Divergence and Curl Operators in Skew Coordinates
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Pure Torsion Problem in Tensor Notation [3] Sual I
56 Budownictwo i Architektura 18(1) (2019) 57-6961-73 DOI: 10.24358/Bud-Arch_19_181_06 7. References: [1] EN 206-1:2013, Concrete – Part 1: Specification, performance, production and conformity. [2] Holicky, M., Vorlicek. M., Fractile estimation and sampling inspection in building, Acta Polytechnica, CVUT Praha, 1992.Vol. 32. Issue 1, 87-96. Pure torsion problem in tensor notation [3] Sual I. Gass, and Arjang A. Assad, An annotated timeline of operation research. An informal history. Springer Science and Business Media, 2005. Sławomir Karaś [4] Harrison. T.A., Crompton, S., Eastwood. C., Richardson. G., Sym, R., Guidance on the application of the EN 206-1 conformity rules, Quarry Products Association, 2001. Department of Roads and Bridges, Faculty of Civil Engineering and Architecture, Lublin University of Technology, e–mail: [email protected] [5] Czarnecki, L. et al., Concrete according to PN EN 206-1- The comment - Collective work, Polish ORCID: 0000-0002-0626-5582 Cement and PKN, Cracow, Poland, 2004. [6] Lynch, B., Kelly, J., McGrath, M., Murphy, B., Newell, J., The new Concrete Standards, An Abstract: The paper examines the application of the tensor calculus to the classic Introduction to EN 206-1, The Irish Concrete Society, 2004. problem of the pure torsion of prismatic rods. The introduction contains a short description [7] Montgomery, D.C., Introduction to Statistical Quality Control, 5th Ed., Wiley, 2005. of the reference frames, base vectors, contravariant and covariant vector coordinates when [8] Taerwe, L., Evaluation of compound compliance criteria for concrete strength, Materials and applying the Einstein summation convention. Torsion formulas were derived according to Structures, 1998, 21, pp. -
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. -
Appendix a Orthogonal Curvilinear Coordinates
Appendix A Orthogonal Curvilinear Coordinates Given a symmetry for a system under study, the calculations can be simplified by choosing, instead of a Cartesian coordinate system, another set of coordinates which takes advantage of that symmetry. For example calculations in spherical coordinates result easier for systems with spherical symmetry. In this chapter we will write the general form of the differential operators used in electrodynamics and then give their expressions in spherical and cylindrical coordi- nates.1 A.1 Orthogonal curvilinear coordinates A system of coordinates u1, u2, u3, can be defined so that the Cartesian coordinates x, y and z are known functions of the new coordinates: x = x(u1, u2, u3) y = y(u1, u2, u3) z = z(u1, u2, u3) (A.1) Systems of orthogonal curvilinear coordinates are defined as systems for which locally, nearby each point P(u1, u2, u3), the surfaces u1 = const, u2 = const, u3 = const are mutually orthogonal. An elementary cube, bounded by the surfaces u1 = const, u2 = const, u3 = const, as shown in Fig. A.1, will have its edges with lengths h1du1, h2du2, h3du3 where h1, h2, h3 are in general functions of u1, u2, u3. The length ds of the line- element OG, one diagonal of the cube, in Cartesian coordinates is: ds = (dx)2 + (dy)2 + (dz)2 1The orthogonal coordinates are presented with more details in J.A. Stratton, Electromagnetic The- ory, McGraw-Hill, 1941, where many other useful coordinate systems (elliptic, parabolic, bipolar, spheroidal, paraboloidal, ellipsoidal) are given. © Springer International Publishing Switzerland 2016 189 F. Lacava, Classical Electrodynamics, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-3-319-39474-9 190 Appendix A: Orthogonal Curvilinear Coordinates Fig. -
Del in Cylindrical and Spherical Coordinates - Wikipedia, the Free Encycl
Del in cylindrical and spherical coordinates - Wikipedia, the free encycl... http://en.wikipedia.org/wiki/Nabla_in_cylindrical_and_spherical_coordi... Make a donation to Wikipedia and give the gift of knowledge! Del in cylindrical and spherical coordinates From Wikipedia, the free encyclopedia (Redirected from Nabla in cylindrical and spherical coordinates) This is a list of some vector calculus formulae of general use in working with standard coordinate systems. Table with the del operator in cylindrical and spherical coordinates Operation Cartesian coordinates (x,y,z) Cylindrical coordinates (ρ,φ,z) Spherical coordinates (r,θ,φ) Definition of coordinates A vector field Gradient Divergence Curl Laplace operator or Differential displacement Differential normal area Differential volume Non-trivial calculation rules: 1. (Laplacian) 2. 3. 