Lecture 10: Vector Fields, Curl and Divergence
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Arxiv:1507.07356V2 [Math.AP]
TEN EQUIVALENT DEFINITIONS OF THE FRACTIONAL LAPLACE OPERATOR MATEUSZ KWAŚNICKI Abstract. This article discusses several definitions of the fractional Laplace operator ( ∆)α/2 (α (0, 2)) in Rd (d 1), also known as the Riesz fractional derivative − ∈ ≥ operator, as an operator on Lebesgue spaces L p (p [1, )), on the space C0 of ∈ ∞ continuous functions vanishing at infinity and on the space Cbu of bounded uniformly continuous functions. Among these definitions are ones involving singular integrals, semigroups of operators, Bochner’s subordination and harmonic extensions. We collect and extend known results in order to prove that all these definitions agree: on each of the function spaces considered, the corresponding operators have common domain and they coincide on that common domain. 1. Introduction We consider the fractional Laplace operator L = ( ∆)α/2 in Rd, with α (0, 2) and d 1, 2, ... Numerous definitions of L can be found− in− literature: as a Fourier∈ multiplier with∈{ symbol} ξ α, as a fractional power in the sense of Bochner or Balakrishnan, as the inverse of the−| Riesz| potential operator, as a singular integral operator, as an operator associated to an appropriate Dirichlet form, as an infinitesimal generator of an appropriate semigroup of contractions, or as the Dirichlet-to-Neumann operator for an appropriate harmonic extension problem. Equivalence of these definitions for sufficiently smooth functions is well-known and easy. The purpose of this article is to prove that, whenever meaningful, all these definitions are equivalent in the Lebesgue space L p for p [1, ), ∈ ∞ in the space C0 of continuous functions vanishing at infinity, and in the space Cbu of bounded uniformly continuous functions. -
Symmetry and Tensors Rotations and Tensors
Symmetry and tensors Rotations and tensors A rotation of a 3-vector is accomplished by an orthogonal transformation. Represented as a matrix, A, we replace each vector, v, by a rotated vector, v0, given by multiplying by A, 0 v = Av In index notation, 0 X vm = Amnvn n Since a rotation must preserve lengths of vectors, we require 02 X 0 0 X 2 v = vmvm = vmvm = v m m Therefore, X X 0 0 vmvm = vmvm m m ! ! X X X = Amnvn Amkvk m n k ! X X = AmnAmk vnvk k;n m Since xn is arbitrary, this is true if and only if X AmnAmk = δnk m t which we can rewrite using the transpose, Amn = Anm, as X t AnmAmk = δnk m In matrix notation, this is t A A = I where I is the identity matrix. This is equivalent to At = A−1. Multi-index objects such as matrices, Mmn, or the Levi-Civita tensor, "ijk, have definite transformation properties under rotations. We call an object a (rotational) tensor if each index transforms in the same way as a vector. An object with no indices, that is, a function, does not transform at all and is called a scalar. 0 A matrix Mmn is a (second rank) tensor if and only if, when we rotate vectors v to v , its new components are given by 0 X Mmn = AmjAnkMjk jk This is what we expect if we imagine Mmn to be built out of vectors as Mmn = umvn, for example. In the same way, we see that the Levi-Civita tensor transforms as 0 X "ijk = AilAjmAkn"lmn lmn 1 Recall that "ijk, because it is totally antisymmetric, is completely determined by only one of its components, say, "123. -
A Brief Tour of Vector Calculus
A BRIEF TOUR OF VECTOR CALCULUS A. HAVENS Contents 0 Prelude ii 1 Directional Derivatives, the Gradient and the Del Operator 1 1.1 Conceptual Review: Directional Derivatives and the Gradient........... 1 1.2 The Gradient as a Vector Field............................ 5 1.3 The Gradient Flow and Critical Points ....................... 10 1.4 The Del Operator and the Gradient in Other Coordinates*............ 17 1.5 Problems........................................ 21 2 Vector Fields in Low Dimensions 26 2 3 2.1 General Vector Fields in Domains of R and R . 26 2.2 Flows and Integral Curves .............................. 