DOCSLIB.ORG

C:\Book\Booktex\C1s3.DVI

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. Here the gij are functions of the x coordinates and is sometimes written as gij = gij (x). Further, the metrices gij are symmetric in the indices i and j so that gij = gji for all values of i and j over the range of the indices. If we transform to another coordinate system, say xi,i=1,...,N, then the element of arc length squared is expressed in terms 2 i j of the barred coordinates and ds = gij dx dx , where gij = gij (x) is a function of the barred coordinates. The following example demonstrates that these metrices are second order covariant tensors. 66 EXAMPLE 1.3-1. Show the metric components gij are covariant tensors of the second order. Solution: In a coordinate system xi,i=1,...,N the element of arc length squared is 2 i j ds = gij dx dx (1.3.6) while in a coordinate system xi,i=1,...,N the element of arc length squared is represented in the form 2 m n ds = gmndx dx . (1.3.7) The element of arc length squared is to be an invariant and so we require that m n i j gmndx dx = gij dx dx (1.3.8) Here it is assumed that there exists a coordinate transformation of the form defined by equation (1.2.30) together with an inverse transformation, as in equation (1.2.32), which relates the barred and unbarred coordinates. In general, if xi = xi(x), then for i =1,...,N we have ∂xi ∂xj dxi = dxm and dxj = dxn (1.3.9) ∂xm ∂xn Substituting these differentials in equation (1.3.8) gives us the result ∂xi ∂xj ∂xi ∂xj g dxmdxn = g dxmdxn or g − g dxmdxn =0 mn ij ∂xm ∂xn mn ij ∂xm ∂xn ∂xi ∂xj For arbitrary changes in dxm this equation implies that g = g and consequently g transforms mn ij ∂xm ∂xn ij as a second order absolute covariant tensor. EXAMPLE 1.3-2. (Curvilinear coordinates) Consider a set of general transformation equations from rectangular coordinates (x, y, z) to curvilinear coordinates (u, v, w). These transformation equations and the corresponding inverse transformations are represented x = x(u, v, w) u = u(x, y, z) y = y(u, v, w) v = v(x, y, z) (1.3.10) z = z(u, v, w). w = w(x, y, z) Here y1 = x, y2 = y, y3 = z and x1 = u, x2 = v, x3 = w are the Cartesian and generalized coordinates and N =3. The intersection of the coordinate surfaces u = c1,v = c2 and w = c3 define coordinate curves of the curvilinear coordinate system. The substitution of the given transformation equations (1.3.10) into the position vector r = x e1 + y e2 + z e3 produces the position vector which is a function of the generalized coordinates and r = r(u, v, w)=x(u, v, w) e1 + y(u, v, w) e2 + z(u, v, w) e3 67 ∂r ∂r ∂r and consequently dr = du + dv + dw, where ∂u ∂v ∂w ∂r ∂x ∂y ∂z E1 = = e1 + e2 + e3 ∂u ∂u ∂u ∂u ∂r ∂x ∂y ∂z E2 = = e1 + e2 + e3 (1.3.11) ∂v ∂v ∂v ∂v ∂r ∂x ∂y ∂z E3 = = e1 + e2 + e3. ∂w ∂w ∂w ∂w are tangent vectors to the coordinate curves. The element of arc length in the curvilinear coordinates is ∂r ∂r ∂r ∂r ∂r ∂r ds2 = dr · dr = · dudu + · dudv + · dudw ∂u ∂u ∂u ∂v ∂u ∂w ∂r ∂r ∂r ∂r ∂r ∂r + · dvdu + · dvdv + · dvdw (1.3.12) ∂v ∂u ∂v ∂v ∂v ∂w ∂r ∂r ∂r ∂r ∂r ∂r + · dwdu + · dwdv + · dwdw. ∂w ∂u ∂w ∂v ∂w ∂w Utilizing the summation convention, the above can be expressed in the index notation. Define the quantities ∂r ∂r ∂r ∂r ∂r ∂r g11 = · g12 = · g13 = · ∂u ∂u ∂u ∂v ∂u ∂w ∂r ∂r ∂r ∂r ∂r ∂r g21 = · g22 = · g23 = · ∂v ∂u ∂v ∂v ∂v ∂w ∂r ∂r ∂r ∂r ∂r ∂r g31 = · g32 = · g33 = · ∂w ∂u ∂w ∂v ∂w ∂w and let x1 = u, x2 = v, x3 = w. Then the above element of arc length can be expressed as 2 i j i j ds = Ei · Ej dx dx = gij dx dx ,i,j=1, 2, 3 where ∂r ∂r ∂ym ∂ym g = E · E = · = ,i,jfree indices (1.3.13) ij i j ∂xi ∂xj ∂xi ∂xj are called the metric components of the curvilinear coordinate system. The metric components may be thought of as the elements of a symmetric matrix, since gij = gji. In the rectangular coordinate system x, y, z, the element of arc length squared is ds2 = dx2 + dy2 + dz2. In this space the metric components are 100 gij = 010 . 001 68 EXAMPLE 1.3-3. (Cylindrical coordinates (r, θ, z)) The transformation equations from rectangular coordinates to cylindrical coordinates can be expressed as x = r cos θ, y = r sin θ, z = z. Here y1 = x, y2 = y, y3 = z and x1 = r, x2 = θ, x3 = z, and the position vector can be expressed r = r(r, θ, z)=r cos θ e1 + r sin θ e2 + z e3. The derivatives of this position vector are calculated and we find ∂r ∂r ∂r E1 = =cosθ e1 +sinθ e2, E2 = = −r sin θ e1 + r cos θ e2, E3 = = e3. ∂r ∂θ ∂z From the results in equation (1.3.13), the metric components of this space are 100 2 gij = 0 r 0 . 001 We note that since gij =0wheni = j, the coordinate system is orthogonal. Given a set of transformations of the form found in equation (1.3.10), one can readily determine the metric components associated with the generalized coordinates. For future reference we list several differ- ent coordinate systems together with their metric components. Each of the listed coordinate systems are orthogonal and so gij =0fori = j. The metric components of these orthogonal systems have the form 2 h1 00 2 gij = 0 h2 0 2 00h3 and the element of arc length squared is 2 2 1 2 2 2 2 2 3 2 ds = h1(dx ) + h2(dx ) + h3(dx ) . 1. Cartesian coordinates (x, y, z) x = x h1 =1 y = y h2 =1 z = z h3 =1 The coordinate curves are formed by the intersection of the coordinate surfaces x =Constant, y =Constant and z =Constant. 69 Figure 1.3-1. Cylindrical coordinates. 2. Cylindrical coordinates (r, θ, z) x = r cos θ r ≥ 0 h1 =1 y = r sin θ 0 ≤ θ ≤ 2π h2 = r z = z −∞<z<∞ h3 =1 The coordinate curves, illustrated in the figure 1.3-1, are formed by the intersection of the coordinate surfaces x2 + y2 = r2, Cylinders y/x =tanθ Planes z = Constant Planes. 3. Spherical coordinates (ρ, θ, φ) x = ρ sin θ cos φ ρ ≥ 0 h1 =1 y = ρ sin θ sin φ 0 ≤ θ ≤ π h2 = ρ z = ρ cos θ 0 ≤ φ ≤ 2π h3 = ρ sin θ The coordinate curves, illustrated in the figure 1.3-2, are formed by the intersection of the coordinate surfaces x2 + y2 + z2 = ρ2 Spheres x2 + y2 =tan2 θz2 Cones y = x tan φ Planes. 4. Parabolic cylindrical coordinates (ξ,η,z) 2 2 x = ξη −∞<ξ<∞ h1 = ξ + η 1 2 2 2 2 y = (ξ − η ) −∞<z<∞ h2 = ξ + η 2 z = z η ≥ 0 h3 =1 70 Figure 1.3-2. Spherical coordinates. The coordinate curves, illustrated in the figure 1.3-3, are formed by the intersection of the coordinate surfaces ξ2 x2 = −2ξ2(y − ) Parabolic cylinders 2 η2 x2 =2η2(y + ) Parabolic cylinders 2 z = Constant Planes. Figure 1.3-3. Parabolic cylindrical coordinates in plane z =0. 5. Parabolic coordinates (ξ,η,φ) 2 2 x = ξη cos φ ξ ≥ 0 h1 = ξ + η 2 2 y = ξη sin φ η ≥ 0 h2 = ξ + η 1 2 2 z = (ξ − η ) 0 <φ<2π h3 = ξη 2 71 The coordinate curves, illustrated in the figure 1.3-4, are formed by the intersection of the coordinate surfaces ξ2 x2 + y2 = −2ξ2(z − ) Paraboloids 2 η2 x2 + y2 =2η2(z + ) Paraboloids 2 y = x tan φ Planes.
