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14.2.2 Covariant and Contravariant Coordinates 14. Vector Algebra and Vector Analysis 2016-07-07 $Version 10.0 for Mac OS X x86 (64-bit) (December 4, 2014) Mathematica 9: The commands for vector analysis are contained in the kernel in contrast to earlier versions. Mathematica 7 or 8: The package must be loaded by the command << "VectorAnalysis`" . Some of the commands differ from those given in version 9, so in these lecture notes. In particular, the arguments of the operators do not contain the list of the independent variables as arguments. Ph. Boylard: A Guide to Standard Mathematica Packages. Mathematica Technical Report (1991). E. Martin: Mathematica 3.0 Standard Add-On Packages. Cambridge University Press. pp.74-82. 14.1 Rectangular, Cylindrical and Spherical Coordinates Rectangular (as well as cylindrical and spherical) coordinates have been treated at the end of Chap.3. Knowledge of these subjects is pressuposed in this chapter. 14.2 Curvilinear Orthogonal Coordinates 14.2.1 Literature Murray R. Spiegel, Schaum's Mathemtical Handbook of Formulas and Tables, McGraw-Hill, New York 1968, pp.116-130 Philip M. Morse, Herman Feshbach: Methods of Theoretical Physics, McGraw-Hill, New York, 1953. (MF), 656ff. George B. Arken, Mathematical Methods for Pysicists, 3 rd ed., Academic Press, New York, 1985. B. Schnizer, Analytical methods in applied theoretrical physics. Lecture notes (in German). Chaps. 6 and 7. http://itp.tugraz.at/~schnizer/AnalyticalMethods/ P. Moon, D.E. Spencer, Field Theory Handbook. 2 nd ed., Springer, 1988. (MS) G. Wendt, Statische Felder und Stationäre Ströme. Hb. d. Physik XVI Springer, 1958, S.1 ff. W.R. Smith, Static and Dynamic Electricity, McGraw-Hill, 1950 14.2.2 Covariant and contravariant coordinates A general non-singular transformation from Cartesian coordinates x1, x2, x3 to more general curvilinear coordinates u1, u2, u3 is given by: xi = xi (u1, u2, u3), uk = uk (x1, x2, x3); i, k = 1, 2, 3. (1) So a triple of numbers for the uk's gives a point in space in the same way as a triple of Cartesian coordinates x1, x2, x3. Keeping two uk's constant while varying the third one gives a curve in space. There are three such curves passing through each point. The three tangents to these curves are given by: k τi = ∂xi/ ∂uk; (2) k numbering the curves; i numbering the components of each vector. These three vectors are orthogonal to each other, but they are not normalized; in addition, their directions and lengths may change from point to point. Varying two uk's for a fixed third us gives a surface in space. There are three surfaces passing through a given point with coordinates u1, u2, u3. The three gradients k γi = ∂uk/ ∂xi; (3) (k numbering the surface; i numbering the components of each vector) make up a second triple of orthogonal non-normalized vectors. The two triples are reciprocal to each other, i.e. p q τiγi = δpq. (4) The inner products of the tangent vectors give the covariant metric tensor p q gpq := τiτi = (∂xi/ ∂up) (∂xi/ ∂uq) = gqp . (5) The inner products of the gradient vectors give the contravariant metric tensor pq p q qp g := γiγi = (∂up/ ∂xi) (∂uq/ ∂xi) = g . (6) So in non-orthogonal coordinate systems there are two systems, whose base vectors may change their orientations and lengths from point to point. The components of any vector a , whose Cartesian components are ai: a = ai ei may be taken with respect to each of these systems. The covariant components Ap are: p k ai = Ap γi = Ap ∂up/ ∂xi, Ap := Aiτi = Ai ∂xi/ ∂uk; (7) while the contravariant components Ap are: p p p p k ai = A τi = A ∂xi/ ∂uk, A := aiγi = ai ∂uk/ ∂xi. (8) The arc element may be expressed by the covariant metric tensor (5): 2 ds := dxidxi = gpq dup duq. (9) So a triple of numbers for the uk's gives a point in space in the same way as a triple of Cartesian coordinates x1, x2, x3. Keeping two uk's constant while varying the third one gives a curve in space. There are three such curves passing through each point. The three tangents to these curves are given by: k τi = ∂xi/ ∂uk; (2) 2 math14a.nbk numbering the curves; i numbering the components of each vector. These three vectors are orthogonal to each other, but they are not normalized; in addition, their directions and lengths may change from point to point. Varying two uk's for a fixed third us gives a surface in space. There are three surfaces passing through a given point with coordinates u1, u2, u3. The three gradients k γi = ∂uk/ ∂xi; (3) (k numbering the surface; i