31 2.3 Conservative Vector Fields and Potentials...................... 32 2.4 Vector Fields from Frames*.............................. 37 2.5 Divergence, Curl, Jacobians, and the Laplacian................... 41 2.6 Parametrized Surfaces and Coordinate Vector Fields*............... 48 2.7 Tangent Vectors, Normal Vectors, and Orientations*................ 52 2.8 Problems........................................ 58 3 Line Integrals 66 3.1 Defining Scalar Line Integrals............................. 66 3.2 Line Integrals in Vector Fields ............................ 75 3.3 Work in a Force Field................................. 78 3.4 The Fundamental Theorem of Line Integrals .................... 79 3.5 Motion in Conservative Force Fields Conserves Energy .............. 81 3.6 Path Independence and Corollaries of the Fundamental Theorem......... 82 3.7 Green's Theorem.................................... 84 3.8 Problems........................................ 89 4 Surface Integrals, Flux, and Fundamental Theorems 93 4.1 Surface Integrals of Scalar Fields........................... 93 4.2 Flux........................................... 96 4.3 The Gradient, Divergence, and Curl Operators Via Limits* . 103 4.4 The Stokes-Kelvin Theorem..............................108 4.5 The Divergence Theorem ...............................112 4.6 Problems........................................114 List of Figures 117 i 11/14/19 Multivariate Calculus: Vector Calculus Havens 0. -
Curl, Divergence and Laplacian
Curl, Divergence and Laplacian What to know: 1. The definition of curl and it two properties, that is, theorem 1, and be able to predict qualitatively how the curl of a vector field behaves from a picture. 2. The definition of divergence and it two properties, that is, if div F~ 6= 0 then F~ can't be written as the curl of another field, and be able to tell a vector field of clearly nonzero,positive or negative divergence from the picture. 3. Know the definition of the Laplace operator 4. Know what kind of objects those operator take as input and what they give as output. The curl operator Let's look at two plots of vector fields: Figure 1: The vector field Figure 2: The vector field h−y; x; 0i: h1; 1; 0i We can observe that the second one looks like it is rotating around the z axis. We'd like to be able to predict this kind of behavior without having to look at a picture. We also promised to find a criterion that checks whether a vector field is conservative in R3. Both of those goals are accomplished using a tool called the curl operator, even though neither of those two properties is exactly obvious from the definition we'll give. Definition 1. Let F~ = hP; Q; Ri be a vector field in R3, where P , Q and R are continuously differentiable. We define the curl operator: @R @Q @P @R @Q @P curl F~ = − ~i + − ~j + − ~k: (1) @y @z @z @x @x @y Remarks: 1. -
Electromagnetic Fields and Energy
MIT OpenCourseWare http://ocw.mit.edu Haus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, 1989. ISBN: 9780132490207. Please use the following citation format: Haus, Hermann A., and James R. Melcher, Electromagnetic Fields and Energy. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed [Date]). License: Creative Commons Attribution-NonCommercial-Share Alike. Also available from Prentice-Hall: Englewood Cliffs, NJ, 1989. ISBN: 9780132490207. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms 8 MAGNETOQUASISTATIC FIELDS: SUPERPOSITION INTEGRAL AND BOUNDARY VALUE POINTS OF VIEW 8.0 INTRODUCTION MQS Fields: Superposition Integral and Boundary Value Views We now follow the study of electroquasistatics with that of magnetoquasistat ics. In terms of the flow of ideas summarized in Fig. 1.0.1, we have completed the EQS column to the left. Starting from the top of the MQS column on the right, recall from Chap. 3 that the laws of primary interest are Amp`ere’s law (with the displacement current density neglected) and the magnetic flux continuity law (Table 3.6.1). � × H = J (1) � · µoH = 0 (2) These laws have associated with them continuity conditions at interfaces. If the in terface carries a surface current density K, then the continuity condition associated with (1) is (1.4.16) n × (Ha − Hb) = K (3) and the continuity condition associated with (2) is (1.7.6). a b n · (µoH − µoH ) = 0 (4) In the absence of magnetizable materials, these laws determine the magnetic field intensity H given its source, the current density J. -
