ENGI 4430 Non-Cartesian Coordinates Page xi Fy j Fzk from Cartesian coordinates z to another orthonormal coordinate system u, v, ˆ i ˆ ˆi

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1 ENGI 44 Non-Catesian Coodinates Page 7-7. Conesions between Coodinate Systems In geneal, the conesion of a ecto F F xi Fy j Fzk fom Catesian coodinates x, y, z to anothe othonomal coodinate system u,, w in (whee othonomal means that the new basis ectos u,, w ae mutually othogonal and of unit length) is gien by F F i F j F k F u F F w. x y z u w Howee, F and w In matix fom F F u F i F j F k u i u F j u F k u F. u x y z x y z F ae defined similaly in tems of the Catesian components F, F, F. F u i u j u k u Fx F i j k Fy. F w F i w j w k w z x y z The matices on the ight hand side of the equation will contain a mixtue of expessions in the new u,, w and old x, y, z coodinates. This needs to be coneted into a set of expessions in u,, w only. Example 7. Expess the ecto F y i xj zk in cylindical pola coodinates. x y plane: coodinates x y plane: basis ectos x cos i i cos cos j co y sin z z s sin k c os

2 ENGI 44 Non-Catesian Coodinates Page 7- Example 7. (continued) i cos sin ** ik j cos cos jk os k c kk The coefficient conesion matix fom Catesian to cylindical pola is theefoe cos sin sin cos Letting ccos, s sin : F yi xj zk si cj zk F pola c s s cs sc s c c ss cc z z z Theefoe F y i xj zk zk ** This esult can be obtained fom the tigonometic identity cos A B cos Acos B sin Asin B Setting A and B, cos cos cos sin sin sin

3 ENGI 44 Non-Catesian Coodinates Page 7- We can also geneate the coodinate tansfomation matix fom Catesian coodinates x, y, z to spheical pola coodinates,,. [ is the declination (angle down fom the noth pole, ) and is the azimuth (angle aound the equato ).] [Vetical] Plane containing z-axis and adial ecto : z cos The pojection of the adial ecto onto the plane z cos has length sin The angle between and k is k cos sin The angle between and k is k cos [Hoizontal] Plane z cos : The shadow of the adius has length sin. The pojection of onto the x axis ( î ) is x sincos i i sin cos T x y z sin cos sin sin cos T The pojection of onto the y axis ( ĵ ) is y sinsin j j sin sin The angle between and ĵ is j cos The angle between and î is i cos sin The emaining thee enties in the coodinate conesion matix can be found in a simila way.

4 ENGI 44 Non-Catesian Coodinates Page 7-4 The conesion matix fom Catesian to spheical pola coodinates is then i j k sin cos sin sin cos cos cos cos sin sin i j k sin cos i j k Example 7. Conet F yi xj to spheical pola coodinates. Let c cos, s sin, c cos, s sin y s s, x s c F F s c s s c s s F c c c s s s c F s c s c s s c s c c s s c s s c s Theefoe F sin Expessions fo the gadient, diegence, cul and Laplacian opeatos in any othonomal coodinate system will follow late in this chapte.

5 ENGI 44 Non-Catesian Coodinates Page 7-5 Summay fo Coodinate Conesion: To conet a ecto expessed in Catesian components i j k into the x y z equialent ecto expessed in cylindical pola coodinates zk, expess in tems of,, z the Catesian components x, y, z x cos, y sin, z z ; then ealuate using cos sin x sin cos y z z Use the inese matix to tansfom back to Catesian coodinates: x cos sin y sin cos z z To conet a ecto expessed in Catesian components i j k into the x y z equialent ecto expessed in spheical pola coodinates, expess the Catesian components x, y, z,, using x sin cos, y sin sin, z cos ; then ealuate in tems of sin cos sin sin cos x cos cos cos sin sin y sin cos z Use the inese matix to tansfom back to Catesian coodinates: x sin cos cos cos sin y sin sin cos sin cos cos sin z Note that, in both cases, the tansfomation matix A is othogonal, so that A = A T. This is not tue fo most squae matices A, but it is geneally tue fo tansfomations between othonomal coodinate systems.

