Vector Analysis in Space

High order finite volume approximations of differential operators on nonuniform grids

Mathematical physics problems are often formulated using differential operators of vector analysis, i.e. invariant operators of first order, namely, divergence, gradient and rotor (curl) operators. In approximation of such problems it is natural to employ similar operator formulations for grid problems. The VAGO (Vector Analysis Grid Operators) method is based on such a methodology. In this paper the vector analysis difference operators are constructed using the Delaunay triangulation and the Voronoi diagrams. Further the VAGO method is used to solve approximately boundary value problems for the general elliptic equation of second order. In the convection‐diffusion‐reaction equation the diffusion coefficient is a symmetric tensor of second order.

Applied Electromagnetics and Electromagnetic Compatibility

An arbitrary-order method for magnetostatics on polyhedral meshes based on a discrete de Rham sequence

The separation of wave modes for isotropic elastic wavefields is typically done using Helmholtz decomposition. However, Helmholtz decomposition using conventional divergence and curl operators is not satisfactory for anisotropic media and leaves the different wave modes only partially separated. The separation of anisotropic waveflelds requires more sophisticated operators which depend on local material parameters. Wavefleld separation operators for TI (transversely isotropic) models can be constructed based on the polarization vectors evaluated at each point of the medium by solving the Christoffel equation using local medium parameters. These polarization vectors can be represented in the space domain as localized filters, which resemble conventional derivative operators. The spatially‐variable “pseudo” derivative operators perform well in 2D heterogeneous TI media even at places of rapid variation. Wave separation for 3D TI media can be performed in a similar way. In 3D TI media, P and SV waves are polarized only in symmetry planes, and SH waves are polarized orthogonal to symmetry planes. Using the mutual orthogonality property between these modes, we only need to solve for the P wave polarization vectors from the Christoffel equation, and SV and SH wave polarizations can be constructed using the relationship between these three modes. Synthetic results indicate that the operators can be used to separate wavefields for TI media with arbitrary strength of anisotropy.

The divergence and curl operators are fundamental mathematical tools introduced in university-level electromagnetism courses. Although students generally understand the operational use of these concepts, developing a clear physical intuition of what divergence and curl are can be challenging. This difficulty often arises due to the superficial or confusing treatment found in many standard textbooks. The goal of this article is to provide a clear demonstration, based on examples, that two-dimensional visualizations can be misleading when used to interpret these operators. A combination of graphical and analytical (symbolic) representations was found to be essential for developing an accurate physical intuition about divergence and curl. A deeper comprehension of these operators not only improves proficiency in electromagnetism, but also benefits a range of other fields, from fluid dynamics to meteorology and to technical applications.

P and S decomposition is an essential step in isotropic viscoelastic reverse time migration (VRTM). Separation using divergence and curl operators doesn’t preserve the phase or amplitude of the input viscoelastic wavefield, so an alternative decomposition that preserves the vector components of the Pand S-waves in the input wavefield is desired. With a decoupled assumption, we rearrange the memory variables in the elastodynamic equations based on the standard linear solid. The vector components of the propagating Pand S-wave can be can be decomposed in isotropic viscoelastic media without losing accuracy. Synthetic tests show that the Pand S-waves can be decomposed with no distortion of the amplitude or phase, and have the same vector components of particle velocity that exist in the viscoelastic wavefield before decomposition.

The paper considered the signal processing of simplicial complexes. Hodge decomposition formula for discrete fields is given, which is similar to the Hodge decomposition formula for smooth vector fields. The construction and the estimation of the gradient, divergence and curl operators and Laplace matrices for discrete vector fields are considered.

Maxwell’s equations are usually formulated as differential equations (1.1)–(1.4), where we have derivatives with respect to space (represented by the curl and divergence operators) and derivatives with respect to time. The sought electromagnetic field is a function of space and time that must satisfy Maxwell’s equations. A rather simple way to pursue a numerical solution is to represent the electromagnetic field by its function values at a (finite) set of discrete grid points, where we would normally use a uniform Cartesian grid with respect to space and time. Then, it is quite natural to approximate the derivatives of the electromagnetic field in Maxwell’s equations by finite-difference approximations that involve the fields at neighboring points on the grid. This type of approach is referred to as a finite-difference method.KeywordsGrid PointStagger GridFinite Difference ApproximationComplex ExponentialDifference FormulaThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Atrial Fibrillation (AF) is the most common supraventricular arrhythmia and has different underlying activation mechanism, including functional rotors (FR) and ectopic foci (EF).In this work we propose an approach for locating FR and EF in potential maps, which were tested with mathematical phantoms.12 phantoms were created (128x128 array, 4 s, Fs 500 Hz), simulating the motion of: FR (4 maps), EF (4 maps) and superpositions of these (4 maps).These were downsampled to different grids from 16x16 to 8x8, simulating electrode acquisition.Noise (SNR from 2 to 60) was added.To locate the mechanism, the signals were filtered and interpolated.Farneback optical flow was applied to compute the motion vector field (MVF).The MVF was normalized and its temporal average was calculated.Finally, we computed curl and divergence, by using a x and y oriented 11 11 Sobel filter as a estimation of the of the partial derivatives.The location of extrema in the curl and divergence maps were used for locating FR and EF respectively.Method robustness was tested by comparing the algorithm performance on different grid sizes and SNR.The mechanism was considered to be detected accurately if the position was within a normalized error of 5% from its respective phantom location.The results showed that our approach was able to locate both mechanisms, but revealed a dependency on spatial resolution.

