Vectors, Vector Calculus, and Coordinate Systems

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1 ! Revised Apil 11, :48 PM! 1 Vectos, Vecto Calculus, and Coodinate Systems David Randall Physical laws and coodinate systems Fo the pesent discussion, we define a coodinate system as a tool fo descibing positions in space. Coodinate systems ae human inventions, and theefoe ae not pat of physics, although they can be used in a discussion of physics. Any physical law should be expessible in a fom that is invaiant with espect to ou choice of coodinate systems; we cetainly do not expect that the laws of physics change when we switch fom spheical coodinates to catesian coodinates! It follows that we should be able to expess physical laws without making efeence to any coodinate system. Nevetheless, it is useful to undestand how physical laws can be expessed in diffeent coodinate systems, and in paticula how vaious quantities tansfom as we change fom one coodinate system to anothe. Scalas, vectos, and tensos Tensos can be defined without efeence to any paticula coodinate system. A tenso is simply out thee, and has a meaning that is the same whethe we happen to be woking in spheical coodinates, o Catesian coodinates, o whateve. Tensos ae, theefoe, just what we need to fomulate physical laws. The simplest kind of tenso, called a tenso of ank 0, is a scala, which is epesented by a single numbe -- essentially a magnitude with no diection. An example of a scala is tempeatue. Not all quantities that ae epesented by a single numbe ae scalas, because not all of them ae defined without efeence to any paticula coodinate system. An example of a (single) numbe that is not a scala is the longitudinal component of the wind, which is defined with espect to a paticula coodinate system, i.e., spheical coodinates. A scala is expessed in exactly the same way egadless of what coodinate system may be in use to descibe non-scalas in a poblem. Fo example, if someone tells you the tempeatue in Fot Collins, you don t have to ask whethe they ae using spheical coodinates o some othe coodinate system, because it makes no diffeence at all. Vectos ae tensos of ank 1; a vecto can be epesented by a magnitude and one diection. An example is the wind vecto. In atmospheic science, vectos ae nomally eithe

2 ! Revised Apil 11, :48 PM! 2 thee dimensional o two dimensional, but in pinciple they have any numbe of dimensions. A scala can be consideed to be a vecto in a one-dimensional space. A vecto can be expessed in a paticula coodinate system by an odeed list of numbes, which ae called the components of the vecto. The components have meaning only with espect to the paticula coodinate system. Moe o less by definition, the numbe of components needed to descibe a vecto is equal to the numbe of dimensions in which the vecto is embedded. We can define unit vectos that point in each of the coodinate diections. A vecto can then be witten as the vecto sum of each of the unit vectos times the component associated with the unit vecto. In geneal, the diections in which the unit vectos point depend on position. Unit vectos ae always non-dimensional; note that hee we ae using the wod dimension to efe to physical quantities, such as length, time, and mass. Because the unit vectos ae non-dimensional, all components of a vecto must have the same dimensions as the vecto itself. Spatial coodinates may o may not have the dimensions of length. In the familia Catesian coodinate system, the thee coodinates, ( x, y, z), each have dimensions of length. In spheical coodinates, ( λ,ϕ, ), whee λ is longitude, ϕ is latitude, and is distance fom the oigin, the fist two coodinates ae non-dimensional angles, while the thid has units of length. When we change fom one coodinate system to anothe, an abitay vecto V tansfoms accoding to V = MV. (1) Hee V is the epesentation of the vecto in the fist coodinate system (i.e., V is the list of the components of the vecto in the fist coodinate system), V is the epesentation the vecto in the second coodinate system, and M is a otation matix. that maps V onto V. The otation matix used to tansfom a vecto fom one coodinate system to anothe is a popety of the two coodinate systems in question; it is the same fo all vectos, but it does depend on the paticula coodinate systems involved, so it is not a tenso. The tansfomation ule (1) is actually pat of the definition of a vecto, i.e., a vecto must, by definition, tansfom fom one coodinate system to anothe via a ule of the fom (1). It follows that not all odeed lists of numbes ae vectos. The list is not a vecto. (mass of the moon, distance fom Fot Collins to Denve) Let V be the a vecto epesenting the thee-dimensional velocity of a paticle in the atmosphee. The Catesian and spheical epesentations of V ae

