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In this paper, we improve an earlier iterative successive approximation method for solving the Hamilton–Jacobi equation (HJE) arising in deterministic optimal control of affine nonlinear systems. T...

This paper presents an exact analytical solution to the 3D transient dynamics of a linear elastic, isotropic homogeneous, curved beam, with uniform rectangular cross-section. The solution technique uses vector identities to decouple the governing equations. The decoupled equations are then solved by method of separation of variables. The solutions to the decoupled equations can be recombined to form a new equation. Solving this new equation yields the displacement field. To demonstrate the capabilities of the proposed solution technique, a generic case study was modeled and computed. A curved beam is subjected to a longitudinal impulse loading and the transient displacement field is calculated. This solution technique is valid for any curvature so long as the curvature is the same throughout the beam. This includes the limiting case where the inner radius of the beam goes to zero and the curved beam becomes a section of a disk. The presented solution results were compared with an approximate solution from the literature and experimental data from the literature. The solution is also compared with an explicit FEM solution conducted by the Authors. The presented solution agrees with the results from the literature and from the FEM solutions conducted by the Authors. This paper demonstrates that the accuracy and robustness of the proposed solution technique meets the needs of many potential applications.

The potential formulation has significant advantages over field formulation in solving complicated problems in electromagnetic field theory. One essential part of electromagnetic field theory's potential formulation is gauge invariance and gauge theories because it provides an extra degree of freedom. By using this extra degree of freedom, we can solve complicated electromagnetic problems quickly. Thus, it is necessary to include a systematic explanation of gauge theories in teaching electromagnetic theory. However, textbooks usually formulate gauge theories by using Maxwell's equations of electromagnetism, by using vector calculus identities. However, this method of formulation of gauge theories does not give a clear idea about the origin of gauge theories and gauge invariance in electromagnetism. Here the author formulates gauge theories from wave equations of the electric and magnetic fields instead of directly using Maxwell's equations. This method generalizes all gauge theories like Lorenz gauge theory, Coulomb gauge theory, Etc. Gauge theory, because of the way the author derives it, gives a distinct idea about the mathematical origin of the gauge theories and gauge invariance in electromagnetic field theory. Thus, the author reviews the origin of gauge theories in electromagnetic field theory and develops a distinct and effective method to introduce gauge theory in the teaching of electromagnetic field theory that can provide better understanding of the topic to undergraduate students.

The aim of this work is to introduce a purely order-based Kantorovich–Wright type integration of scalar functions with respect to a vector measure defined on a δ-ring and taking values in a Dedekind σ-complete vector lattice and use it for obtaining general representation theorems for Dedekind complete vector lattices.

Tensor calculus is critical in the study of the vector calculus of the surface of a body. Indeed, tensor calculus is a natural step-up for vector calculus. This paper presents some pitfalls of a traditional course in vector calculus in transitioning to tensor calculus. We show how a deeper emphasis on traditional topics such as the Jacobian can serve as a bridge for vector calculus into tensor calculus.

International audience; The main mathematical difficulty in crystal geometry and X-ray diffraction stems from oblique system of coordinatesas soon as we move on from the simple cubic to the body-centered cubic case without mentioning thenon-cubic systems. We show in this work that there exists a simple way for dealing with obliquegeometry for all crystal systems by combining direct and reciprocal space geometrical elements with the help of a small set of vector identities that are not usually taught in a general physics curriculum. This allows us to find a set of compact formulae applicable to all crystalline systems for directions and planes, moduli, angles and scalar products of basis vectors belonging to the direct and reciprocal lattices as well as interplanar distances and angles.; La difficulté principale dans la géométrie cristalline provient de la nécessité de passer en géométrie oblique aussitôt que l'on passe au cristal Cubique à Corps Centré. Nous montrons dans ce travail comment en combinant les éléments géométriques provenant de l'espace direct et de l'espace réciproque avec l'aide de quelqus identités vectorielles non enseignées de manière usuelle en physique, l'on peut facilement, rapidement et sans commetre d'erreurs calculer des volumes, distances et angles pour tout système cristallin.

In this paper, we study the equation , where L is a -valued vector field in with a homogeneous singularity. The properties of the solutions are linked to the number theoretic properties of a pair of complex numbers attached to the vector field. As an application, we obtain results about an associated Riemann–Hilbert problem for the vector field L.

