Topic
Elliptic coordinate system
About: Elliptic coordinate system is a research topic. Over the lifetime, 670 publications have been published within this topic receiving 11135 citations. The topic is also known as: elliptical coordinate system & elliptic coordinates.
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01 Jan 1958
TL;DR: In this introductory chapter, the basic notions which are essential to a geometrical picture of the significance of the differential calculus are reviewed.
Abstract: In this introductory chapter we briefly review the basic notions which are essential to a geometrical picture of the significance of the differential calculus; thus this chapter is not concerned explicitly with the calculus and is intended for revision and reference.
1 citations
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TL;DR: In this article, the equations of motion for the spatial circular restricted three-body problem in sidereal spherical coordinates system were established, and an initial value procedure that can be used to compute both the spherical and Cartesian sidereal coordinates and velocities was also developed.
Abstract: In this paper of the series, the equations of motion for the spatial circular restricted three-body problem in sidereal spherical coordinates system were established. Initial value procedure that can be used to compute both the spherical and Cartesian sidereal coordinates and velocities was also developed. The application of the procedure was illustrated by numerical example and graphical representations of the variations of the two sidereal coordinate systems.
1 citations
01 Jan 2009
1 citations
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18 Aug 2008
TL;DR: The Laplace transform analytic element method (LT-AEM) as discussed by the authors was used to derive analytic solutions to the modified Helmholtz equation and then back-transforms the results with a numerical inverse Laplace transformation algorithm.
Abstract: The Laplace transform analytic element method
(LT-AEM), applies the traditionally steady-state
analytic element method (AEM) to the
Laplace-transformed diffusion equation (Furman and
Neuman, 2003). This strategy preserves the accuracy
and elegance of the AEM while extending the method to
transient phenomena. The approach taken here
utilizes eigenfunction expansion to derive analytic
solutions to the modified Helmholtz equation, then
back-transforms the LT-AEM results with a numerical
inverse Laplace transform algorithm. The
two-dimensional elements derived here include the
point, circle, line segment, ellipse, and infinite
line, corresponding to polar, elliptical and
Cartesian coordinates. Each element is derived for
the simplest useful case, an impulse response due to
a confined, transient, single-aquifer source. The
extension of these elements to include effects due to
leaky, unconfined, multi-aquifer, wellbore storage,
and inertia is shown for a few simple elements (point
and line), with ready extension to other elements.
General temporal behavior is achieved using
convolution between these impulse and general time
functions; convolution allows the spatial and
temporal components of an element to be handled
independently.
1 citations
15 May 2011
TL;DR: In this paper, the potential vorticity equation and expression in the isobaric and isoentropic coordinates are obtained via coordinate transformation with the two methods, starting from the three-dimensional vector motion equation, and then combining with the thermodynamic equation.
Abstract: The potential vorticity theory and diagnostic techniques are based on the potential vorticity equation and expression in the common meteorological coordinate systems. In this paper, the potential vorticity equation and expression in the isobaric and isoentropic coordinates are gotten via coordinate transformation with the two methods. First, starting from the three-dimensional vector motion equation, the potential vorticity equations and expressions are gotten by the combination of the three-dimensional vorticity equation, continuity equation, and thermodynamic equation. Second, the potential vorticity equations and expressions are directly gotten from the corresponding scalar motion equations in the isobaric and isoentropic coordinates. The results show that potential vorticity expression is different with one method from that with the other in the isobaric coordinate system, and it is the same as each other in the isoentropic coordinate system. It was found, based on further analysis of the physical nature of the coordinates, that the isobaric and isoentropic coordinates are essentially treated as a mathematical coordinate system with the first method despite the coordinate transformation made for the term of pressure gradient force in the vector motion equation. From the procedure for the second method it is clearly seen that the isobaric and isoentropic coordinate systems are the physical coordinate system under the assumption of static equilibrium, which are not simply used as a mathematical coordinate system. As far as the isobaric coordinate is concerned, only the potential vorticity equation obtained from the scalar motion equations is the strict potential vorticity equation. As for the isoentropic coordinate, owing to the potential temperature gradient perpendicular to the isoentropic plane, the potential vorticity equation and expression are the same regardless of the coordinate being viewed as the physical or the mathematical.
1 citations