3 ! Revised Apil 11, :48 PM! 3 V =!xi +!yj +!zk, (2) V =! λ cosϕe λ +!ϕe ϕ +!e. Hee a dot denotes a Lagangian time deivative, i.e., a time deivative following a moving paticle, i, j, and k ae unit vectos in the catesian coodinate system, and e λ, e ϕ, and e ae unit vectos in the spheical coodinate system. Eqs. (6) and (7) both descibe the same vecto, V, i.e., the meaning of V is independent of the coodinate system that is chosen to epesent it. Vectos ae consideed to be tensos of ank one, and scalas ae tensos of ank zeo. The numbe of diections associated with a tenso is called the ank of the tenso. In pinciple, the ank can be abitaily lage, but we aely meet tensos with anks highe than two in atmospheic science. A tenso of ank 2 that is impotant in atmospheic science is the flux of momentum. The momentum flux, also called a stess, and equivalent to a foce pe unit aea, has a magnitude and two diections. One of the diections is associated with the foce vecto itself, and the othe is associated with the nomal vecto to the unit aea in question. The momentum flux tenso can be witten as ρv V, whee ρ is the density of the ai, V is the wind vecto, and is the oute o dyadic poduct that accepts two vectos as input and delives a ank-2 tenso as output. Like a vecto, a tenso of ank 2 can be expessed in a paticula coodinate system, i.e., we can define the components of the tenso with espect to a paticula coodinate system. The components of a tenso of ank 2 can be aanged in the fom of a two-dimensional matix, in contast to the components of a vecto, which fom an odeed one-dimensional list. When we change fom one coodinate system to anothe, a tenso of ank 2 tansfoms accoding to T = MTM 1, whee T is the epesentation of a ank-2 tenso in the fist coodinate system, T is the epesentation of the same tenso in the second coodinate system, M is the matix intoduced in Eq. (1) above, and M 1 is its invese. (3) (4) Diffeential opeatos Seveal familia diffeential opeatos can be defined without efeence to any coodinate system. These opeatos ae moe fundamental than, fo example, x, whee x is a paticula spatial coodinate. The coodinate-independent opeatos that we need most often fo atmospheic science (and fo most othe banches of physics too) ae:

4 ! Revised Apil 11, :48 PM! 4 the gadient, denoted by A whee A is an abitay scala; the divegence, denoted by Q, whee Q is an abitay vecto; and the cul, denoted by Q, the Laplacian, given by 2 A ( A). Note that the gadient and cul ae vectos, while the divegence is a scala. The gadient opeato accepts scalas as input, while the divegence and cul opeatos consume vectos. In discussions of two-dimensional motion, it is often convenient to intoduce an additional opeato called the Jacobian, denoted by ( ) k ( α β ) = k ( α β ) J α,β = k ( β α ). Hee the gadient opeatos ae undestood to poduce vectos in the two-dimensional space, α and β ae abitay scalas, and k is a unit vecto pependicula to the two-dimensional suface. The second and thid lines of (9) can be deived with the use of vecto identities found in a table late in this QuickStudy. (5) (6) (7) (8) (9) is: A definition of the gadient opeato that does not make efeence to any coodinate system 1 A lim S 0 V! S nads, whee S is the suface bounding a volume V, and n is the outwad nomal on S. Hee the tems volume and bounding suface ae used in the following genealized sense. In a theedimensional space, volume is liteally a volume, and bounding suface is liteally a suface. In a two-dimensional space, volume means an aea, and bounding suface means the cuve bounding the aea. In a one-dimensional space, volume means a cuve, and bounding suface means the end points of the cuve. The limit in (10) is one in which the volume and the aea of its bounding suface shink to zeo. (10)

5 ! Revised Apil 11, :48 PM! 5 As an example, conside a Catesian coodinate system ( x, y) on a plane, with unit vectos i and j in the x and y diections, espectively. Conside a box of width Δx and height Δy, y x Δy ( x, y) = ( x 0, y 0 ) Δx Figue 1: Diagam illustating a ectangula box in a plana two-dimensional space, with cente at x 0, y 0 ( ), width Δx, and height Δy. as shown in Figue 1. We can wite A lim ( Δx,Δy) 0 1 A x 0 + Δx ΔxΔy 2, y 0 Δyi + A x 0, y 0 + Δy Δxj 2 A x 0 Δx 2, y 0 Δyi A x 0, y 0 Δy Δxj 2 = A x i + A y j. (11) This is the answe that we expect. Definitions of the divegence and cul opeatos that do not make efeence to any coodinate system ae: 1 Q lim S 0 V 1 Q lim S 0 V! S! S n QdS, n QdS. (12) (13)