In this paper, we present iterative or successive approximation methods for solving the coupled Hamilton–Jacobi–Isaacs equations (HJIEs) arising in nonzero-sum differential game for affine nonlinear systems. We particularly consider the ones arising in mixed $${\mathcal H}_{2}/{\mathcal H}_{\infty }$$ control. However, the approach is perfectly general and can be applied to any others including those arising in the N-player case. The convergence of the method is established under fairly mild assumptions, and examples are solved to demonstrate the utility of the method. The results are also specialized to the coupled algebraic Riccati equations arising typically in mixed $${\mathcal H}_{2}/{\mathcal H}_{\infty }$$ linear control. In this case, a bound within which the optimal solution lies is established. Finally, based on the iterative approach developed, a local existence result for the solution of the coupled-HJIEs is also established.

The scalar and vector Laplacians are basic operators in physics and engineering. In applications, they show up frequently perturbed by lower-order terms. The effect of such perturbations on mixed finite element methods in the scalar case is well-understood, but that in the vector case is not. In this paper, we first show that surprisingly for certain elements there is degradation of the convergence rates with certain lower-order terms even when both the solution and the data are smooth. We then give a systematic analysis of lower-order terms in mixed methods by extending the Finite Element Exterior Calculus (FEEC) framework, which contains the scalar, vector Laplacian, and many other elliptic operators as special cases. We prove that stable mixed discretization remains stable with lower-order terms for sufficiently fine discretization. Moreover, we derive sharp improved error estimates for each individual variable. In particular, this yields new results for the vector Laplacian problem which are useful in applications such as electromagnetism and acoustics modeling. Further our results imply many previous results for the scalar problem and thus unifies them all under the FEEC framework.

Summary We have developed a new divergence preserving method for the reconstruction of the Cartesian components of a vector field from the orthogonal projection of a vector field to the normals to edges in two dimensional. In this method, discrete divergences computed from the nodal components and from the normal ones are exactly the same. Our new method consists of two stages. At the first stage, we use an extended version of the local procedure described in [J. Comput. Phys., 139:406–409, 1998] to obtain a ‘reference’ nodal vector. This local procedure is exact for linear vector fields; however, the discrete divergence is not preserved. Then, we formulate a constrained optimization problem, in which this reference vector plays the role of a target, and the divergence constraints are enforced by using Lagrange multipliers. It leads to the solution of ‘elliptic’ like discrete equations for the cell-centered Lagrange multipliers. The new global divergence preserving method is exact for linear vector fields. We describe all details of our new method and present numerical results, which confirm our theory. Copyright © 2016 John Wiley & Sons, Ltd.

Lattice differential operators are known to preserve key properties of their analytical counterpart, such as isotropy, fundamental vector identities due to the symmetries of the discrete kinetic lattice. Here, we present the idea of discrete lattice operators derived on a body-centered-cubic (BCC) lattice. These operators show quite a high degree of accuracy and isotropy as compared to the earlier simple cubic (SC) representations of the same while maintaining a relatively smaller stencil. To illustrate the usefulness of these schemes, we have considered a couple of examples, such as passive scalar transport and fluctuating hydrodynamics.

Tyson (J Math Econ 49(4): 266–277, 2013) introduces the notion of symmetry vector field for a smooth preference relation, and establishes necessary and sufficient conditions for a vector field on consumption space to be a symmetry vector field. The structure of a such a condition is discussed on both geometric and economic grounds. It is established that symmetry vector fields do commute (i.e. have vanishing Lie bracket) for additive and joint separability. The marginal utility of money is employed as a normalization of the expansion vector field (Mantovi, J Econ 110(1): 83–105, 2013) which results in the fundamental (expansion-) symmetry vector field. Finally, a characterization of symmetry vector fields is given in terms of their action on the distance function, and a pattern of complete response is discussed for additive preferences. Examples of such constructions are explicitly worked out. Potential implications of the results are discussed.