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]
  • A Scaling Technique for the Design Of
    A SCALING TECHNIQUE FOR THE DESIGN OF IDEALIZED ELECTROMAGNETIC LENSES Tj:lesis by Carl E. Baum Capt . , USAF In Partial Fulfillment of the Requi rements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1969 (Submitte d August 22 , 1968) -ii- ACKNOWLEDGEMENT We would like to thank Professor C. H. Papas for his guidance and encouragement and Mrs. R. Stratton for typing the text. -iii- ABSTRACT A technique is developed for the design of lenses for transi­ tioning TEM waves between conical and/or cylindrical transmission lines, ideally with no reflection or distortion of the waves. These lenses utilize isotropic but inhomogeneous media and are based on a soluti on of Maxwell's equations instead of just geometrical optics. The technique employs the expression of the constitutive parameters, £ and µ , plus Maxwell's equations, in a general orthogonal curvilinear coordinate system in tensor form, giving what we term as formal quantities. Solving the problem for certain types of formal constitutive parameters, these are transformed to give £ and µ as functions of position. Several examples of such l enses are considered in detail. -iv- TABLE OF CONTENTS I. Introduction l II. Formal Vectors and Operators 5 III. Formal Electromagnetic Quantities lO IV. Restriction of Constitutive Parameters to Scalars l4 V. General Case with Field Components in All Three 16 Coordinate Directions VI . Three- Dimensional TEM Waves 19 VII. Three-Dimensional TEM Lenses 29 A. Modified Spherical Coordinates 29 B. Mo dif i ed Bispherical Coordinates 33 c. Modified Toroidal Coordinates 47 D. Modified Cylindrical Coordinates 57 VIII.
    [Show full text]
  • Transformation of Axes
    19 Chapter 2 Transformation of Axes 2.1 Transformation of Coordinates It sometimes happens that the choice of axes at the beginning of the solution of a problem does not lead to the simplest form of the equation. By a proper transformation of axes an equation may be simplified. This may be accom- plished in two steps, one called translation of axes, the other rotation of axes. 2.1.1 Translation of axes Let ox and oy be the original axes and let o0x0 and o0y0 be the new axes, parallel respectively to the old ones. Also, let o0(h, k) referred to the origin of new axes. Let P be any point in the plane, and let its coordinates referred to the old axes be (x, y) and referred to the new axes be (x0y0) . To determine x and y in terms of x0, y0, h and x = h + x0 y = k + y0 20 Chapter 2. Transformation of Axes these equations represent the equations of translation and hence, the coeffi- cients of the first degree terms are vanish. 2.1.2 Rotation of axes Let ox and oy be the original axes and ox0 and oy0 the new axes. 0 is the origin for each set of axes. Let the angle x0 ox through which the axes have been rotated be represented by q. Let P be any point in the plane, and let its coordinates referred to the old axes be (x, y) and referred to the new axes be (x0, y0), To determine x and y in terms of x0, y0 and q : x = OM = ON − MN = x0 cos q − y0 sin q and y = MP = MM0 + M0P = NN0 + M0P = x0 sin q + y0 cos q 2.1.
    [Show full text]
  • Computer Facilitated Generalized Coordinate Transformations of Partial Differential Equations with Engineering Applications
    Computer Facilitated Generalized Coordinate Transformations of Partial Differential Equations With Engineering Applications A. ELKAMEL,1 F.H. BELLAMINE,1,2 V.R. SUBRAMANIAN3 1Department of Chemical Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1 2National Institute of Applied Science and Technology in Tunis, Centre Urbain Nord, B.P. No. 676, 1080 Tunis Cedex, Tunisia 3Department of Chemical Engineering, Tennessee Technological University, Cookeville, Tennessee 38505 Received 16 February 2008; accepted 2 December 2008 ABSTRACT: Partial differential equations (PDEs) play an important role in describing many physical, industrial, and biological processes. Their solutions could be considerably facilitated by using appropriate coordinate transformations. There are many coordinate systems besides the well-known Cartesian, polar, and spherical coordinates. In this article, we illustrate how to make such transformations using Maple. Such a use has the advantage of easing the manipulation and derivation of analytical expressions. We illustrate this by considering a number of engineering problems governed by PDEs in different coordinate systems such as the bipolar, elliptic cylindrical, and prolate spheroidal. In our opinion, the use of Maple or similar computer algebraic systems (e.g. Mathematica, Reduce, etc.) will help researchers and students use uncommon transformations more frequently at the very least for situations where the transformations provide smarter and easier solutions. ß2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 19: 365À376, 2011; View this article online at wileyonlinelibrary.com; DOI 10.1002/cae.20318 Keywords: partial differential equations; symbolic computation; Maple; coordinate transformations INTRODUCTION usual Cartesian, polar, and spherical coordinates. For example, Figure 1 shows two identical pipes imbedded in a concrete slab.
