Search results for "Direction vector"

showing 4 items of 4 documents

Vectors and Vector Fields

2012

The purpose of this book is to explain in a rigorous way Stokes’s theorem and to facilitate the student’s use of this theorem in applications. Neither of these aims can be achieved without first agreeing on the notation and necessary background concepts of vector calculus, and therein lies the motivation for our introductory chapter.

AlgebraSolenoidal vector fieldStandard basisPhysics::Physics EducationVector fieldCross productDirection vectorVector calculusComplex lamellar vector fieldCauchy–Schwarz inequalityMathematics
researchProduct

Biorthonormal-basis method for the vector description of optical-fiber modes

1998

This paper gives the theoretical basis for the development of real vector modal methods to describe optical-fiber modes. To this end, the vector wave equations, which determine the electromagnetic fields, are written in terms of a pair of linear, nonself-adjoint operators, whose eigenvectors satisfy biorthogonality relations. The key of our method is to obtain a matrix representation of the vector wave equations in a basis that is defined by the modes of an auxiliary system. Our proposed technique can be applied to fibers with any profile, even those with a complex refractive index. An example is discussed to illustrate our approach.

Electromagnetic fieldNormal modeMathematical analysisMatrix representationCalculusPolarization (waves)Wave equationDirection vectorAtomic and Molecular Physics and OpticsEigenvalues and eigenvectorsVector potentialMathematicsJournal of Lightwave Technology
researchProduct

A poincar�-bendixson theorem for analytic families of vector fields

1995

We provide a characterization of the limit periodic sets for analytic families of vector fields under the hypothesis that the first jet is non-vanishing at any singular point. Also, applying the family desingularization method, we reduce the complexity of some of these sets.

Pure mathematicsVector measureSolenoidal vector fieldJet (mathematics)General MathematicsMathematical analysisVector fieldSingular point of a curveDirection vectorPoincaré–Bendixson theoremMathematicsVector potentialBoletim da Sociedade Brasileira de Matem�tica
researchProduct

A method of desingularization for analytic two-dimensional vector field families

1991

It is well known that isolated singularities of two dimensional analytic vector fields can be desingularized: after a finite number of blowing up operations we obtain a vector field that exhibits only elementary singularities. In the present paper we introduce a similar method to simplify the periodic limit sets of analytic families of vector fields. Although the method is applied here only to reduce to families in which the zero set has codimension at least two, we conjecture that it can be used in general. This is related to the famouss Hibert's problem about planar vector fields.

Vector calculus identitiesCurl (mathematics)Solenoidal vector fieldVector operatorGeneral MathematicsMathematical analysisFundamental vector fieldDirection vectorComplex lamellar vector fieldMathematicsVector potentialBoletim da Sociedade Brasileira de Matem�tica
researchProduct