In a numerical solution of the Maxwell-Vlasov system, the consistency with the charge conservation and divergence conditions has to be kept solving the hyperbolic evolution equations of the Maxwell system, since the vector identity ? ? ( ? i? u ? ) = 0 and/or the charge conservation of moving particles may be not satisfied completely due to discretization errors. One possible method to force the consistency is the hyperbolic divergence cleaning. This hyperbolic constraint formulation of Maxwell's equations has been proposed previously, coupling the divergence conditions to the hyperbolic evolution equations, which can then be treated with the same numerical method. We pick up this method again and show that electrostatic limit may be obtained by accentuating the divergence cleaning sub-system and converging to steady state. Hence, the electrostatic case can be treated by the electrodynamic code with reduced computational effort. In addition, potential boundary conditions as often given in practical applications can be coupled in a similar way to get appropriate boundary conditions for the field equations. Numerical results are shown for an electric dipole, a parallel-plate capacitor, and a Langmuir wave. The use of potential boundary conditions is demonstrated in an Einzel lens simulation.

In this paper, we apply recently proposed generalized integral and differential operators to develop generalized vector calculus and generalized variational calculus for problems defined over a convex domain. In particular, we present some generalization of Green’s and Gauss divergence theorems involving some new operators, and apply these theorems to generalized variational calculus. For fractional power kernels, the formulation leads to fractional vector calculus and fractional variational calculus for problems defined over a convex domain. In special cases, when certain parameters take integer values, we obtain formulations for integer order problems. Two examples are presented to demonstrate applications of the generalized variational calculus which utilize the generalized vector calculus developed in the paper. The first example leads to a generalized partial differential equation and the second example leads to a generalized eigenvalue problem, both in two dimensional convex domains. We solve the generalized partial differential equation by using polynomial approximation. A special case of the second example is a generalized isoperimetric problem. We find an approximate solution to this problem. Many physical problems containing integer order integrals and derivatives are defined over arbitrary domains. We speculate that future problems containing fractional and generalized integrals and derivatives in fractional mechanics will be defined over arbitrary domains, and therefore, a general variational calculus incorporating a general vector calculus will be needed for these problems. This research is our first attempt in that direction.

A system of equations with respect to a pair (V, p) of a scalar field and a vector field in a torus D is considered. The system consists of the Euler equation with a given vector field f and the solenoidality equation for the field V. We seek for solutions (V, p) of this system such that the lines of the vector field V inside D coincide with meridians of tori embedded in D with the same circular axis. Conditions on the vector field f under which the problem is solvable are established, and the whole class of such solutions is described.

The additional equalities (additional differential equations) for the Navier - Stokes and other vector PDE are established in this paper. These equations are obligatory requirements (properties) of three functions forming a vector field on Euclidean space. Therefore all solutions of the vector PDE should satisfy these requirements. Without these equalities the Navier-Stokes equations with so called a continuity equation are underdetermined as vector system and any "exact solution" is not solution of the true Navier - Stokes vector equation.

The aim of this paper is to represent the trajectory’s curvature related to the power density of the electromagnetic wave that has been widely described as the Poynting vector. The paper starts from the infinitesimal work equation and transformed by using the vector identities and Maxwell’s equations. The result of this transformation leads to a decoupled representation of the Poynting vector as a separate contribution of the electrical and magnetic fields. The decoupled form of the Poynting vector is emphasized by the presence of the charged particle velocity and its curvature. The discussion is conducted symbolically in order to conclude that the power density produced by a charged particle traveling in an electromagnetic field will decrease as the trajectory’s curvature increases.

We construct a functional calculus on unital uniformly complete f-algebras for continuous center-valued functions of polynomial growth and study the connection with the Minkowski duality.

In spaces Rn, n ≥ 2, it has been proved that a solenoidal vector field and its rotor satisfy the series of new integral identities which have covariant form. The interest in them is explained by hydrodynamics problems for an ideal fluid.

This paper provides derivation of some basic identities for complex four-component vectors defined in a complex four-dimensional spacetime frame specified by an imaginary temporal axis. The resulting four-vector identities take exactly the same forms of the standard vector identities established in the familiar three-dimensional space, thereby confirming the consistency of the definition of the complex four-vectors and their mathematical operations in the general complex spacetime frame. Contravariant and covariant forms have been defined, providing appropriate definitions of complex tensors, which point to the possibility of reformulating differential geometry within a spacetime frame.