    [Show full text]
  • In George Warner Swenson, Jr. B.S., Michigan College of Mining And
    SOLUTION OF LAPLACE'S EQUATION in INVERTED COORDINATE SYSTEMS by George Warner Swenson, Jr. B.S., Michigan College of Mining and Technology (1944) Submitted in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY 1948 Signature of Author ..0 Q .........0 .. .. Department of Electrical Engineering Certified by .. S....uper. ... .... .. Thesis Supervisor Chairman, Departmental Committee on Graduate Students ACKNOWLEDGMENT The author wishes to express his sincerest apprecia- tion to Professor Parry Moon, of the Department of Electrical Engineering, who suggested the general topic of this thesis. His lectures in electrostatic field theory have provided the essential background for the investigation and his interest in the problem has prompted many helpful suggestions. In addition, the author is indebted to Dr. R. M. Redheffer, of the Department of Mathematics, without whose patient interpretation of his Doctorate thesis and frequent assistance in overcoming mathematical difficulties the present work could hardly have been accomplished. 2982i19 ii CONTENTS ABSTRACT.............................................. iv I ORIENTATION .............. ,............ *,*.*..*** 1 II THE GEOMETRICAL PROPERTIES OF INVERTED SYSTEMS.... 7 A. The Process of Inversion ..................... 7 B. Practical Methods of Inversion................ 8 (1' Graphical Method ... ................... 8 2 Optical Method ......... ................. 10 C. The Inverted Coordinate Systems .............. 19 1) Inverse Rectangular Coordinates ......... 19 2) Inverse Spheroidal Coordinates .......... 19 3) Inverse Parabolic CooDrdinates ........... 21 III LAPLACE'S EQUATION IN INVERTED COORDINATE SYSTEMS. 30 A. The Form of the Equation ..................... 30 B. Separation of Variables ...................... 30 C. Tabulation of the Separated Equations for the Inverted Coordinate Systems *..........*......35 IV TBE APPLICATION OF THE INVERTED COORDINATE SYSTEMS TO BOUNDARY VALUE PROBLEMS ......................
    [Show full text]
  • General Theory of Conicoids 1
    GENERAL THEORY OF CONICOIDS Structure 1.1 Introduction Objectives 1.2 What is a Conicoid? 1.3 Change of Axes 1.3.1 Translat~on of Axes 1.3.2 Projection L 1.3.3 Rotation of Axes 1.4 Reduction to Standard Form 1.5 Summary 1.6 Solutions and Answers I 1.1 INTRODUCTION You have seen in Block 7 that the general equation of second degree in two variables x and y represents a conic. In analogy with this we can ask: what will a general second degree equation in three variables represent? In Block 8 you have studied some particular forms of second degree equations in three variables, namely, those representing spheres, cones and cylinders. In this unit we study the most general form of a second degree equation in three variables. The surface generated by these equations are called quadrics or conicoids. This name is apt because, as you will see in Unit 8, they can be formed by revolving a conic about a line called an axis. Alexis Clairaut (17 13- 1765), a French mathematician, was one of the pioneers to study quadric surfaces. He specified that a surface, in .general, can be represented by an equation in three variables. He presented his ideas in his book 'Recherche Sur Les Courbes a Double Courbure' in which he gave the equations of several conicoids like the sphere, cylinder, hyperboloid and ellipsoid. We start this unit with a small section in which we define a conicoid: In the next section we discuss rigid body motions in a three-dimensional system.
    [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]
  • Vector Analysis: MS4613
    Vector Analysis: MS4613 Prof. S.B.G. O’Brien Main Building, B3041. References Advanced engineering mathematics, Kreyszig. Mathematics applied to continuum mechanics, Segel. Mathematical methods for physicists, Arfken. Mathematics applied to the deterministic sciences, Lin and Segel. Vectors and cartesian tensors, Bourne and Kendall. Vector Analysis (Schaum series), Murray and Spiegel. GREEK ALPHABET For each Greek letter, we illustrate the form of the capital letter and the form of the lower case letter. In some cases, there is a popular variation of the lower case letter. Greek Greek English Greek Greek English letter name equivalent letter name equivalent A α Alpha a N ν Nu n B β Beta b Ξ ξ Xi x Γ γ Gamma g O o Omicron o ∆ δ Delta d Π π ̟ Pi p E ǫ ε Epsilon e P ρ ̺ Rho r Z ζ Zeta z Σ σ ς Sigma s H η Eta e T τ Tau t Θ θ ϑ Theta th Υ υ Upsilon u I ι Iota i Φ φ ϕ Phi ph K κ Kappa k X χ Chi ch Λ λ Lambda l Ψ φ Psi ps M µ Mu m Ω ω Omega o Exam Papers: http://www.ul.ie/portal/students Some papers are also available at: http://www.staff.ul.ie/obriens/index.html 1 1 Rectangular cartesian coordinates and rotation of axes 1.1 Rectangular cartesian coordinates We wish to be able to describe locations and directions in space. From a fixed point O, called the origin (of coordinates), draw three fixed lines Ox,Oy,Oz at right angles to each other as in fig.1.1 and forming a right handed system (rectangular cartesian axes) (use RH screw rule as in fig.1.2).
    [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]
  • 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.
    [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]