4.1 Discrete Differential Geometry
Spring 2015 CSCI 599: Digital Geometry Processing 4.1 Discrete Differential Geometry Hao Li http://cs599.hao-li.com 1 Outline • Discrete Differential Operators • Discrete Curvatures • Mesh Quality Measures 2 Differential Operators on Polygons Differential Properties! • Surface is sufficiently differentiable • Curvatures → 2nd derivatives 3 Differential Operators on Polygons Differential Properties! • Surface is sufficiently differentiable • Curvatures → 2nd derivatives Polygonal Meshes! • Piecewise linear approximations of smooth surface • Focus on Discrete Laplace Beltrami Operator • Discrete differential properties defined over (x) N 4 Local Averaging Local Neighborhood ( x ) of a point !x N • often coincides with mesh vertex vi • n-ring neighborhood ( v ) or local geodesic ball Nn i 5 Local Averaging Local Neighborhood ( x ) of a point !x N • often coincides with mesh vertex vi • n-ring neighborhood ( v ) or local geodesic ball Nn i Neighborhood size! • Large: smoothing is introduced, stable to noise • Small: fine scale variation, sensitive to noise 6 Local Averaging: 1-Ring (x) N x Barycentric cell Voronoi cell Mixed Voronoi cell (barycenters/edgemidpoints) (circumcenters) (circumcenters/midpoint) tight error bound better approximation 7 Barycentric Cells Connect edge midpoints and triangle barycenters! • Simple to compute • Area is 1/3 o triangle areas • Slightly wrong for obtuse triangles 8 Mixed Cells Connect edge midpoints and! • Circumcenters for non-obtuse triangles • Midpoint of opposite edge for obtuse triangles • Better approximation, -
A Doubly Nonlocal Laplace Operator and Its Connection to the Classical Laplacian Petronela Radu University of Nebraska - Lincoln, [email protected]
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications, Department of Mathematics Mathematics, Department of 2019 A Doubly Nonlocal Laplace Operator and Its Connection to the Classical Laplacian Petronela Radu University of Nebraska - Lincoln, [email protected] Kelseys Wells University of Nebraska - Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathfacpub Part of the Applied Mathematics Commons, and the Mathematics Commons Radu, Petronela and Wells, Kelseys, "A Doubly Nonlocal Laplace Operator and Its Connection to the Classical Laplacian" (2019). Faculty Publications, Department of Mathematics. 215. https://digitalcommons.unl.edu/mathfacpub/215 This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Faculty Publications, Department of Mathematics by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. A DOUBLY NONLOCAL LAPLACE OPERATOR AND ITS CONNECTION TO THE CLASSICAL LAPLACIAN PETRONELA RADU, KELSEY WELLS Abstract. In this paper, motivated by the state-based peridynamic frame- work, we introduce a new nonlocal Laplacian that exhibits double nonlocality through the use of iterated integral operators. The operator introduces addi- tional degrees of flexibility that can allow for better representation of physical phenomena at different scales and in materials with different properties. We study mathematical properties of this state-based Laplacian, including connec- tions with other nonlocal and local counterparts. Finally, we obtain explicit rates of convergence for this doubly nonlocal operator to the classical Laplacian as the radii for the horizons of interaction kernels shrink to zero. 1. Introduction Physical phenomena that are beset by discontinuities in the solution, or in the domain, have been challenging to study in the context of classical partial differen- tial equations (PDEs). -
The Electrostatic Potential Is the Coulomb Integral Transform of the Electric Charge Density Revista Mexicana De Física, Vol
Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México Medina, L.; Ley Koo, E. Mathematics motivated by physics: the electrostatic potential is the Coulomb integral transform of the electric charge density Revista Mexicana de Física, vol. 54, núm. 2, diciembre, 2008, pp. 153-159 Sociedad Mexicana de Física A.C. Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=57028302013 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative ENSENANZA˜ REVISTA MEXICANA DE FISICA´ E 54 (2) 153–159 DICIEMBRE 2008 Mathematics motivated by physics: the electrostatic potential is the Coulomb integral transform of the electric charge density L. Medinaa and E. Ley Koo a;b aFacultad de Ciencias, Universidad Nacional Autonoma´ de Mexico,´ Ciudad Universitaria, Mexico´ D.F., 04510, Mexico,´ e-mail: [email protected], bInstituto de F´ısica, Universidad Nacional Autonoma´ de Mexico,´ Apartado Postal 20-364, 01000 Mexico D.F., Mexico, e-mail: eleykoo@fisica.unam.mx Recibido el 19 de octubre de 2007; aceptado el 11 de marzo de 2007 This article illustrates a practical way to connect and coordinate the teaching and learning of physics and mathematics. The starting point is the electrostatic potential, which is obtained in any introductory course of electromagnetism from the Coulomb potential and the superposition principle for any charge distribution. The necessity to develop solutions to the Laplace and Poisson differential equations is also recognized, identifying the Coulomb potential as the generating function of harmonic functions. -
Divergence, Gradient and Curl Based on Lecture Notes by James
Divergence, gradient and curl Based on lecture notes by James McKernan One can formally define the gradient of a function 3 rf : R −! R; by the formal rule @f @f @f grad f = rf = ^{ +| ^ + k^ @x @y @z d Just like dx is an operator that can be applied to a function, the del operator is a vector operator given by @ @ @ @ @ @ r = ^{ +| ^ + k^ = ; ; @x @y @z @x @y @z Using the operator del we can define two other operations, this time on vector fields: Blackboard 1. Let A ⊂ R3 be an open subset and let F~ : A −! R3 be a vector field. The divergence of F~ is the scalar function, div F~ : A −! R; which is defined by the rule div F~ (x; y; z) = r · F~ (x; y; z) @f @f @f = ^{ +| ^ + k^ · (F (x; y; z);F (x; y; z);F (x; y; z)) @x @y @z 1 2 3 @F @F @F = 1 + 2 + 3 : @x @y @z The curl of F~ is the vector field 3 curl F~ : A −! R ; which is defined by the rule curl F~ (x; x; z) = r × F~ (x; y; z) ^{ |^ k^ = @ @ @ @x @y @z F1 F2 F3 @F @F @F @F @F @F = 3 − 2 ^{ − 3 − 1 |^+ 2 − 1 k:^ @y @z @x @z @x @y Note that the del operator makes sense for any n, not just n = 3. So we can define the gradient and the divergence in all dimensions. However curl only makes sense when n = 3. Blackboard 2. The vector field F~ : A −! R3 is called rotation free if the curl is zero, curl F~ = ~0, and it is called incompressible if the divergence is zero, div F~ = 0. -
De Vectores Al´Algebra Geométrica
De Vectores al Algebra´ Geom´etrica Sergio Ramos Ram´ırez, Jos´eAlfonso Ju´arezGonz´alez, Volkswagen de M´exico 72700 San Lorenzo Almecatla, Cuautlancingo, Pue., M´exico Garret Sobczyk Universidad de las Am´ericas-Puebla Departamento de F´ısico-Matem´aticas 72820 Puebla, Pue., M´exico February 4, 2018 Abstract El Algebra´ Geom´etricaes la extensi´onnatural del concepto de vector y de la suma de vectores. Despu´esde revisar las propiedades de la suma vec- torial, definimos la multiplicaci´onde vectores de tal manera que respete el famoso Teorema de Pit´agoras. Varias demostraciones sint´eticasde teo- remas en geometr´ıaeuclidiana pueden as´ıser reemplazadas por elegantes demostraciones algebraicas. Aunque en este art´ıculonos limitamos a 2 y 3 dimensiones, el Algebra´ Geom´etricaes aplicable a cualquier dimensi´on, tanto en geometr´ıaseuclidianas como no-euclidianas. 