A general model for the coupled analysis of magneto-mechanical systems is developed by minimizing the continuum energy functional of the system using the calculus of variation. This approach, which is in contrast with the traditional approach of minimizing after discretization, allows the use of strain and stress tensors, vector identities and the divergence theorem, and results in coupled governing equations of the system with three coupling terms; the magnetic stress tensor, the magnetostriction stress tensor, and the magnetostriction reluctivity. The model uses the information contained in the set of experimental magnetostriction curves dependent on stress to calculate the permeability variation due to stress. The governing equations are then discretized using the Galerkin method resulting in methods for the calculation of nodal magnetic and magnetostriction forces including the coupling effects. Finally the model is applied to a simple 2D problem and the flux density distributions using the proposed method and the traditional method of using experimental magnetization curves are compared.

The steady-state two-dimensional potential flow of an inviscid fluid past a circular cylinder is considered. The resulting Laplace’s equation for the velocity potential is converted to cylindrical coordinates by the Calculus of Variations method of transformation of coordinates introduced in a pastoral interlude and that equation solved by a separation of variables technique. Then integration of the pressure determined by Bernoulli’s relation about the cylinder yields D’Alembert’s paradox for a two-dimensional situation that the drag on the cylinder is zero which is a consequence of the assumption that the small viscosity coefficient can be neglected. The problems fill in some details involving vector identities employing the alternating tensor introduced in Chap. 9, examine the properties of the orthogonal Hermite polynomials similar in behavior to the Legendre polynomials discussed below, and consider the corresponding companion situation of three-dimensional potential flow past a sphere. This requires that the resulting Laplace’s equation for the velocity potential be converted to spherical coordinates by the Calculus of Variations transformation method. Then the separation of variables technique of solution gives rise to Legendre polynomials the properties of which have been deduced by means of the two pastoral interludes that conclude the chapter. Integration of the pressure about the sphere yields D’Alembert’s paradox for a three-dimensional situation that the force on the sphere is zero.

Vector field is mostly linearly approximated for the purpose of classification and description. This approximation gives us only basic information of the vector field. We will show how to approximate the vector field with second order derivatives, i.e. Hessian and Jacobian matrices. This approximation gives us much more detailed description of the vector field. Moreover, we will show the similarity of this approximation with conic section formula.

The system variable vector \( {\bar{\text{X}}}_{{\text{sys}}} \) of an optical system is, nearly always, different from the boundary variable vector \( {\bar{\text{X}}}_{\text{i}} \) of a boundary surface. Furthermore, changes in the system variable vector may have a profound effect on the behavior of the rays as they propagate through the system. Therefore, the Jacobian matrix \( \text{d}\bar{\text{X}}_{\text{i}} /\text{d}\bar{\text{X}}_{{\text{sys}}} \) of the boundary variable vector \( {\bar{\text{X}}}_{\text{i}} \) with respect to the system variable vector \( {\bar{\text{X}}}_{{\text{sys}}} \) is of crucial concern to optical systems designers.

We study two classes of vector fields on the path space over a closed manifold with a Wiener Riemannian measure. By adopting the viewpoint of Yang-Mills field theory, we study a vector field defined by varying a metric connection. We prove that the vector field obtained in this way satisfies a Jacobi field equation which is different from that of classical one by taking in account that a Brownian motion is invariant under the orthogonal group action, so that it is a geometric vector field on the space of continuous paths, and induces a quasi-invariant solution flow on the path space. The second object of this paper is vector fields obtained by varying area. Here we follow the idea that a continuous semimartingale is indeed a rough path consisting of not only the path in the classical sense, but also its Levy area. We prove that the vector field obtained by parallel translating a curve in the initial tangent space via a connection is just the vector field generated by translating the path along a direction in the Cameron-Martin space in the Malliavin calculus sense, and at the same time changing its Levy area in an appropriate way. This leads to a new derivation of the integration by parts formula on the path space.

We study conditions under which the characteristic vector of a normal lcQS-manifold is a torsion-forming or even a concircular vector field. We prove that the following assertions are equivalent: An lcQS-structure is normal, and its characteristic vector is a torsion-forming vector field. An lcQS-structure is normal, and its characteristic vector is a concircular vector field. An lcQS-structure is locally conformally cosymplectic and has a closed contact form.

