The generalized stokes integral theorem for a covariant pseudotensor field

Ориентируемые континуумы играют важную роль в микрополярной теории упругости, все реализации которой возможны только в рамках псевдотензорного формализма и представления об ориентируемом многообразии. Особенно это касается теории микрополярных гемитропных упругих сред. В настоящей работе рассматриваются различные формулировки интегральной теоремы Стокса для асимметричного ковариантного пседотензорного поля, заданного веса. Тем самым достигается распространение известной интегральной формулы Стокса на случай псевдотензоров. Последнее обстоятельство позволяет использовать, указанное обобщение для микрополярных континуумов. Исследование существенно опирается на класс специальных координатных систем. Oriented continua play an important role in the micropolar theory of elasticity, all realizations of which are possible only within the framework of the pseudotensor formalism and the orientable manifold concept. This especially concerns the theory of micropolar hemitropic elastic media. In this paper, we consider various formulations of the Stokes integral theorem for an asymmetric covariant pseudotensor field of a given weight. This extends the well-known Stokes integral formula to the case of pseudotensors. The latter circumstance makes it possible to use the manifistated generalization for micropolar continua. The study relies heavily on the class of special coordinate systems.

Comparison of spectral methods for the evaluation of Stokes integral

<p>The spherical approximation of the fundamental equation of geodesy defines the boundary value problems. Stokes’s integral provides the solution of boundary value problems that enables the computation of geoid from the properly reduced gravity measurements to the geoid. The stokes integral can be evaluated by brute-force numerical integration, spectral methods, and least-squares collocation. There is a trade-off between computation time and accuracy when we chose numerical integration technique or any spectral method. This research will compare time complexity and the accuracy of different spectral methods (1D-FFT, 2D-FFT, Multi-band FFT) and numerical integration technique for the region in the lower Himalaya, around Nainital, Uttarakhand, India. </p>

Comparison among Gravimetric, Astrogravimetric and Astrogeodetic Geoids: Case Study for Austria

<p>It is used to state that all geoid determination techniques should yield to the same geoid if the indirect effect is properly taken into account (Heiskanen and Moritz, 1967). The current study compares different geoid determination techniques for Austria. The used techniques are the gravimetric, astrogravimetric and astrogeodetic geoid determination techniques. The available data sets (gravity, deflections of the vertical, height, GPS) are described. The window remove-restore technique (Abd-Elmotaal and Kuehtreiber, 2003) has been used. The available gravity anomalies and the deflections of the vertical have been topographically-isostatically reduced using the Airy isostatic hypothesis. The reduced deflections have been used to interpolate deflections on a relatively dense grid covering the data window. These gridded reduced deflections have been used to compute an astrogeodetic geoid for Austria using least-squares collocation technique within the remove-restore scheme. The Vening Meinesz formula has been used to compute an astrogravimetric geoid for Austria. Another gravimetric geoid for Austria has been determined in the framework of the window remove-restore technique using Stokes integral with modified Stokes kernel. All computed geoids have been validated using GNSS/levelling derived geoid. A wide comparison among the derived geoids computed within the current investigation has been carried out.</p>

GEOID MODELLING USING INTEGRATION AND FFT ASSOCIATED WITH DIFFERENT GRAVIMETRIC REDUCTION METHODS

ABSTRACT. A vertical reference system is characterized by a vertical datum and a set of scientific altitudes. In the case of orthometric altitudes, the geoid is used as a reference surface, equipotential surface of the gravity field of the Earth that better fits, in the sense of the Least Square Method, to the mean sea level. This study aimed to determine the geoid by applying two processes for calculation of residual ondulation, the integration and the Fast Fourier Transform. These techniques were applied to the values of the residual anomalies obtained from different methods of gravimetric reduction, the Helmert’s Second Method of Condensation, Bouguer and Rudzki. Two test areas were used. For area 1, the best gravimetric geoid was obtained by applying 1D planar FFT with the Helmert’s SecondMethod of Condensation. For area 2, the best gravimetric geoid was obtained through the application of integration and the Rudzki’s reduction. It can be concluded that the physical characteristics of both areas are relevant in the determination of the geoid and that additional procedures must be applied to improve the geoid determination, mainly, in area 2 whose physical characteristics are more heterogeneous than in area 1.Keywords: Geoid, GeoFis 1.0, Gravimetric Reduction, FFT, Stokes Integral. RESUMO. Um sistema vertical de referência é caracterizado por um datum vertical e pelo conjunto de altitudes científicas. No caso das altitudes científicas adotadas serem as ortométricas utiliza-se como superfície de referência o geoide, superfície equipotencial do campo da gravidade da Terra que melhor se ajusta, no sentido do método dos mínimos quadrados, ao nível médio do mar. O objetivo desse trabalho foi determinar o geoide aplicando dois processos de cálculo da ondulação residual, a integração e a Transformada Rápida de Fourier. Essas técnicas foram empregadas aos valores de anomalias residuais obtidas a partir de diferentes métodos de redução gravimétrica, Segundo Método de Condensação de Helmert, Bouguer e Rudzki. Foram utilizadas duas áreas de teste. Verificou-se que para a área 1 o melhor geoide gravimétrico foi obtido pela aplicação da FFT planar 1D juntamente com o Segundo Método de Condensação de Helmert. Para a área 2 o melhor geoide gravimétrico foi obtido pela aplicação da integração e da redução de Rudzki. Conclui-se que as características físicas das duas áreas são relevantes na determinação do geoide e que procedimentos complementares devem ser aplicados para melhorar a determinação do geoide, principalmente, na área 2 cujas características físicas são mais heterogêneas do que da área 1. Palavras-chave: Geoide, GeoFis 1.0, Redução gravimétrica, FFT, Integral de Stokes.

The Newtonian Potential and the Demagnetizing Factors of the General Ellipsoid

The objective of this paper is to present a modern and concise new derivation for the explicit expression of the interior and exterior Newtonian potential generated by homogeneous ellipsoidal domains in $\mathbb{R}^N$ (with $N \geqslant 3$). The very short argument is essentially based on the application of Reynolds transport theorem in connection with Green-Stokes integral representation formula for smooth functions on bounded domains of$\mathbb{R}^N$, which permits to reduce the N-dimensional problem to a 1-dimensional one. Due to its physical relevance, a separate section is devoted to the derivation of the demagnetizing factors of the general ellipsoid which are one of the most fundamental quantities in ferromagnetism.