6 ! Revised Apil 11, :48 PM! 6 It is possible to wok though execises simila to (11) fo these opeatos too. You might want to ty it youself, to see if you undestand. Finally, the Jacobian on a two-dimensional suface can be defined by ( ) = lim J A, B A 0! C A B t dl, (14) whee t is a unit vecto that is tangent to the bounding cuve C. Vecto identities Many useful identities elate the divegence, cul, and gadient opeatos. Most of the following identities can be found in any mathematics efeence manual, e.g., Beye (1984). As befoe, let α and β be abitay scalas, let V, A, B, and C be abitay vectos, and let T be an abitay tenso of ank 2. Then: ( α ) = 0, ( V) = 0, A B = B A, ( αv) = α ( V) + V α, ( A B) = ( A) B ( B) A ( αv) = α V + α ( V), A ( B C) = ( A B) C = B ( C A), A ( B C) = B( C A) C( A B), ( A B) = A( B) B( A) ( A )B + ( B )A, ( A B) = ( A )B + ( B )A + A ( B) + B ( A), (15) (16) (17) (18) (19) (20) (21) (22) (23) (24)

7 ! Revised Apil 11, :48 PM! 7 J ( α,β ) k ( α β ) = k ( α β ) = k ( β α ) = k ( β α ), 2 V ( )V = ( V) ( V), ( A B) = ( A )B + ( B )A, ( αt) = ( α ) T + α ( T). In (27), A B denotes the oute poduct of two vectos, which yields a tenso of ank 2. (25) (26) (27) (28) A special case of (24) is 1 2 ( V V) (29) This identity is used to wite the advection tems of the momentum equation in altenative foms. Identity (26) says that the Laplacian of a vecto is the gadient of the divegence of the vecto, minus the cul of the cul of the vecto. The fist tem involves only the divegent pat of the wind field, and the second tem involves only the otational pat. Eq. (26) can be used, fo example, in a paameteization of momentum diffusion. Spheical coodinates Fo obvious easons, spheical coodinates ae of special impotance in geophysics. The unit vectos in spheical coodinates ae denoted by e λ pointing towads the east, e ϕ pointing towads the noth, and e pointing outwad fom the oigin (in geophysics, outwad fom the cente of the Eath). A useful esult that is a special case of (23) is e ( e H) = H, whee e is the unit vecto pointing upwad, and H is an abitay hoizontal vecto. In wods, the cul of e H is equal to the divegence of H. Similaly, a useful special case of (19) is (30) ( e H) = e ( H). (31)

8 ! Revised Apil 11, :48 PM! 8 This means that the divegence of e H is equal to minus the cul of H. The gadient, divegence, cul, Laplacian, and Jacobian opeatos can be expessed in spheical coodinates as follows: V = α = V = 1 V ϕ ( V ϕ ), 1 1 α cosϕ λ, 1 α ϕ, α, 1 V λ cosϕ λ + 1 cosϕ ϕ V cosϕ ϕ ( V λ ) ( ) + 1 ( ), 2 cosϕ 1 V cosϕ λ, 1 V 2 V ϕ λ ( ϕ V cosϕ) λ, (32) (33) (34) 1 1 α 2 α = 2 cosϕ λ cosϕ λ + ϕ 2 cosϕ α, J ( α,β ) = 1 2 cosϕ ϕ β α λ ϕ. α β λ Hee α is an abitay scala, and V is an abitay vecto. (35) (36) If V is sepaated into a hoizontal vecto and a vetical vecto, as in V = V h + V e, then (34) can be witten as ( V h + V e ) = V h ( ) + e 1 V h ( ) V In case V is the velocity, then the fist tem on the ight-hand side of (37) is the vetical component of the voticity, and the second tem is the hoizontal voticity vecto. Eq. (37) shows that the cul of a puely vetical vecto is minus e cossed with the hoizontal gadient of the magnitude of that vecto. The thee-dimensional cul of a puely hoizontal vecto has both a vetical pat, given by V h., and a hoizontal pat, given by e 1 (37) ( V h ). The twodimensional cul of a hoizontal vecto has only a vetical component, namely V h.

9 ! Revised Apil 11, :48 PM! 9 Conside how the two-dimensional vesion of (32) can be deived fom (10). Fig. 2 illustates the poblem. Hee we have eplaced by a, the adius of the Eath. The angle θ depicted in the figue aises fom the gadual otation of e λ and e ϕ, the unit vectos associated with the spheical coodinates, as the longitude changes; the diections of e λ and e ϕ in the cente of the aea element, whee A is defined, ae diffeent fom thei espective diections on e ϕ acos( ϕ + dϕ)dλ e λ Figue 2: A patch of the sphee, with longitudinal width a cosϕdλ, and latitudinal height adϕ. adϕ A 4 A 3 A 1 A 2 θ a cosϕdλ eithe east-west wall of the aea element. Inspection of Fig. 2 shows that θ satisfies 1 acos ϕ + dϕ tanθ = 2 adϕ ( ) acosϕ 1 2 ϕ cosϕ dλ = 1 2 sinϕdλ sinθ. (38) The angle θ is of diffeential o infinitesimal size. Nevetheless, it is needed in the deivation of (32). The line integal in (10) can be expessed as dλ