A new Green's function formulation is developed systematically for modeling general homogeneous (dielectric or magnetic) objects in a layered medium. The dyadic form of the Green's function is first derived based on the pilot vector potential approach. The matrix representation in the moment method implementation is then derived by applying integration by parts and vector identities. The line integral issue in the matrix representation is investigated, based on the continuity property of the propagation factor and the consistency of the primary term and the secondary term. The extinction theorem is then revisited in the inhomogeneous background and a surface integral equation for general homogeneous objects is set up. Different from the popular mixed potential integral equation formulation, this method avoids the artificial definition of scalar potential. The singularity of the matrix representation of the Green's function can be made as weak as possible. Several numerical results are demonstrated to validate the formulation developed in this paper. Finally, the duality principle of the layered medium Green's function is discussed in the appendix to make the formulation succinct.

The second derivative of two vector functions is related to the divergence of the vector functions with first order operators. Namely, 𝐏 ⋅ ∇ 2 𝐐 − 𝐐 ⋅ ∇ 2 𝐏 = ∇ ⋅ [ 𝐏 ( ∇ ⋅ 𝐐 ) − 𝐐 ( ∇ ⋅ 𝐏 ) + 𝐏 × ∇ × 𝐐 − 𝐐 × ∇ × 𝐏 ] .

Deformation is an important research topic in graphics. There are two key issues in mesh deformation: (1) self-intersection and (2) volume preserving. In this paper, we present a new method to construct a vector field for volume-preserving mesh deformation of free-form objects. Volume-preserving is an inherent feature of a curl vector field. Since the field lines of the curl vector field will never intersect with each other, a mesh deformed under a curl vector field can avoid self-intersection between field lines. Designing the vector field based on curl is useful in preserving graphic features and preventing self-intersection. Our proposed algorithm introduces distance field into vector field construction; as a result, the shape of the curl vector field is closely related to the object shape. We define the construction of the curl vector field for translation and rotation and provide some special effects such as twisting and bending. Taking into account the information of the object, this approach can provide easy and intuitive construction for free-form objects. Experimental results show that the approach works effectively in real-time animation.

Considering the notion of Jacobi type vector fields for a real hypersurface in a complex two-plane Grassmannian, we prove that if a structure vector field is of Jacobi type it is Killing. As a consequence we classify real hypersurfaces whose structure vector field is of Jacobi type.

Very general univariate Ostrowski type inequalities are presented regarding Banach space valued functions. They provide norm estimates for the deviation of the average of the function from its vector values, and they involve the n th order vector derivative, n ≥ 1 , with respect to ‖ ⋅ ‖ p , 1 ≤ p ≤ ∞ . On the way to prove our results we establish useful functional vector identities. Using our vector Ostrowski inequalities we derive the vector Landau inequalities.

We advocate the use of an alternative calculus in biomedical image analysis, known as multiplicative (a.k.a. non-Newtonian) calculus. It provides a natural framework in problems in which positive images or positive definite matrix fields and positivity preserving operators are of interest. Indeed, its merit lies in the fact that preservation of positivity under basic but important operations, such as differentiation, is manifest. In the case of positive scalar functions, or in general any set of positive definite functions with a commutative codomain, it is a convenient, albeit arguably redundant framework. However, in the increasingly important non-commutative case, such as encountered in diffusion tensor imaging and strain tensor analysis, multiplicative calculus complements standard calculus in a truly nontrivial way. The purpose of this article is to provide a condensed review of multiplicative calculus and to illustrate its potential use in biomedical image analysis. Keywords: Multiplicative calculus – Non-Newtonian calculus – Diffusion tensor imaging – Cardiac strain tensor analysis – Positivity

In this paper, a generalized $\epsilon-$subdifferential, which was defined by a norm, is first introduced for a vector valued mapping. Some existence theorems and the properties of the generalized $\epsilon-$subdifferential are discussed. A relationship between the generalized $\epsilon-$subdifferential and a directional derivative is investigated for a vector valued mapping. Then, the calculus rules of the generalized $\epsilon-$subdifferential for the sum and the difference of two vector valued mappings were given. The positive homogeneity of the generalized $\epsilon-$subdifferential is also provided. Finally, as applications, necessary and sufficient optimality conditions are established for vector optimization problems.

