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Vector analysis
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Exercise 4.3. A Gaussian surface M is a piecewise-smooth, oriented closed surface in ℝ3 that encloses the origin. Evaluate the flux integral and explain its significance. What would the answer be if the origin is outside M? ∬Mxdydz+ydzdx+zdxdy(x2+y2+z2)32.
Electric Flux Density
Published in Ahmad Shahid Khan, Saurabh Kumar Mukerji, Electromagnetic Fields, 2020
Ahmad Shahid Khan, Saurabh Kumar Mukerji
The Gaussian surface is an imaginary (closed) surface that encloses a symmetrical charge distribution. This surface has to be such that D is either normal or tangential to the segments of this surface. When it is normal D ∙ ds = D·ds and when tangential, D ∙ ds = 0. Thus this surface is to be so chosen that some symmetry is exhibited vis-à-vis the charge distribution. Equation 6.9 is the key relation for evaluating the flux caused due to different charge configurations and passing through surfaces of different shapes. Some of these are discussed in the following subsections.
The static and quasistatic electromagnetic fields
Published in Edward J. Rothwell, Michael J. Cloud, Electromagnetics, 2018
Edward J. Rothwell, Michael J. Cloud
When the spatial distribution of charge is highly symmetric, Gauss’s law (in either integral or point form) may be solved directly for the electric field. Any of the following symmetry conditions is appropriate:The charge depends only on the spherical coordinate r so that D(r)=Dr(r)r^ $ \mathbf{D}(\mathbf{r})=D_r(r)\hat{\mathbf{r}} $ .The charge depends only on the cylindrical coordinate ρ $ \rho $ so that D(r)=Dρ(ρ)ρ^ $ \mathbf{D}(\mathbf{r})=D_\rho (\rho ){\hat{\boldsymbol{\rho }}} $ .The charge depends on a single rectangular coordinate, e.g., x, so that D(r)=Dx(x)x^ $ \mathbf{D}(\mathbf{r})=D_x(x)\hat{\mathbf{x}} $ .When employing the integral form of Gauss’s law, the procedure is to choose a flux surface (called a Gaussian surface) over which the electric field is either constant in magnitude and parallel to the surface normal, or perpendicular to the normal (or over which some combination of these conditions holds). Then the electric field may be removed from the integral and determined. The point form is employed by separating the field volume into regions in which the partial differential equation may be reduced to an ordinary differential equation solvable by direct integration. The solutions are then connected across the adjoining surfaces using boundary conditions.
Fabric defect detection algorithm using RDPSO-based optimal Gabor filter
Published in The Journal of The Textile Institute, 2019
Yueyang Li, Haichi Luo, Miaomiao Yu, Gaoming Jiang, Honglian Cong
Elliptical Gabor filter (EGF) is the modification of traditional Gabor filter. Its center, frequency, and orientation can all be tuned freely to improve the detection performance. In the spatial domain, a 2D-EGF is defined by (Hu, 2015) where, is the spatial central frequency of the filter in the spatial-frequency domain. and represent the shape factor of the Gaussian surface. is the aspect ratio of the Gaussian. defines the central spatial-frequency of the sinusoid. is defined by where, the rotation angle of the filter.
Calculation of the statistical characteristics of the light reflected by a rough random cylindrical homogeneous Gaussian surface
Published in Journal of Modern Optics, 2018
Rauf Gardashov, Gökhan Kara, E. Gül Emecen Kara
Further consideration of the problem was made under the assumption of independence random variables: and . Note, that the strong theoretical estimation of the correlations between is a difficult problem. But some rough estimation of correlations can be made, for example, in the case of Gaussian correlation function, , of the surface ; where is a radius of correlation, which is defined from: . Then the radiuses of correlation of and are: and , respectively. For a Gaussian surface from Equation (8) for we have: . Using the relations and , we obtain: .
Study on Sliding Wear Characteristics of Non-Gaussian Rough Surface in Mixed Lubrication
Published in Tribology Transactions, 2022
Jiang Zhao, Zhengminqing Li, Hong Zhang, Rupeng Zhu
This study aimed to investigate the effect of surface topography evolution on lubrication and wear characteristics during mixed lubrication using non-Gaussian surface reconstruction and mixed EHL wear modeling. A coupling model of surface wear and lubrication was established, and the effectiveness of the lubrication model was verified through film interference experiments. This created a basis to discuss the evolution of surface wear characteristics and lubrication characteristics of the Gaussian surface, positive skewness non-Gaussian surface, and negative skewness non-Gaussian surface. The effects of the kurtosis and skewness of the positive non-Gaussian rough surface on the lubrication and wear characteristics were also discussed.