4. (using Lagrange's formula for the cross product) 5. Remarks 1 of 2 10/20/2007 7:08 PM Del in cylindrical and spherical coordinates - Wikipedia, the free encycl... http://en.wikipedia.org/wiki/Nabla_in_cylindrical_and_spherical_coordi... This page uses standard physics notation; some (American mathematics) sources define φ as the angle from the z-axis instead of θ. The function atan2(y, x) is used instead of the mathematical function arctan(y/x) due to its domain and image. The classical arctan(y/x) has an image of (-π/2, +π/2), whereas atan2(y, x) is defined to have an image of (-π, π]. See also Orthogonal coordinates Curvilinear coordinates Vector fields in cylindrical and spherical coordinates Retrieved from "http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates" Categories: Vector calculus | Coordinate systems This page was last modified 11:17, 11 October 2007. All text is available under the terms of the GNU Free Documentation License. -
Orthogonal Curvilinear Coordinates
AN INTRODUCTION TO CURVILINEAR ORTHOGONAL COORDINATES Overview Throughout the first few weeks of the semester, we have studied vector calculus using almost exclusively the familiar Cartesian x,y,z coordinate system. In your past math and physics classes, you have encountered other coordinate systems such as cylindri- cal polar coordinates and spherical coordinates. These three coordinate systems (Cartesian, cylindrical, spherical) are actually only a subset of a larger group of coordinate systems we call orthogonal coordinates. We are familiar that the unit vectors in the Cartesian system obey the relationship xi x j dij where d is the Kronecker delta. In other words, the dot product of any two unit vectors is 0 unless they are the same vector (in which case the dot product is one). This is the orthogonality property of vectors, and orthogonal coordinate systems are those in which all the unit vectors obey this property. We will learn general techniques for for translating vector operations into any orthogonal coordinate system, although we will be most concerned with the three systems used most frequently in basic physics (Cartesian, cylindrical polar, and spherical). We will start with very basic geometric properties of coordinate systems, and use those results as the basis for a more detailed analysis. Scale Factors Let' s start by considering a point in the x-y plane. We know we can describe the location of this point by the ordered pair (x,y) if we are using Cartesian coordinates, or by the ordered pair r,f) if we are using polar coordinates. You may also be familiar with the use of the symbols (r,q) for polar coordinates; either usage is fine, but I will try to be consistent in the use of (r,f) for plane polar coordinates, and (r,f,z) for cylindrical polar coordinates. -
A Review: Operators Involving ∇In Curvilinear Orthogonal Coordinates
A Review: Operators involving in Curvilinear Orthogonal Coordinates r and Vector Identities and Product Rules r Markus Selmke Student at the Universität Leipzig, IPSP,[email protected] 22.7.2006, updated 15.02.2008 Contents Abstract I 1 The Laplacian in Spherical Coordinates - The Tedious Way 1 2 The Angular Momentum Operators in Spherical Coordinates 4 3 Operators in Arbitrary Orthogonal Cuvilinear Coordinate Systems 5 r Summary of Important Formulas 11 4 Examples using the General Expressions 12 5 Proofs for the Product Rules 16 r 6 Gauss Divergence Theorem variants 18 References 19 Abstract This document provides easily understandable derivations for all important operators applicable to arbitrary orthog- 3 r onal coordinate systems in R (like spherical, cylindrical, elliptic, parabolic, hyperbolic,..). These include the laplacian (of a scalar function and a vector field), the gradient and its absolute value (of a scalar function), the divergence (of a vector field), the curl (of a vector field) and the convective operator (acting on a scalar function and a vector field) (Section 3). For the first two mentioned special coordinate systems the operators are calculated using these expressions along with some remarks on their simplifications in cartesian coordinates (Section 4). Alongside these calculations, some useful derivations are presented, including the volume element and the velocity components. The Laplacian as well as the absolute value of the gradient are also calculated the clumsy way (ab)using the chain-rule (Section 1). Using the same tedious way as in section 1, the angular momentum operators arising in quantum mechanics are derived for spherical coodinates (Section 2) only. -
Tensor Analysis and Curvilinear Coordinates.Pdf