10 ! Revised Apil 11, :48 PM! αndl Aea! = a 2 cosϕdλdϕ e ϕα 1 α cosϕdλ + e λ α 2 adϕ cosθ + e ϕ α 2 adϕ sinθ +e ϕ α 3 cos( ϕ + dϕ )dλ e λ α 4 adϕ cosθ + e ϕ α 4 adϕ sinθ ( α = e 2 α 4 )cosθ α λ + e 3 acos( ϕ + dϕ ) α 1 acosϕ dλ + ( α 2 + α 4 )sinθadϕ ϕ acosϕdλ a 2 cosϕdλdϕ. Note how the angle θ has enteed hee. Put cosθ 1 and sinθ 1 sinϕdλ, to obtain 2 (39) 1 Aea ( ) ( ) α 1 acosϕ α! αndl = e 2 α 4 λ acosϕdλ + e α 3 acos ϕ + dϕ ϕ acosϕdϕ + α + α sinϕ acosϕ 1 α e λ acosϕ λ + e 1 ( ϕ acosϕ ϕ α cosϕ) + α sinϕ acosϕ 1 α = e λ acosϕ λ + e 1 α ϕ a ϕ, (40) which agees with the two-dimensional vesion of (32). Simila (but moe staightfowad) deivations can be given fo (33) - (36). Using (15) - (17), we can pove the following about the unit vectos in spheical coodinates: e λ = 0, e ϕ = tanϕ, e = 2, e λ = e ϕ + tanϕ e, e ϕ = e λ, e = 0. (41) The following elations ae useful when woking with the momentum equation in spheical coodinates: (42)

11 ! Revised Apil 11, :48 PM! 11 ( V h )e λ = usinϕ ( V h )e ϕ = usinϕ e ϕ u cosϕ e, e λ vsinϕ e, ( V h )e = V h. (43) Hee V h ue λ + ve ϕ is the hoizontal wind vecto. As an example of the application of (34), the vetical component of the voticity is 1 v ζ = cosϕ λ ( ϕ u cosϕ) Fo the case of pue solid body otation of the atmosphee about the Eath s axis of otation, we have (44) u =! λ cosϕ and v = 0, (45) whee! λ is independent of ϕ. Substitution gives 1 ζ = (!λ cos 2 ϕ) cosϕ ϕ = 2! λ sinϕ. This is the expected fom of the voticity associated with the vetical component of the Eath s otation vecto. In the conventional notation,! λ is eplaced by Ω. Conside the special case of two-dimensional flow. Two useful identities ae ( e A) = e 2 A, and ( e A) = 0. Fo two-dimensional flow, the Laplacian of a vecto can be witten in a vey simple way. Let ζ e V h and δ V h. Then (26) educes to (46) (47) (48)

12 ! Revised Apil 11, :48 PM! 12 Using (34), we can wite Then (49) becomes ζ e 2 V h = δ ( ζ e ). ( ) = 1 ζ ϕ, 1 ζ cosϕ λ, 0 = e ζ. 2 V h = δ + e ζ. (49) (50) (51) Conclusions This bief oveview is intended mainly as a efeshe fo students who leaned these concepts once upon a time, but may have not thought about them fo awhile. The efeences below povide much moe infomation. You can also find lots of esouces on the Web. Refeences Abamowitz, M., and I. A. Segun, 1970: Handbook of mathematical functions. Dove Publications, Inc., New Yok, 1046 pp. Afken, G., 1985: Mathematical methods fo physicists. Academic Pess, 985 pp. Ais, R., 1962: Vectos, tensos, and the basic equations of fluid mechanics. Dove Publ. Inc., 286 pp. Beye, W. H., 1984: CRC Standad Mathematical Tables, 27th Edition. CRC Pess Inc., pp Hay, G. E., 1953: Vecto and tenso analysis. Dove Publications, 193 pp. Jeevanjee, N., 2011: An Intoduction to Tensos and Goup Theoy fo Physicists. Bikhäuse, 258 pp. Schey, H. M., 1973: Div, gad, cul, and all that. An infomal text on vecto calculus. W. W. Noton and Co., Inc., 163 pp. Wede, R. C., 1972: Intoduction to vecto and tenso analysis. Dove Publications, 418 pp.

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