numbering the components of each vector) make up a second triple of orthogonal non-normalized vectors. The two triples are reciprocal to each other, i.e. p q τiγi = δpq. (4) The inner products of the tangent vectors give the covariant metric tensor p q gpq := τiτi = (∂xi/ ∂up) (∂xi/ ∂uq) = gqp . (5) The inner products of the gradient vectors give the contravariant metric tensor pq p q qp g := γiγi = (∂up/ ∂xi) (∂uq/ ∂xi) = g . (6) So in non-orthogonal coordinate systems there are two systems, whose base vectors may change their orientations and lengths from point to point. The components of any vector a , whose Cartesian components are ai: a = ai ei may be taken with respect to each of these systems. The covariant components Ap are: p k ai = Ap γi = Ap ∂up/ ∂xi, Ap := Aiτi = Ai ∂xi/ ∂uk; (7) while the contravariant components Ap are: p p p p k ai = A τi = A ∂xi/ ∂uk, A := aiγi = ai ∂uk/ ∂xi. (8) The arc element may be expressed by the covariant metric tensor (5): 2 ds := dxidxi = gpq dup duq. (9) 14.2.3 Introducing physical coordinates for orthogonal curvilinear systems p p In orthogonal curvilinear systems, the base vectors γi and τi have the same directions. One introduces one set of normalized base vectors i: p p i := τi/hp = γihp In place of the covariant and contravariant components of a vector one gets one set, the physical coordinates i such that : p p p a = ai ei = ii = A τ = Ap γ; (10) i i = hiA = Ai/ hi. (No summation in this line !) For example, in cylindrical coordinates ρ, ϕ, z the transformation from rectangular coordinates, the three tangents to the coordinate curves and the gradients of the coordinate surfaces are: x = x = cos , y = x = sin , z = x ; = x2 y2 , = tan y , z = x . 1 ρ ϕ 2 ρ ϕ 3 ρ + ϕ x 3 k 1 2 3 τi = ∂xi/ ∂uk: τi= ( cosϕ, sinϕ, 0), τi= ( -ρ sinϕ, ρ cosϕ, 0), τi= (0, 0, 1), with ρ = x2 + y2 , k 1 2 2 2 3 γi = ∂uk/ ∂xi; γi= (x/ρ, y/ρ, 0),γi= (-y, x)/x + y , γi = (0, 0, 1). The metric coefficients hiare: hρ = hz = 1, hϕ= ρ. The various coordinates of the position (r) and of the velocity vector (v) are: ρ ϕ z r = rρρ + rϕϕ + rz z z , r ρ= ρ, rϕ= 0, rz= z. A = Aρ= 0, A = Aϕ= 0, A = Az= z. - - - ( - ) - - z v = vρρ + vϕϕ + vz z z , v ρ= ρ, v )= (), v z= z. A = A(= (, A = ), A)= (), A = Az= z. math14a.nb 3 For example, in cylindrical coordinates (, ), z the transformation from rectangular coordinates, the three tangents to the coordinate curves and the gradients of the coordinate surfaces are: x = x = cos , y = x = sin , z = x ; = x2 y2 , = tan y , z = x . 1 ( ) 2 ( ) 3 ( * ) x 3 k 1 2 3 !i = "xi/ "uk: !i= ( cos), sin), 0), !i= ( -( sin), ( cos), 0), !i= (0, 0, 1), with ( = x2 * y2 , k 1 2 2 2 3 #i = "uk/ "xi; #i= (x/(, y/(, 0),#i= (-y, x)/+x * y ,, #i = (0, 0, 1). The metric coefficients hiare: h( = hz = 1, h)= (. The various coordinates of the position (r) and of the velocity vector (v) are: ( ) z r = r(&( + r)&) + rz z &z , r (= (, r)= 0, rz= z. A = A(= 0, A = A)= 0, A = Az= z. - - - ( - ) - - z v = v(&( + v)&) + vz z &z , v (= (, v )= (), v z= z. A = A(= (, A = ), A)= (), A = Az= z. The square of the arc element is: 2 2 2 2 2 2 2 2 2 2 ds % dx * dy * dz = h1 du1 * h2 du2 * h3 du3 . (11) The surface elements and the volume elment are: df1 = h2h3du2du3, df2 = h1h3 du1du3, df3 = h1h2 du1du2; (12) dV = h1 h2 h3 du1 du2 du3. (13) 14.2.4 Vector Analysis implementations in Mathematica The vector analysis impemented in Mathematica deals only with orthogonal coordinate systems, so only with physical coordinates as explained in the previous subsection. As of Mathematica 9 there are available two versions of implementations : 1. The old version as implemented in version 8 and previous ones with some modifications is available as a package, which must be called by the command: Needs["VectorAnalysis`"] This implementaton is created for 14 curvilinear orthogonal coordinate systems in three-dimen- sional Euclidean space given below under 14.3, in particular 14.3.5. When the package is called then the new vector analysis functionality implemented in the kernel (see 2.) is blocked. 2. The new version of Mathematica Vector Analysis is now implemented in the kernel; so it is ready automatically if not blocked (see previous paragraph 1.) It used new commands completely different from those employed in the package. It has implementations for two-dimensional curvilinear orthogo- nal coordinate systems. It treats the three-dimensional systems listed in 1.) In addition it deals with some n-dimensional orthogonal systems.
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