Tensor fields of arbitrary rank in multidimensional space are naturally extended to the concepts of divergence, gradient, curl, deformer operators, as well as their second-order superpositions. Two options for generalizing the rotor as an external product are presented. Differential operators of the second order that do not change the rank of the tensor to which they are applied are considered in detail. Square matrices are introduced, consisting of differential operators $${\text{Di}}{{{\text{v}}}_{{(l)}}}{\text{Gra}}{{{\text{d}}}_{{(k)}}}$$ , $${\text{Gra}}{{{\text{d}}}_{{(k)}}}{\text{Di}}{{{\text{v}}}_{{(l)}}}$$ , and their relationship is established. An explicit expression is written for the repeated operator rotor. All introduced generalized operators in particular cases agree in their properties with the corresponding classical operators in vector and tensor analysis.

That the advantage of using vector analysis in mathematical physics has met with so little recognition by British applied mathematicians must appear rather strange to those of other countries. There are probably not more than half a dozen among us who habitually use vector methods and notation, at any rate in published work. The fact is still more surprising when we recall the work of Maxwell and Heaviside, seeing that the former gave a sort of authority to the curl and divergence , while the latter had so much to do with the systematic development of the vectorial calculus. Perhaps the chief reason why vector analysis has not come into more general favour with us is that our leaders in applied mathematics have not felt the need of it. It has often been remarked, and perhaps with some degree of truth, that nothing can be accomplished by vector methods that cannot also be done by Cartesian analysis ; and therefore, it is argued, the change is unnecessary and useless. If we had to deal only with minds of special mathematical ability and analytical insight, this conclusion might be accepted. But with the average student so much of his attention is occupied in dealing with the complex array of symbols of partial differentiation to which he is often led in Cartesian analysis, that he is unable to grasp the inner meaning of the work. It is difficult for him in many cases even to see exactly what is expressed in the formulae obtained, involving as they do the three components of a vector quantity in combinations not easy to visualise. And even if the student succeeds in following the argument it is often almost impossible for him to remember either the train of reasoning or the result arrived at. He does not perhaps see why his equations should be differentiated partially with respect to x, y, z and added, or with respect to z, y and subtracted.

Symbolic vector analysis in plasma physics

As a common type of refractive error, astigmatism has always been one of the important factors affecting visual quality in clinical practice and it is quite an important public health burden. This study aims to evaluate astigmatism research and the clinical application of the vector method of the past half-century through a broad scientometric analysis, and to explore its reference value for guiding clinical treatment. The literature search was conducted on the Web of Science for astigmatic vector studies published from 1965 to August 2024. Retrieved publications were analyzed by the number of annual publications, prolific countries, and researchers, core publications and journals, and the number of citations through descriptive statistics. Collaboration networks and keyword analysis were visualized by VOSviewer and CiteSpace. One thousand and fifty-nine publications were included for a visualized analysis. Vector methods help researchers to describe astigmatism specialized, and clinicians are mainly focused on the correction of astigmatism in both corneal and cataract surgery. Journal of Cataract and Refractive Surgery was the core journal for this field. The United States accounts for the largest proportion of publications and Australia had the highest citation ratio. Through the keyword analysis, the network identified 4 major research trends of corneal refractive surgery, penetrating keratoplasty, cataract surgery, and epidemiological surveys of astigmatism. "Photorefractive keratectomy," "postoperative astigmatism" and "surgery" had significant burst strength and continuous attention to the astigmatism vector analysis. Vector analysis is the most commonly used method to evaluate astigmatism and could significantly improve the accuracy of astigmatism correction, particularly in areas of refractive and cataract surgery. The application of vector analysis is beneficial in guiding the design of surgical incisions, determining the nomogram, optimizing the surgical protocol, and improving the accuracy of astigmatism correction. Meanwhile, the popularization of vector method will help to improve the accuracy of astigmatism analysis and promote the benign development of public health.

In this work, we develop a graphical calculus for multi-qudit computations with generalized Clifford algebras, building off the algebraic framework developed in our prior work. We build our graphical calculus out of a fixed set of graphical primitives defined by algebraic expressions constructed out of elements of a given generalized Clifford algebra, a graphical primitive corresponding to the ground state, and also graphical primitives corresponding to projections onto the ground state of each qudit. We establish many properties of the graphical calculus using purely algebraic methods, including a novel algebraic proof of a Yang-Baxter equation and a construction of a corresponding braid group representation. Our algebraic proof, which applies to arbitrary qudit dimension, also enables a resolution of an open problem of Cobanera and Ortiz on the construction of self-dual braid group representations for even qudit dimension. We also derive several new identities for the braid elements, which are key to our proofs. Furthermore, we demonstrate that in many cases, the verification of involved vector identities can be reduced to the combinatorial application of two basic vector identities. Additionally, in terms of quantum computation, we demonstrate that it is feasible to envision implementing the braid operators for quantum computation, by showing that they are 2-local operators. In fact, these braid elements are almost Clifford gates, for they normalize the generalized Pauli group up to an extra factor ζ , which is an appropriate square root of a primitive root of unity.