wavelet transform. Three-dimensional flow features are also clearly revealed by this technique. Furthermore, proper orthogonal decomposition was used to investigate regions of vortex production and breakdown. The results show clearly that the investigation of phase-averaged data hides several key flow features. Careful data post-processing is therefore necessary to investigate the flow around the vibrating cantilever and similar highly transient periodic flows.

Limit cycles of three-dimensional polynomial vector fields

Nonlinearity ◽  
2004 ◽  
Vol 18 (1) ◽  
pp. 175-209 ◽  
Author(s):  
Marcin Bobie ski ◽  
Henryk o a dek
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Three Dimensional ◽  
Polynomial Vector ◽  
Polynomial Vector Fields

Reconstructing Three-Dimensional Fluid Velocity Vector Fields from Acoustic Transmission Measurements

1977 ◽  
pp. 307-326 ◽  
Author(s):  
S. A. Johnson ◽  
J. F. Greenleaf ◽  
C. R. Hansen ◽  
W. F. Samayoa ◽  
M. Tanaka ◽  
...  
Keyword(s):  
Velocity Vector ◽  
Vector Fields ◽  
Fluid Velocity ◽  
Three Dimensional ◽  
Acoustic Transmission ◽  
Transmission Measurements

Unitary vector fields are Fermi–Walker transported along Rytov–Legendre curves

2015 ◽  
Vol 12 (10) ◽  
pp. 1550111 ◽  
Author(s):  
Mircea Crasmareanu ◽  
Camelia Frigioiu
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Killing Vector ◽  
Three Dimensional ◽  
Ricci Soliton ◽  
Conformal Killing ◽  
Space Forms ◽  
Potential Vector ◽  
Legendre Curves ◽  
Complex Space Forms

Fix ξ a unitary vector field on a Riemannian manifold M and γ a non-geodesic Frenet curve on M satisfying the Rytov law of polarization optics. We prove in these conditions that γ is a Legendre curve for ξ if and only if the γ-Fermi–Walker covariant derivative of ξ vanishes. The cases when γ is circle or helix as well as ξ is (conformal) Killing vector filed or potential vector field of a Ricci soliton are analyzed and an example involving a three-dimensional warped metric is provided. We discuss also K-(para)contact, particularly (para)Sasakian, manifolds and hypersurfaces in complex space forms.

Twelve Limit Cycles in 3D Quadratic Vector Fields with Z3 Symmetry

2018 ◽  
Vol 28 (11) ◽  
pp. 1850139 ◽  
Author(s):  
Laigang Guo ◽  
Pei Yu ◽  
Yufu Chen
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Center Manifold ◽  
Three Dimensional ◽  
Center Manifold Theory ◽  
Normal Form Theory ◽  
Quadratic Vector Fields ◽  
Fine Focus ◽  
Form Theory ◽  
Manifold Theory

This paper is concerned with the number of limit cycles bifurcating in three-dimensional quadratic vector fields with [Formula: see text] symmetry. The system under consideration has three fine focus points which are symmetric about the [Formula: see text]-axis. Center manifold theory and normal form theory are applied to prove the existence of 12 limit cycles with [Formula: see text]–[Formula: see text]–[Formula: see text] distribution in the neighborhood of three singular points. This is a new lower bound on the number of limit cycles in three-dimensional quadratic systems.

scholarly journals On the symmetries of three-dimensional generalized symmetric spaces

2021 ◽  
Vol 13(62) (2) ◽  
pp. 451-462
Author(s):  
Lakehal Belarbi
Keyword(s):  
Symmetric Spaces ◽  
Vector Fields ◽  
Killing Vector ◽  
Three Dimensional ◽  
Riemannian Metric ◽  
Killing Vector Fields ◽  
Generalized Symmetric Space ◽  
Generalized Symmetric Spaces ◽  
Full Classification

In this work we consider the three-dimensional generalized symmetric space, equipped with the left-invariant pseudo-Riemannian metric. We determine Killing vector fields and affine vectors fields. Also we obtain a full classification of Ricci, curvature and matter collineations

scholarly journals ARBTools: A Tricubic Spline Interpolator for Three-Dimensional Scalar or Vector Fields

2019 ◽  
Author(s):  
Paul Walker ◽  
Ulrich Krohn ◽  
Carty David
Keyword(s):  
Magnetic Field ◽  
Vector Field ◽  
Vector Fields ◽  
Three Dimensional ◽  
Spline Method ◽  
Particle Trajectories ◽  
The Core ◽  
Numerical Integrators ◽  
Data Points ◽  
System Requirements

ARBTools is a Python library containing a Lekien-Marsden type tricubic spline method for interpolating three-dimensional scalar or vector fields presented as a set of discrete data points on a regular cuboid grid. ARBTools was developed for simulations of magnetic molecular traps, in which the magnitude, gradient and vector components of a magnetic field are required. Numerical integrators for solving particle trajectories are included, but the core interpolator can be used for any scalar or vector field. The only additional system requirements are NumPy.

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