| dc.contributor | SABINO CHAVEZ CERDA | |
| dc.creator | JOSE ADAN HERNANDEZ NOLASCO | |
| dc.date | 2012-06 | |
| dc.date.accessioned | 2018-11-19T14:26:19Z | |
| dc.date.available | 2018-11-19T14:26:19Z | |
| dc.identifier | http://inaoe.repositorioinstitucional.mx/jspui/handle/1009/289 | |
| dc.identifier.uri | http://repositorioslatinoamericanos.uchile.cl/handle/2250/2258434 | |
| dc.description | The Scalar Helmholtz Equation is separable in eleven orthogonal coordinate systems. Despite
being one of the most studied equations in Mathematical Physics, this fact is not mentioned
and the whole story is hard to find in the classical textbooks or even specialized literature.
Moreover, even when mentioned, the geometry of some of those coordinate systems is not
properly illustrated in the literature.
In order to transform the differential operator of the Helmholtz equation the
corresponding metric for the curvilinear coordinates is used. For some of these, after
performing the transformation, the resulting equation can be very cumbersome and the
method of separation of variables does not apply in a simple and straightforward way. This
leads to look for an alternative method also barely known that is the Stackel determinant, this
allows to get the three separated differential equations for each of the curvilinear coordinates.
The method can be used for the eleven coordinate systems. From which emerge fifteen
different differential equations to solve.
To represent the whole families of scalar wave fields it is necessary to solve each of the
three equations for each one of the curvilinear coordinates. In most of the cases the solutions
are given in terms of not widely known special functions and in some cases their numerical
evaluation is not an easy task.
We will present in an understandable and visual way the eleven coordinate systems
in which the Helmholtz equation is separable. We will show the surfaces associated with
them, whose intersections determine univocally a point in a three dimensional space. We
will present the normalization in a canonical form, the corresponding sets of ordinary
differential equations resulting from the separation of the Helmholtz equation, providing
a graphical picture of the fundamental solutions that enable the representation of radiating
electromagnetic wave fields for each of the curvilinear coordinate systems. | |
| dc.format | application/pdf | |
| dc.language | spa | |
| dc.publisher | Instituto Nacional de Astrofísica, Óptica y Electrónica | |
| dc.relation | citation:Hernández-Nolasco J.A. | |
| dc.rights | info:eu-repo/semantics/openAccess | |
| dc.rights | http://creativecommons.org/licenses/by-nc-nd/4.0 | |
| dc.subject | info:eu-repo/classification/Óptica de onda/Wave optics | |
| dc.subject | info:eu-repo/classification/Ecuaciones de onda/Wave equations | |
| dc.subject | info:eu-repo/classification/Matemáticas/Mathematics | |
| dc.subject | info:eu-repo/classification/Propagación de onda/Wave propagation | |
| dc.subject | info:eu-repo/classification/cti/1 | |
| dc.subject | info:eu-repo/classification/cti/22 | |
| dc.subject | info:eu-repo/classification/cti/2209 | |
| dc.title | Familias de campos ondulatorios fundamentales de la ecuación de Helmholtz en sistemas de coordenadas curvilíneas ortogonales | |
| dc.type | Tesis | |
| dc.type | info:eu-repo/semantics/acceptedVersion | |
| dc.audience | students | |
| dc.audience | researchers | |
| dc.audience | generalPublic | |
