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Concept# Vector calculus

Summary

Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb{R}^3. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of
electromagnetic fields, gravitational fields, and fluid flow.
Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis. In the conventional form using cross products, vector calculus does not generalize to higher dimensio

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Curl (mathematics)

In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in t

Cross product

In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented

Divergence

In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the di

Related courses (91)

Related lectures (155)

MATH-203(a): Analysis III

Le cours étudie les concepts fondamentaux de l'analyse vectorielle et de l'analyse de Fourier-Laplace en vue de leur utilisation pour résoudre des problèmes pluridisciplinaires d'ingénierie scientifique.

MATH-203(c): Analysis III

Le cours étudie les concepts fondamentaux de l'analyse vectorielle et l'analyse de Fourier en vue de leur utilisation pour
résoudre des problèmes pluridisciplinaires d'ingénierie scientifique.

ME-372: Finite element method

L'étudiant acquiert une initiation théorique à la méthode des éléments finis qui constitue la technique la plus courante pour la résolution de problèmes elliptiques en mécanique. Il apprend à appliquer cette méthode à des cas simples et à l'exploiter pour résoudre les problèmes de la pratique.

The Monge problem [23], [27], as reformulated by Kantorovich [19], [20] is that of the transportation, at a minimum "cost", of a given mass distribu- tion from an initial to a
final position during a given time interval. It is an optimal transport problem [28, sects. 1, 2]. Following the fluid mechanical solution provided by Benamou and Brenier for quadratic cost functions [4] ,[28, sects. 5.4, 8.1] and, by analogy with the fi
xed end problem in Analytical Mechanics, Lagrangian formulations are needed to solve this boundary value problem in time. They are also needed to determine the Actions as time in- tegral of Lagrangians, that are measures of the "cost"of the transportations [4, proposition 1.1]. Four versions of explicit constructions of Lagrangians are proposed in section 3. They are associated to the Hamiltonians of perfect and self-interacting systems presented in section 2. These Hamiltonians are ex- pressed in function of pairs of the well known canonically conjugated Clebsch variables, namely mass densities and velocity potentials [14], [15]. The fi
rst version consists in the elimination of the velocity potentials as a function of the densities and their time derivatives by inversion of the continuity equations de- rived from given Hamiltonians. The second version consists in the elimination of the gradient of the velocity potentials from the continuity equations thanks to the introduction of vector valued applications such that their divergences give the mass densities. It turns out that, up to a sign factor, these vector
elds are canonically conjugated to Euler velocity
elds. The third version is a generalization in nD of Gelfand mass coordinate, a constant of the motion in 1 D [17], by the introduction of n-dimensional vector valued applications that enable to determine the mass densities as the determinant of their Jaco- bian matrices. Comparison of this set of mass coordinates with other sets of constants of the motion familiar in Fluid Dynamics is made in sub-section 3.3. Note that version two and three are identical for one-dimensional problems. The fourth version is based on the introduction of the Lagrangian coordinates that describe the characteristics of the different models and are parametrized by the former auxiliary vector fields. As illustrations, weak solutions of several models of Coulombian and Newtonian systems known in Plasma Physics and in Cosmology, respectively, with spherically symmetric boundary densities are given in section 4. However, and up to one exception given in the sub-section 3.3, calculations of the actions associated to these illustrations are not reported in this paper, nor the important analysis of the convexity-concavity properties of our Lagrangians. Lastly, and for the same models as those evoked above, a survey of past work concerning weak solutions of the Cauchy problem obeying the Hopf-Lax variational principle extended to negative time and having cor- related initial conditions is given in the Introduction as well as the derivation of the continuum fluid limit from many particle Hamiltonians.

In this thesis we study calculus of variations for differential forms. In the first part we develop the framework of direct methods of calculus of variations in the context of minimization problems for functionals of one or several differential forms of the type, $\int_{\Omega} f(d\omega), \quad \int_{\Omega} f(d\omega_{1}, \ldots, d\omega_{m}) \quad \text{ and } \int_{\Omega} f(d\omega, \delta\omega).$ We introduce the appropriate convexity notions in each case, called \emph{ext. polyconvexity}, \emph{ext. quasiconvexity} and \emph{ext. one convexity} for functionals of the type $\int_{\Omega} f(d\omega),$ \emph{vectorial ext. polyconvexity}, \emph{vectorial ext. quasiconvexity} and \emph{vectorial ext. one convexity} for functionals of the type $\int_{\Omega} f(d\omega_{1}, \ldots, d\omega_{m})$ and \emph{ext-int. polyconvexity}, \emph{ext-int. quasiconvexity} and \emph{ext-int. one convexity} for functionals of the type $\int_{\Omega} f(d\omega, \delta\omega).$ We study their interrelationships and the connections of these convexity notions with the classical notion of polyconvexity, quasiconvexity and rank one convexity in classical vectorial calculus of variations. We also study weak lower semicontinuity and weak continuity of these functionals in appropriate spaces, address coercivity issues and obtain existence theorems for minimization problems for functionals of one differential forms.\smallskip In the second part we study different boundary value problems for linear, semilinear and quasilinear Maxwell type operator for differential forms. We study existence and derive interior regularity and $L^{2}$ boundary regularity estimates for the linear Maxwell operator $\delta (A(x)d\omega) = f$ with different boundary conditions and the related Hodge Laplacian type system $\delta (A(x)d\omega) + d\delta\omega = f,$ with appropriate boundary data. We also deduce, as a corollary, some existence and regularity results for div-curl type first order systems. We also deduce existence results for semilinear boundary value problems \begin{align*} \left\lbrace \begin{gathered} \delta ( A (x) ( d\omega ) ) + f( \omega ) = \lambda\omega \text{ in } \Omega, \ \nu \wedge \omega = 0 \text{ on } \partial\Omega. \end{gathered} \right. \end{align*} Lastly, we briefly discuss existence results for quasilinear Maxwell operator \begin{align*} \delta ( A ( x, d \omega ) ) = f , \end{align*} with different boundary data.

We study integrals of the form integral(Omega) f(d omega(1), ..., d omega(m)), where m >= 1 is a given integer, 1