DOCSLIB.ORG

A Review: Operators Involving ∇In Curvilinear Orthogonal Coordinates

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. Furthermore, all important product rules for the operator, which are utilized throughout the derivations in section 3, are proven using the Levi-Civita-Symbol and the Kronecker-Deltar (Section 5). I 1 THE LAPLACIAN IN SPHERICAL COORDINATES - THE TEDIOUS WAY 1 1 The Laplacian in Spherical Coordinates - The Tedious Way In this section the Laplacian operator acting on a scalar function is derived for the special case of spherical coordinates only. The way presented here and often described in physic books is very tedious and involved, though only the very basic concepts of partial differential calculus are used here. However, the operator may be calculated way easier using the method decribed in (Section 3) and carried out in (Section 4)! Recall the definition of the Laplacian in cartesian coordonates: @ 2 f @ 2 f @ 2 f ∆f := + + (1) @ x 2 @ y 2 @ z 2 Some things we need and the transforms: x = r sinθ cosφ y = r sinθ sinφ z = r cosθ (2) ! z y p 2 2 2 θ = arccos φ = arctan r = x + y + z (3) p 2 2 2 x x + y + z f = f x r,θ ,φ ,y r,θ ,φ ,z r,θ ,φ (4) @ f @ f @ r @ f @ θ @ f @ φ = + + (5) @ x @ r @ x @ θ @ x @ φ @ x @ f @ f @ r @ f @ θ @ f @ φ = + + (6) @ y @ r @ y @ θ @ y @ φ @ y @ f @ f @ r @ f @ θ @ f @ φ = + + (7) @ z @ r @ z @ θ @ z @ φ @ z 1 1 1 1 0 1 arccosx x arctanx arctan (8) ( )0 = p p 0 = ( )0 = 2 = 2 − 1 x 2 2px 1 + x x − 1 + x − p y y y x @ x 2 y 2 z 2 1 2x 0 0 + + arctan = 2 2 arctan = 2 2 = (9) x y y x x x x y @ x 2 p 2 2 2 − + + x + y + z Now we begin: 2 3 2 3 @ f @ f 1 2r sinθ cosφ @ f 1 1 2x @ f r sinθ sinφ 6 z 7 4 5 = + 4 Æ 2 2 − 23=2 5 + 2 2 2 2 @ x @ r 2 r @ θ − r cos θ 2 r @ φ − r sin θ cos φ + sin φ 1 r 2 − @ f @ f 1 @ f sinφ = sinθ cosφ + cosθ cosφ (10) @ r @ θ r − @ φ r sinθ 2 3 2 3 @ f @ f 1 2r sinθ sinφ @ f 1 1 2y @ f r sinθ cosφ 6 z 7 4 5 = + 4 Æ 2 2 − 23=2 5 + 2 2 2 2 @ y @ r 2 r @ θ − r cos θ 2 r @ φ − r sin θ cos φ + sin φ 1 r 2 − @ f @ f 1 @ f cosφ = sinθ sinφ + cosθ sinφ + (11) @ r @ θ r @ φ r sinθ 2 1 3 @ f @ f 1 2r cosθ @ f 1 r r cosθ 2r 2r cosθ @ f = + 4 − 2 5 + [0] @ z @ r 2 r @ θ − sinθ r @ φ @ f @ f sinθ = cosθ (12) @ r − @ θ r 1 THE LAPLACIAN IN SPHERICAL COORDINATES - THE TEDIOUS WAY 2 @ 2 f @ @ f @ @ f @ r @ @ f @ θ @ @ f @ φ = = + + (13) @ x 2 @ x @ x @ r @ x @ x @ θ @ x @ x @ φ @ x @ x @ 2 f @ @ f @ @ f @ r @ @ f @ θ @ @ f @ φ = = + + (14) @ y 2 @ y @ y @ r @ y @ x @ θ @ x @ y @ φ @ y @ y @ 2 f @ @ f @ @ f @ r @ @ f @ θ @ @ f @ φ = = + + (15) @ z 2 @ z @ z @ r @ z @ z @ θ @ z @ z @ φ @ z @ z Plugging equations (10),(11),(12) into (13),(14),(15) we get: @ 2 f @ @ f @ f 1 @ f sinφ 2 = sinθ cosφ + cosθ cosφ sinθ cosφ + ... @ x @ r @ r @ θ r − @ φ r sinθ @ @ f @ f 1 @ f sinφ cosθ cosφ sinθ cosφ + cosθ cosφ + ... @ θ @ r @ θ r − @ φ r sinθ r @ @ f @ f 1 @ f sinφ sinφ sinθ cosφ + cosθ cosφ − @ φ @ r @ θ r − @ φ r sinθ r sinθ @ 2 f 1 @ f sinφ @ f = sinθ cosφ 2 cosθ cosφ 2 + 2 sinθ cosφ + ... @ r − r @ θ r sinθ @ φ @ f 1 @ f 1 @ 2 f cosθ sinφ @ f cosθ cosφ cosθ cosφ sinθ cosφ + cosθ cosφ 2 + 2 + ... @ r − r @ θ r @ θ r sin θ @ φ r @ f 1 @ f cosφ @ f sinφ @ 2 f sinφ sinθ sinφ cosθ sinφ 2 − − @ r − r @ θ − r sinθ @ φ − r sinθ @ φ sinθ 2 2 2 2 @ f sinθ cosθ cos φ @ f sinφ cosφ @ f = sin θ cos φ 2 2 + 2 + ... @ r − r @ θ r @ φ cos2 θ cos2 φ @ f sinθ cosθ cos2 φ @ f cos2 θ cos2 φ @ 2 f cos2 θ cosφ sinφ @ f ... 2 + 2 2 + 2 2 + r @ r − r @ θ r @ θ r sin θ @ φ sin2 φ @ f cosθ sin2 φ @ f cosφ sinφ @ f sin2 φ @ 2 f + + + (16) r @ r r 2 sinθ @ θ r 2 sin2 θ @ φ r 2 sin2 θ @ φ2 @ 2 f @ @ f @ f 1 @ f cosφ = sinθ sinφ + cosθ sinφ + sinθ sinφ + ... @ y 2 @ r @ r @ θ r @ φ r sinθ @ @ f @ f 1 @ f cosφ cosθ sinφ sinθ sinφ + cosθ sinφ + + ... @ θ @ r @ θ r @ φ r sinθ r @ @ f @ f 1 @ f cosφ cosφ sinθ sinφ + cosθ sinφ + @ φ @ r @ θ r @ φ r sinθ r sinθ 2 2 2 2 @ f cosθ sinθ sin φ @ f cosφ sinφ @ f = sin θ sin φ 2 2 2 + ... @ r − r @ θ − r @ φ cos2 θ sin2 φ @ f sinθ cosθ sin2 φ @ f cos2 θ sin2 φ @ 2 f cosφ cos2 θ sinφ @ f ... 2 + 2 2 2 2 + r @ r − r @ θ r @ θ − r sin θ @ φ cos2 φ @ f cos2 φ cosθ @ f sinφ cosφ @ f cos2 φ @ 2 f (17) + 2 2 + 2 2 2 r @ r r sinθ @ θ − r sin θ @ φ r sin θ @ φ @ 2 f @ @ f @ f sinθ 2 = cosθ cosθ + ... @ z @ r @ r − @ θ r @ @ f @ f sinθ sinθ cosθ − @ θ @ r − @ θ r r 2 2 @ f sinθ cosθ @ f = cos θ + + ... @ r 2 r 2 @ θ sin2 θ @ f cosθ sinθ @ f sin2 θ @ 2 f + + (18) r @ r r 2 @ θ r 2 @ θ 2 1 THE LAPLACIAN IN SPHERICAL COORDINATES - THE TEDIOUS WAY 3 Using equation (16),(17),(18) we obtain the Laplacian: @ 2 f @ 2 f @ 2 f ∆f = + + @ x 2 @ y 2 @ z 2 @ 2 f sinθ cosθ @ f cos2 θ @ f sinθ cosθ @ f cos2 θ @ 2 f 1 @ f cosθ @ f = 2 2 + 2 + 2 2 + + 2 + ... @ r − r @ θ r @ r − r @ θ r @ θ r @ r r sinθ @ θ 1 @ 2 f sinθ cosθ @ f sin2 θ @ f cosθ sinθ @ f sin2 θ @ 2 f + + + + r 2 sin2 θ @ φ2 r 2 @ θ r @ r r 2 @ θ r 2 @ θ 2 @ 2 f 2 @ f 1 @ 2 f 1 @ 2 f cosθ @ f = + + + + @ r 2 r @ r r 2 @ θ 2 r 2 sin2 θ @ φ2 r 2 sinθ @ θ @ 2 f 2 @ f 1 @ 2 f 1 @ 2 f @ f = + + + sinθ + cosθ @ r 2 r @ r r 2 sin2 θ @ φ2 r 2 sinθ @ θ 2 @ θ @ 2 f 2 @ f 1 @ 2 f 1 @ @ f = + + + sinθ @ r 2 r @ r r 2 sin2 θ @ φ2 r 2 sinθ @ θ @ θ @ 2 2 @ 1 @ @ 1 @ 2 ∆ = + + sinθ + (19) @ r 2 r @ r r 2 sinθ @ θ @ θ r 2 sin2 θ @ φ2 We also note the following relations: 2 2 1 @ 2 @ f 1 @ f 2 @ f 2 @ f @ f r = 2r + r = + (20) r 2 @ r @ r r 2 @ r @ r 2 r @ r @ r 2 2 2 2 1 @ 1 @ @ f 1 @ f @ f @ f 2 @ f @ f r f (r ) = f (r ) + r = + + r = + (21) r @ r 2 r @ r @ r r @ r @ r @ r 2 r @ r @ r 2 And hence obtain the Laplacian operator in spherical coordinates: 2 1 @ 2 @ 1 @ @ 1 @ ∆ = r + sinθ + (22) r 2 @ r @ r r 2 sinθ @ θ @ θ r 2 sin2 θ @ φ2 Or equivalently: 1 @ 2 1 @ @ 1 @ 2 ∆ = r + sinθ + (23) r @ r 2 r 2 sinθ @ θ @ θ r 2 sin2 θ @ φ2 The absolute value of the gradient in cartesian coordinates is: 2 2 2 2 @ f @ f @ f f = + + (24) jr j @ x @ y @ z 2 2 2 2 Using equations (10),(11),(12) and x + y + z = x + y + z + 2x y + 2xz + 2y z we get: 2 2 2 @ f @ f 1 @ f sinφ @ f @ f 1 @ f cosφ f = sinθ cosφ + cosθ cosφ + sinθ sinφ + cosθ sinφ + + ..
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Divergence and Curl Operators in Skew Coordinates
    International Journal of Modeling and Optimization, Vol. 5, No. 3, June 2015 Divergence and Curl Operators in Skew Coordinates Rajai S. Alassar and Mohammed A. Abushoshah where r is the position vector in any three dimensional space, Abstract—The divergence and curl operators appear in i.e. r r(,,) u1 u 2 u 3 . numerous differential equations governing engineering and physics problems. These operators, whose forms are well r Note that is a tangent vector to the ui curve where known in general orthogonal coordinates systems, assume ui different casts in different systems. In certain instances, one the other two coordinates variables remain constant. A unit needs to custom-make a coordinates system that my turn out to be skew (i.e. not orthogonal). Of course, the known formulas for tangent vector in this direction, therefore, is the divergence and curl operators in orthogonal coordinates are not useful in such cases, and one needs to derive their rr counterparts in skew systems. In this note, we derive two uuii r formulas for the divergence and curl operators in a general ei or hiei (3) coordinates system, whether orthogonal or not. These formulas r hi ui generalize the well known and widely used relations for ui orthogonal coordinates systems. In the process, we define an orthogonality indicator whose value ranges between zero and unity. The Divergence of a vector field over a control volume V bounded by the surface S , denoted by F , is defined by Index Terms—Coordinates systems, curl, divergence, Laplace, skew systems. F n dS F lim S (4) I.