Tensor Analysis and Curvilinear Coordinates Phil Lucht Rimrock Digital Technology, Salt Lake City, Utah 84103 last update: May 19, 2016 Maple code is available upon request. Comments and errata are welcome. The material in this document is copyrighted by the author. The graphics look ratty in Windows Adobe PDF viewers when not scaled up, but look just fine in this excellent freeware viewer: http://www.tracker-software.com/pdf-xchange-products-comparison-chart . The table of contents has live links. Most PDF viewers provide these links as bookmarks on the left. Overview and Summary.........................................................................................................................7 1. The Transformation F: invertibility, coordinate lines, and level surfaces..................................12 Example 1: Polar coordinates (N=2)..................................................................................................13 Example 2: Spherical coordinates (N=3)...........................................................................................14 Cartesian Space and Quasi-Cartesian Space.......................................................................................15 Pictures A,B,C and D..........................................................................................................................16 Coordinate Lines.................................................................................................................................16 Example 1: Polar coordinates, coordinate lines -
IX. ORTHOGONAL SYSTEMS Orthogonal Coordinate Systems
IX. ORTHOGONAL SYSTEMS Orthogonal coordinate systems produce fewer additional terms in transformed partial differential equations, and thus reduce the amount of computation required. Also, as has been noted in Chapter V, severe departure from orthogonality will introduce truncation error in difference expressions. A general discussion of orthogonal systems on planes and curved surfaces is given in Ref. [42], and various generation procedures are surveyed in Ref. [42] and Ref. [1]. In numerical solutions, the concept of numerical orthogonality, i.e., that the off-diagonal metric coefficients vanish when evaluated numerically, is usually more important than strict analytical orthogonality, especially when the equations to be solved on the system are in the conservative law form. There are basically two types of orthogonal generation systems, those based on the construction of an orthogonal system from a non-orthogonal system, and those involving field solutions of partial differential equations. The first approach involves the construction of orthogonal trajectories on a given non-orthogonal system. Here one set of coordinate lines of the non-orthogonal system is retained, while the other set is replaced by lines emanating from a boundary and constructed by integration across the field so as to cross each line of the retained set orthogonally. Control of the line spacing is exercised through the generation of the non-orthogonal system and through the point distribution on the boundary from which the trajectories start. The point distributions on only three of the four boundaries can be specified. Several methods for the construction of orthogonal trajectories are discussed in Ref. [42] and Ref. -
Orthogonal Curvilinear Coordinates 28.3
® Orthogonal Curvilinear Coordinates 28.3 Introduction The derivatives div, grad and curl from Section 28.2 can be carried out using coordinate systems other than the rectangular Cartesian coordinates. This Section shows how to calculate these derivatives in other coordinate systems. Two coordinate systems - cylindrical polar coordinates and spherical polar coordinates - will be illustrated. • be able to find the gradient, divergence and Prerequisites curl of a field in Cartesian coordinates Before starting this Section you should ... • be familiar with polar coordinates • find the divergence, gradient or curl of a Learning Outcomes vector or scalar field expressed in terms of On completion you should be able to ... orthogonal curvilinear coordinates HELM (2008): 37 Section 28.3: Orthogonal Curvilinear Coordinates 1. Orthogonal curvilinear coordinates The results shown in Section 28.2 have been given in terms of the familiar Cartesian (x, y, z) co- ordinate system. However, other coordinate systems can be used to better describe some physical situations. A set of coordinates u = u(x, y, z), v = v(x, y, z) and w = w(x, y, z) where the direc- tions at any point indicated by u, v and w are orthogonal (perpendicular) to each other is referred to as a set of orthogonal curvilinear coordinates. With each coordinate is associated a scale factor q ∂x 2 ∂y 2 ∂z 2 hu, hv or hw respectively where hu = ∂u + ∂u + ∂u (with similar expressions for hv and hw). The scale factor gives a measure of how a change in the coordinate changes the position of a point. Two commonly-used sets of orthogonal curvilinear coordinates are cylindrical polar coordinates and spherical polar coordinates. -