6 ENGI 44 Non-Catesian Coodinates Page 7-6 Basis Vectos in Othe Coodinate Systems In the Catesian coodinate system, all thee basis ectos ae absolute constants: d d d i j k dt dt dt The deiatie of a ecto is then staightfowad to calculate: d f i f j f k i df j df k df dt dt dt dt But most non-catesian basis ectos ae not constant. Cylindical Pola: cos i sin j sin icos j k k d Let then sin i cos j dt cos i sin j k Theefoe if a ecto F is descibed in cylindical pola coodinates F F F F z k, then F F F F F z F k F F F F Fz k

7 ENGI 44 Non-Catesian Coodinates Page 7-7 In paticula, the displacement ecto is t t z t elocity ecto is d d d d z k dt dt dt dt k, so that the Example 7. Find the elocity and acceleation in cylindical pola coodinates fo a paticle taelling along the helix x cos t, y sin t, z t. Cylindical pola coodinates: x cos, y sin, z z y x y, tan x 9cos t 9sin t 9 sin t tan tan t t [pincipal solution fo ] cos t z t z d k 6 k dt [The elocity has no adial component the helix emains the same distance fom the z axis at all times.] d a 6 k 6 dt [The acceleation ecto points diectly at the z axis at all times.]

8 ENGI 44 Non-Catesian Coodinates Page 7-8 Altenatie deiation of cylindical pola basis ectos On page 7. we deied the coodinate conesion matix A to conet a ecto expessed in Catesian components i j k into the equialent ecto expessed x y z in cylindical pola coodinates zk and its inese A - x cos sin x A y sin cos y z z z x cos sin y A sin cos z z z Fo any matix M, M the fist column of M; M the thid column of M M the second column of M; and Note that k, so that the Catesian fom of is Similaly cos sin cos sin cos sin cos sin i j cos sin sin sin cos cos sin cos i j Theefoe the columns of A - (and theefoe the ows of A) ae just the Catesian components of the thee cylindical pola basis ectos.

9 ENGI 44 Non-Catesian Coodinates Page 7-9 Spheical Pola Coodinates The coodinate conesion matix also poides a quick oute to finding the Catesian components of the thee basis ectos of the spheical pola coodinate system. sph x sin cos cos cos sin sin cos y cos sin cos z A sin sin cos sin cos sin sin sin cos i sin sin j cos k x sin cos cos cos sin cos cos y cos sin sin z A sin sin cos sin cos cos sin cos cos i cos sin j sin k x sin cos cos cos sin sin y cos sin z A sin sin cos sin cos cos sin i cos j Again, the Catesian components of the basis ectos ae just the columns of ae also the ows of A ). This is tue fo any othonomal coodinate system. A (which These expessions fo,, can also be found geometically [aailable on the couse web site].

10 ENGI 44 Non-Catesian Coodinates Page 7- Deiaties of the Spheical Pola Basis Vectos sin cos i sin sin j cos k d d d cos cos sin sin i dt dt dt d d cos sin sin cos d sin j k dt dt dt d d sin d dt dt dt cos cos i cos sin j sin k d d d sin cos cos sin i dt dt dt d d sin sin cos cos d cos j k dt dt dt d d cos d dt dt dt sin i cos j d d cos d sin i j dt dt dt But sin cos sin cos i sin sin j sin cos k cos cos i cos sin j sin cos k cos i sin j d dt sin cos d dt In paticula, the displacement ecto is, so that the elocity ecto is d d d d d sin d dt dt dt dt dt dt d d sin d dt dt dt

11 ENGI 44 Non-Catesian Coodinates Page 7- It can be shown that the acceleation ecto in the spheical pola coodinate system is d d d d a sin dt dt dt dt dt dt dt sin d d d d d sin sin dt dt Compae this to the Catesian equialent d x d y d z a i j k! dt dt dt Example 7.4 Find the elocity ecto fo a paticle whose displacement ecto, in spheical pola 4, t, t, t. coodinates, is gien by d d d 4, t, t,, dt dt dt d d sin d dt dt dt 4 4 sin t t 4 8sin t [This descibes a path spialling aound a sphee of adius 4, fom pole to pole.]

