A. Globus,C. Levit,and T. Lasinski RNR Technical Report RNR-91-017
April,1991
We describe a software system,TOPO,that numerically analyzes and graphically displays topological aspects of a three dimensional vector field v to produce a single,relatively simple picture that characterizes v. The topology of v that we consider consists of its critical points (where v+0) their invariant manifolds, and the integral curves connecting these invariant manifolds.Many of the interesting features of v are associated with its critical point as approximated by the Taylor expansion.The coefficients of the first non-zero term of the Taylor expansion around a crucial point are the 3x3 matrix v. Critical points are classified by examining v's eigenvalues. The eigenvectors of v span the invariant manifolds of the linearized field around a critical point.Curves integrated from initial points on the eigenvectors a small distance from a critical point connect with other critical points (or the boundary) to complete the topology. In addition,one class of critical surfaces important in computational fluid dynamics is analysed.
I received this message correcting the references:
I am senior scientist in a french university of south France and I have found your article entitled: "A Tool for Visualizing the Topology of Three-Dimensional Vector Fields" I just would like to tell you that the references for Poincare that you give are wrong.
Henri Poincare:
[1881] Poincare, H., "Sur les courbes definies par une equation differentielle," J. de Math. Pures et Appl., Serie III, 7, 375-422.
[1882] Poincare, H., "Sur les courbes definies par une equation differentielle," J. de Math Pures Appl., Serie III, 8, 251-296.
[1885] Poincare, H., "Sur les courbes definies par une equation differentielle," J. de Math. Pures et Appl., Serie IV, 1, 167-244.
[1886] Poincare, H., "Sur les courbes definies par une equation differentielle," J. de Math. Pures et Appl., Serie IV, 2, 151-217.
This is of great importance since in 1875 Poincare had no Ph-D. He had defended his thesis in 1879.
Sincerely
Dr. Jean-Marc Ginoux
Maitre de Conferences en Mathematiques Appliquees
Departement de Genie Mecanique et Productique
I.U.T. de Toulon, Universite du Sud Toulon Var
Laboratoire PROTEE, equipe EBMA
B.P. 20132, 83957 La Garde cedex, France