C:\Book\Booktex\C1s3.DVI
65 §1.3 SPECIAL TENSORS Knowing how tensors are defined and recognizing a tensor when it pops up in front of you are two different things. Some quantities, which are tensors, frequently arise in applied problems and you should learn to recognize these special tensors when they occur. In this section some important tensor quantities are defined. We also consider how these special tensors can in turn be used to define other tensors. Metric Tensor Define yi,i=1,...,N as independent coordinates in an N dimensional orthogonal Cartesian coordinate system. The distance squared between two points yi and yi + dyi,i=1,...,N is defined by the expression ds2 = dymdym =(dy1)2 +(dy2)2 + ···+(dyN )2. (1.3.1) Assume that the coordinates yi are related to a set of independent generalized coordinates xi,i=1,...,N by a set of transformation equations yi = yi(x1,x2,...,xN ),i=1,...,N. (1.3.2) To emphasize that each yi depends upon the x coordinates we sometimes use the notation yi = yi(x), for i =1,...,N. The differential of each coordinate can be written as ∂ym dym = dxj ,m=1,...,N, (1.3.3) ∂xj and consequently in the x-generalized coordinates the distance squared, found from the equation (1.3.1), becomes a quadratic form. Substituting equation (1.3.3) into equation (1.3.1) we find ∂ym ∂ym ds2 = dxidxj = g dxidxj (1.3.4) ∂xi ∂xj ij where ∂ym ∂ym g = ,i,j=1,...,N (1.3.5) ij ∂xi ∂xj i are called the metrices of the space defined by the coordinates x ,i=1,...,N. -
Appendix a Curvilinear Coordinates
Appendix A Curvilinear coordinates A.1 Lam´e coefficients Consider set of equations ξi = ξi(x1, x2, x3) , i = 1, 2, 3 where ξ1, ξ2, ξ3 independent, single-valued and continuous (x1, x2, x3) : coordinates of point P in Cartesian system with radius-vector x (ξ1, ξ2, ξ3) : coordinates of point P in curvilinear system e1, e2, e3: base unit vectors in Cartesian coordinates eξ1 , eξ2 , eξ3 : base unit vectors in curvilinear system we restrict ourselves to orthogonal curvilinear coordinates systems e e = δ ξi · ξj ij Consider x = x(xi(ξj)) and take the total differential (summation over double appearing indices understood) ∂x dx = dξj ∂ξj ∂x partial derivative keeping ξ2, ξ3 fixed: vector tangential to coordinate line ξ1 ∂ξ1 ∂x = h1 eξ1 −→ ∂ξ1 in general dx = h1 dξ1 eξ1 + h2 dξ2 eξ2 + h3 dξ3 eξ3 hi Lam´e coefficients Those coefficients (might) depend on the coordinates ξi hi = hi(ξ1, ξ2, ξ3) 157 hi dξi (no summation) components of length element along eξi Without restriction to orthogonal systems the length element ds = dx is defined as | | 2 ∂xj ∂xj ds = dx dx = dxj dxj = dξl dξm = glm dξl dξm · ∂ξl ∂ξm Coefficients ∂xj ∂xj glm = ∂ξl ∂ξm are called the elements of the metric tensor For an orthogonal curvilinear system the base vectors eξ1 , eξ2 , eξ3 are mutually perpendic- ular and form a right-handed system We get the length element squared 2 2 2 2 2 2 2 ds = h1 (dξ1) + h2 (dξ2) + h3 (dξ3) the Lam´e coefficients 2 2 2 g11 = h1 , g22 = h2 , g33 = h3 , g = 0 for i = k ik 6 the volume element (elementary rectangular box) 3 d x dV = h1h2h3 dξ1dξ2dξ3 ≡ 158 -
Transformation Optics in Orthogonal Coordinates
Transformation optics in orthogonal coordinates Huanyang Chena, b, * a Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China b Institute of Theoretical Physics, Shanghai Jiao Tong University, Shanghai 200240, China * Corresponding author at: Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China. Tel.: +852 2358 7981; fax: +852 2358 1652. E-mail address: [email protected] (Huanyang Chen). Abstract: The author proposes the methodology of transformation optics in orthogonal coordinates to obtain the material parameters of the transformation media from the mapping in orthogonal coordinates. Several examples are given to show the applications of such a methodology by using the full-wave simulations. PACS number(s): 41.20.Jb, 42.25.Fx Keywords: transformation optics; transformation media; cloaking 1 / 23 References: [1] J.B. Pendry, D. Schurig, and D.R. Smith, Science 312, 1780 (2006). [2] U. Leonhardt, Science 312, 1777 (2006). [3] M. Rahm, D. Schurig, D.A. Roberts, S.A. Cummer, D.R. Smith, and J.B. Pendry, Photon. Nanostruct.: Fundam. Applic. 6, 87 (2008). [4] H.Y. Chen and C. T. Chan, Appl. Phys. Lett. 90, 241105 (2007). [5] A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Phys. Rev. Lett. 99, 183901 (2007). [6] A. V. Kildishev and E. E. Narimanov, Opt. Lett. 32, 3432 (2007). [7] H. Ma, S. Qu, Z. Xu, J. Zhang, B. Chen and J. Wang, Phys. Rev. A 77, 013825 (2008). [8] D.-H. Kwon and D. H. Werner, Appl. Phys. Lett. 92, 013505 (2008).