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • 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).
    [Show full text]
  • A Primer on Tensor Calculus
    A Primer on Tensor Calculus David A. Clarke Saint Mary’s University, Halifax NS, Canada [email protected] June, 2011 Copyright c David A. Clarke, 2011 Contents Preface ii 1 Introduction 1 2 Definition of a tensor 3 3 The metric 9 3.1 Physical components and basis vectors ..................... 11 3.2 The scalar and inner products .......................... 14 3.3 Invariance of tensor expressions ......................... 17 3.4 The permutation tensors ............................. 18 4 Tensor derivatives 21 4.1 “Christ-awful symbols” .............................. 21 4.2 Covariant derivative ............................... 25 5 Connexion to vector calculus 30 5.1 Gradient of a scalar ................................ 30 5.2 Divergence of a vector .............................. 30 5.3 Divergence of a tensor .............................. 32 5.4 The Laplacian of a scalar ............................. 33 5.5 Curl of a vector .................................. 34 5.6 The Laplacian of a vector ............................ 35 5.7 Gradient of a vector ............................... 35 5.8 Summary ..................................... 36 5.9 A tensor-vector identity ............................. 37 6 Cartesian, cylindrical, and spherical polar coordinates 39 6.1 Cartesian coordinates ............................... 40 6.2 Cylindrical coordinates .............................. 40 6.3 Spherical polar coordinates ............................ 40 7 An application to viscosity 42 i Preface These notes stem from my own need to refresh my memory on the
    [Show full text]
  • Class: M. Sc. Subject: Mathematical Methods for Physics
    Class: M. Sc. Subject: Mathematical Methods for Physics I have taken all course materials for Mathematical Methods for Physics from G. B. Arfken and H.J. Weber, Mathematical Methods for Physicists. Students can download this book form given web address; Web address: https://b-ok.cc/book/446170/6e805d All topics have been taken from Chapter-2 & Chapter-4 from above said book ( https://b- ok.cc/book/446170/6e805d ). I am sending pdf file of Chapter 2 and Chapter 4 . Name of Topics Vector Analysis in Curved Coordinates and Tensors, Orthogonal Coordinates in R3, Differential Vector Operators, Special Coordinate Systems, Circular Cylinder Coordinates, Spherical and Polar Coordinates. Tensor Analysis, Contraction, Direct Product, Quotient Rule, Pseudotensors, Dual Tensors, General Tensors, Tensor Derivative Operators. Group Theory, Introduction, Generators of Continuous Group, Orbital Angular Momentum, Angular Momentum Coupling, Homogeneous Lorentz Group, Discrete Groups CHAPTER 2 VECTOR ANALYSIS IN CURVED COORDINATES AND TENSORS In Chapter 1 we restricted ourselves almost completely to rectangular or Cartesian coordi- nate systems. A Cartesian coordinate system offers the unique advantage that all three unit vectors, x, y, and z, are constant in direction as well as in magnitude. We did introduce the radial distanceˆ ˆ r, butˆ even this was treated as a function of x, y, and z. Unfortunately, not all physical problems are well adapted to a solution in Cartesian coordinates. For instance, if we have a central force problem, F rF(r), such as gravitational or electrostatic force, Cartesian coordinates may be unusually= inappropriate.ˆ Such a problem demands the use of a coordinate system in which the radial distance is taken to be one of the coordinates, that is, spherical polar coordinates.
    [Show full text]