Vectors, Vector Calculus, and Coordinate Systems

Transcription

1 Apil 5, 997 A Quick Intoduction to Vectos, Vecto Calculus, and Coodinate Systems David A. Randall Depatment of Atmospheic Science Coloado State Univesity Fot Collins, Coloado Scalas and vectos 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! (Hee we use the tem coodinate system to mean a system fo descibing positions in space; in the pesent discussion, we do not woy about whethe the coodinate system is in motion o is acceleating.) This means that we should be able to expess physical laws without making efeence to any coodinate system. Tenso analysis is the quantitative language that has been invented to make this possible. A tenso is a quantity that is defined without efeence to any paticula coodinate system. A tenso is just out thee, and has a meaning which 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 is a scala, which is epesented by a single numbe -- essentially a magnitude with no diection. An example of a scala is the 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 numbe that is not a scala is the longitudinal component of the wind, which can only be defined though efeence to a paticula spheical coodinate system, in this case latitude-longitude coodinates. Vectos ae tensos that can be epesented by a magnitude and a diection. An example is the wind vecto. In atmospheic science, vectos ae nomally eithe thee dimensional o two dimensional, but in pinciple they have any numbe of dimensions. It is also possible to define highe-ode tensos. An example is the flux (a vecto quantity, which has a diection) of momentum (a second vecto quantity, which has a second, geneally

2 Apil 5, 997 diffeent diection). The momentum flux thus has a magnitude and two diections. It is a tenso of ank two. Vectos ae consideed to be tensos of ank one, and scalas ae tensos of ank zeo. In atmospheic science, we aely meet tensos with anks highe than two. 2. Diffeential opeatos In vecto calculus, we fequently use seveal diffeential opeatos which can be defined without efeence to any coodinate system. These opeatos ae in a sense moe fundamental than, fo example,, whee x is a paticula spatial coodinate. The coodinate-independent x opeatos that we need most often fo atmospheic science (and fo most othe banches of physics too) ae: the gadient, denoted by A, whee A is an abitay scala; () the divegence, denoted by Q, whee Q is an abitay vecto; and (2) the cul, denoted by Q. (3) 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. A definition of the gadient opeato that does not make efeence to any coodinate system is: A S An ds V, (4) 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 it in (4) is one in which the volume and the aea of its bounding suface shink to zeo. As an example, conside a catesian coodinate system (x,y), with unit vectos i and j in the x and y diections, espectively, and a box of width x and height y, as shown in Figue. We can wite S 2

3 Apil 5, 997 y ( x y) ( x, y ), 0 0 x Figue : Diagam illustating a box in a two-dimensional space, of with x and height y, and with cente at (x,y)(x 0,y 0 ). A ( x, y) 0 x A x x y , y 2 0 yi A x0 y y +, xj 0 2 A x x , y y 2 0 yi A x0, y xj 2 (5) A A i + j. x y This is the answe that we expect. Definitions of the divegence and cul opeatos that do not make efeence to any coodinate system ae: Q S Q n ds V S, (6). (7) We can wok though execises simila to (5) fo these opeatos too. You might want to ty this youself, to see if you undestand. 3. Vecto identities Q S Q n ds V S Thee ae many useful identities that elate the divegence, cul, and gadient opeatos. Most of the following identities can be found in any mathematics efeence manual, e.g. Beye (984). Let A and B be abitay scalas, and let V, V 2, V 3 be abitay vectos. Then: 3

4 Apil 5, 997 A 0, V V 2 V 2 V, ( AV ) A( V ) + V A, ( AV ) A V + A( V ), V ( V 2 V 3 ) ( V V 2 ) V 3 V 2 ( V 3 V ), V ( V 2 V 3 ) V 2 ( V 3 V ) V 3 ( V V 2 ), ( V V 2 ) V ( V 2 ) V 2 ( V ) ( V )V 2 + ( V 2 )V. (8) (9) (0) () (2) (3) (4) In discussions of two-dimensional motion, it is often convenient to intoduce a futhe opeato called the Jacobian, denoted by J( A, B) k ( A B) k ( A B) k ( p q) k ( q p) (5) whee A and B ae abitay scalas. Hee the gadient opeatos ae undestood to poduce vectos in the two-dimensional space, and k is a unit vecto pependicula to the two-dimensional suface. 4. Spheical coodinates In spheical coodinates, the gadient, divegence, and cul opeatos can be expessed as follows: A A λ -- A A, , ϕ, (6) V V λ λ ϕ ( V ϕ ) ( V 2 ) 2, (7) V -- V ϕ ---- ( V ϕ ), ( V λ ) V, λ V ϕ λ ( V ϕ λ ), (8) 4

5 Apil 5, A A A A λ λ ϕ ϕ. (9) Hee A is an abitay scala, and V is an abitay vecto. As an example, conside how the two-dimensional vesion of (6) can be deived fom (4). Fig. 2 illustates the poblem. Hee we have eplaced by a, the adius of the Eath. The angle θ acos( ϕ + dϕ) dλ e ϕ A 3 θ adϕ A adλ e ϕ Figue 2: Sketch depicting an aea element on the sphee, with longitudinal width dλ, and latitudinal height dϕ. depicted in the figue aises fom the gadual otation of and, the unit vectos associated with the spheical coodinates, as the longitude changes; the diections of eλ and eϕ in the cente of the aea element, whee A is defined, ae diffeent fom thei espective diections on eithe east-west wall of the aea element. Inspection of Fig. 2 shows that θ satisfies tanθ -- [ acos( ϕ + dϕ) a]dλ dλ -- sinϕdλ sinθ. (20) adϕ 2 ϕ 2 Note that θ is of diffeential o infinitesimal size. Nevetheless, we demonstate below that it cannot be neglected in the deivation of (4). The line integal in (4) can be expessed as 5

6 Apil 5, Andl Aea [ e ϕ A α dλ + adϕcosθ + e ϕ adϕsinθ + e ϕ A 3 cos( ϕ + dϕ)dλ adϕcosθ + e ϕ adϕsinθ ] ( a 2 dλdϕ) ( e ) cosθ λ adλ e {[ A a cos( ϕ + dϕ ) A 3 a]dλ + ( + ) sinθadϕ} ϕ a 2 dλdϕ. (2) Note how the angle θ has enteed hee. Put cosθ and sinθ -- sinϕdλ, to obtain Andl Aea e λ ( ) acosφdλ e A cos( ϕ + dϕ ) A 3 + ϕ adϕ a A e λ ϕ a Asinϕ ( A) ϕ + a a A e λ ϕ -- A , aϕ + A sinϕ a (22) which agees with the two-dimensional vesion of (4). Simila deivations can be given fo (7) and (8). 5. Conclusions This bief discussion is intended mainly as a efeshe, fo students who leaned these concepts once upon a time, but pehaps have not thought about them fo awhile. The efeences below povide much moe infomation. Refeences Abamowitz, M., and I. A. Segun, 970: Handbook of mathematical functions. Dove Publications, Inc., New Yok, 046 pp. Afken, G., 985: Mathematical methods fo physicists. Academic Pess, 985 pp. Beye, W. H., 984: CRC Standad Mathematical Tables, 27th Edition. CRC Pess Inc., pp Hay, G. E., 953: Vecto and tenso analysis. Dove Publications, 93 pp. Schey, H. M., 973: Div, gad, cul, and all that. An infomal text on vecto calculus. W. W. Noton and Co., Inc., 63 pp. Wede, R. C., 972: Intoduction to vecto and tenso analysis. Dove Publications, 48 pp. 6