In this paper, we find a bridge connecting a class of vector fields in ℝ3 with the planar vector fields and give a criterion of the existence of closed orbits, heteroclinic orbits and homoclinic orbits of a class of vector fields in ℝ3. All the possible nonwandering sets of this class of vector fields fall into three classes: (i) singularities; (ii) closed orbits; (iii) graphs of unions of singularities and the trajectories connecting them. The necessary and sufficient conditions for the boundedness of the vector fields are also obtained.

In a three-dimensional bounded possibly multiply-connected domain, we prove the existence and uniqueness of vector potentials in L p -theory, associated with a divergence-free function and satisfying some boundary conditions. We also present some results concerning scalar potentials and weak vector potentials. Furthermore, various Sobolev-type inequalities are given.

In this article we deal with f-conformal Killing vector fields, a generalization of conformal Killing vector fields involving a function f and two real parameters. Under certain conditions on f and the parameters, some non-existence results for such vector fields are proven. Moreover we derive a generalization of the Kazdan-Warner and Bourguignon-Ezin identities to the case of f-conformal Killing vector fields. Based on these, finally we present an application to the f-conformal solitons, a generalization of the Yamabe solitons.

Mimetic discretization methods for the numerical solution of continuum mechanics problems directly use vector calculus and differential forms identities for their derivation and analysis. Fully mimetic discretizations satisfy discrete analogs of the continuum theory results used to derive energy inequalities. Consequently, continuum arguments carry over and can be used to show that discrete problems are well-posed and discrete solutions converge. A fully mimetic discrete vector calculus on three dimensional tensor product grids is derived and its key properties proven. Opinions regarding the future of the field are stated.

A problem of impedance description of magnetic field on the surface of heterogenic Earth is discussed in the article. The concepts of impedance tensor, Vise - Parkinson tensor, generalized impedance tensor, Leontovich's impedance approximation and its generalization are analyzed. It is noticed that these models are of particular character and are real only in cases of special forms of giving primary field and do not always correspond to conditions of conducting real magnet ot elluric experiment. Generalized system of vector identities of impedance type is given for harmonic electromagnetic field on the closed interfaces and generated on their base generalized accurate differential equation of impedances and the system of scalar equations which determine this surface. It is noticed that in case of necessity of using their incomplete, truncated forms these models in contrast to classical ones, have interior criteria of their application and do not demand for their substantiation any additional empirical or heuristic considerations.

This paper is motivated by the desire to formulate a relativistically covariant hidden-variable particle trajectory interpretation of the quantum theory of the vector field that is formulated in such a way as to allow the inclusion of gravity. We present a methodology for calculating the flows of rest energy and a conserved density for the massive vector field using the time-like eigenvectors and eigenvalues of the stress-energy-momentum tensor. Such flows may be used to define particle trajectories which follow the flow. This work extends our previous work which used a similar procedure for the scalar field. The massive, spin-one, complex vector field is discussed in detail and the flows of energy-momentum are illustrated in a simple example of standing waves in a plane.

In this article, we show that complex vector fields on the punctured Riemann sphere are Newton vector fields and as a consequence all meromorphic vector fields and all elliptic vector fields are Newton vector fields. Moreover, for a large class of vector fields, which includes all elliptic vector fields, the proof is constructive in the sense that one can explicitly construct the function characterizing the Newton vector field.

Virtually identical derivations of the expressions for the spherical volume averages of static electric and magnetic fields are presented. These derivations utilize the Coulomb and Biot–Savart laws, and make no use of vector calculus identities or potentials.

It is shown that the Dirac equation with scalar and vector quadratic potentials and a Coulomb-like tensor potential can be solved exactly. The bound state solutions for equal vector and scalar potentials are obtained. The limit of zero tensor coupling is investigated. The case of equal vector and scalar potentials with opposite sign is also studied. The pseudospin symmetry and its breaking by the tensor interaction are discussed.

We present two new families of polynomial differential systems of arbitrary degree with centers, a two--parameter family and a four--parameter family.