0 Introducci´on La evoluci´ondel concepto de n´umero,el cual es una noci´oncentral en Matem´aticas, tiene una larga y fascinate historia, la cual abarca muchos siglos y ha sido tes- tigo del florecimiento y ca´ıdade muchas civilizaciones [4]. En lo que respecta a la introducci´onde los conceptos de n´umerosnegativos e imaginarios, Gauss hac´ıala siguiente observaci´onen 1831: \... este tipo de avances, sin embargo, siempre se han hecho en primera instancia con pasos t´ımidosy cautelosos". En el presente art´ıculo,presentamos al lector las ideas y m´etodos m´aselementales del Algebra´ Geom´etrica,la cual fue descubierta por William Kingdon Clifford (1845-1879) poco antes de su muerte [1], y constituye la generalizaci´onnatural de los sistemas de n´umerosreales y complejos, a trav´esde la introducci´onde nuevos entes matem´aticosdenominados n´umeros con direcci´on. -
Nonlocal Exterior Calculus on Riemannian Manifolds
The Pennsylvania State University The Graduate School Department of Mathematics NONLOCAL EXTERIOR CALCULUS ON RIEMANNIAN MANIFOLDS A Dissertation in Mathematics by Thinh Duc Le c 2013 Thinh Duc Le Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2013 ii The dissertation of Thinh Duc Le was reviewed and approved* by the following: Qiang Du Verne M. Willaman Professor of Mathematics Dissertation Adviser Chair of Committee Long-Qing Chen Distinguished Professor of Material Sciences and Engineering Ping Xu Distinguished Professor of Mathematics Mathieu Stienon Associate Professor of Mathematics Svetlana Katok Director of Graduate Studies, Department of Mathematics *Signatures are on file in the Graduate School. iii Abstract Exterior calculus and differential forms are basic mathematical concepts that have been around for centuries. Variations of these concepts have also been made over the years such as the discrete exterior calculus and the finite element exterior calculus. In this work, motivated by the recent studies of nonlocal vector calculus we develop a nonlocal exterior calculus framework on Riemannian manifolds which mimics many properties of the standard (local/smooth) exterior calculus. However the key difference is that nonlocal \interactions" (functions, operators, fields,...) are not required to be smooth. Also any point/particle can interact directly with any other point/particle in the studied domain (at least in principle). Just as in the standard context, we introduce all necessary elements of exterior calculus such as forms, vector fields, exterior derivatives, etc. We point out the relationships between these elements with the known ones in (local) exterior calcu- lus, discrete exterior calculus, etc. -
Spherical Coordinates
Spherical Coordinates z r^ Transforms ^ " r The forward and reverse coordinate transformations are ! ^ ! r = x2 + y2 + z 2 r x = rsin! cos" y arctan" x2 y2 , z$ y r sin sin ! = # + % = ! " z = rcos! & = arctan(y,x) x " where we formally take advantage of the two argument arctan function to eliminate quadrant confusion. Unit Vectors The unit vectors in the spherical coordinate system are functions of position. It is convenient to express them in terms of the spherical coordinates and the unit vectors of the rectangular coordinate system which are not themselves functions of position. ! r xxˆ + yyˆ + zzˆ rˆ = = = xˆ sin! cos" + yˆ sin! sin " + zˆ cos! r r zˆ #rˆ "ˆ = = $ xˆ sin " + yˆ cos" sin! !ˆ = "ˆ # rˆ = xˆ cos! cos" + yˆ cos! sin " $ zˆ sin! Variations of unit vectors with the coordinates Using the expressions obtained above it is easy to derive the following handy relationships: !rˆ = 0 !r !rˆ = xˆ cos" cos# + yˆ cos" sin # $ zˆ sin " = "ˆ !" !rˆ = $xˆ sin" sin # + yˆ sin " cos# = ($ xˆ sin # + yˆ cos #)sin" = #ˆ sin" !# !"ˆ = 0 !r !"ˆ = 0 !# !"ˆ = $xˆ cos" $ yˆ sin " = $ rˆ sin # + #ˆ cos# !" ( ) !"ˆ = 0 !r !"ˆ = # xˆ sin " cos$ # yˆ sin" sin $ # zˆ cos" = #rˆ !" !"ˆ = # xˆ cos" sin $ + yˆ cos" cos$ = $ˆ cos" !$ Path increment ! We will have many uses for the path increment d r expressed in spherical coordinates: ! $ !rˆ !rˆ !rˆ ' dr = d(rrˆ ) = rˆ dr + rdrˆ = rˆ dr + r& dr + d" + d#) % !r !" !# ( = rˆ dr +"ˆ rd " + #ˆr sin "d# Time derivatives of the unit vectors We will also have many uses for the time derivatives of the unit vectors