12 ENGI 44 Non-Catesian Coodinates Page 7- Summay: Cylindical Pola: d d dt dt d d dt dt d k dt zk zk Spheical Pola: d d sin d dt dt dt d d cos d dt dt dt d sin cos d dt dt sin

13 ENGI 44 Non-Catesian Coodinates Page 7- Gadient Opeato in Othe Coodinate Systems Fo any othogonal cuilinea coodinate system u, u, u in a tangent ecto along the u i cuilinea axis is T i The unit tangent ectos along the cuilinea axes ae whee the scale factos h i. u i u i, e i T i, h u The displacement ecto can then be witten as ue ue ue, whee the unit ectos e i fom an othonomal basis fo [ ij is the Konecke delta.] : ei e j ij i j i j i i The diffeential displacement ecto d is (by the Chain Rule) d du du du h du e h du e h du e u u u which leads to a moe geneal expession fo the elocity ecto (compaed to those of the peceding page): d du du du h e h e h e dt dt dt dt The diffeential ac length ds is ds d d h du h du h du The element of olume dv is dv h h h du du du x, y, z dududu u, u, u x u x u x u y u y u y u z u z u z u du du du

14 ENGI 44 Non-Catesian Coodinates Page 7-4 e e e Gadient opeato h u h u h u e V e V e V Gadient V h u h u h u Diegence Cul Laplacian h h f h h f h h f F h h h u u u h e h e h e F = h h h u u u h f h f h f h h h h e h f u h e h f u h e h f u h h V h h V hh V V h h h u h u u h u u h u Catesian: hx hy hz Cylindical pola: h hz, h Spheical pola: h, h, h sin The familia expessions then follow fo the Catesian coodinate system.

15 ENGI 44 Non-Catesian Coodinates Page 7-5 In cylindical pola coodinates, naming the thee basis ectos as zk z The elationship to the Catesian coodinate system is x cos, y sin, z z x y, tan One scale facto is h T x y z cos sin z cos sin,, k, y x we hae: In a simila way, we can confim that h and h z. x y z h cos sin z sin cos h z x y z z z z z cos sin z z z z

16 ENGI 44 Non-Catesian Coodinates Page 7-6 In cylindical pola coodinates, dv h h h d d dz d dz z d h d h d hz dz d z d d d s V V V V k z F f f f z z f f f f z z F k z f f f z V V V V z z V V V V z All of the aboe ae undefined on the z-axis ( ), whee thee is a coodinate singulaity. Howee, by taking the limit as, we may obtain well-defined alues fo some o all of the aboe expessions.

17 ENGI 44 Non-Catesian Coodinates Page 7-7 Example 7.5 Gien that the gadient opeato in a geneal cuilinea coodinate system is e e e h u h u h u, why isn t the diegence of F F F F F e F e F e equal, in geneal, to? h u h u h u The quick answe is that the diffeential opeatos opeate not just on the components F, F, F, but also on the basis ectos e, e, e. In most othonomal coodinate systems, these basis ectos ae not constant. The diegence theefoe contains additional tems. e e e F e Fe Fe h u h u h u e e F F e e e F F e ee F F e e e e h u h u h u h u h u h u e e F F e e e F F e ee F F e e e e h u h u h u h u h u h u e e F F e e e F F e e e F F e e e h u h u e h u h u h u h u = F F e F e F e e e e h u h u h u h u F e F F e F e e e e h u h u h u h u F e F e F F e e e e h u h u h u h u Fo Catesian coodinates, all deiaties of any basis ecto ae zeo, which leaes the familia Catesian expession fo the diegence. But fo most non-catesian coodinate systems, at least some of these patial deiaties ae not zeo. Moe complicated expessions fo the diegence theefoe aise.

18 ENGI 44 Non-Catesian Coodinates Page 7-8 Example 7.5 (continued) Fo cylindical pola coodinates, we hae F F F F k z F F F F k z F z k F z k F z Fz k k z z But none of the basis ectos aies with o z * and the basis ecto k is absolutely constant. Theefoe the diegence becomes F F F F F z z But F F F F and F So we ecoe the cylindical pola fom fo the diegence, di F F F F Fz z * As shown hee, the basis ectos and clealy ay with but do not change with. k is an absolute constant.

19 ENGI 44 Non-Catesian Coodinates Page 7-9 In spheical pola coodinates, naming the thee basis ectos as The elationship to the Catesian coodinate system is x sin cos, y sin sin, z cos. One of the scale factos is T x y z h sin cos sin sin cos cos cos cos sin sin cos cos sin sin cos sin,,, we hae: In a simila way, we can confim that h and h sin. dv sin d d d sin d d d ds d d sin d d d sin d V V V V sin F sin f sin f f sin sin f f cot f f f sin sin f sin f f

20 ENGI 44 Non-Catesian Coodinates Page 7- Spheical Pola (continued) F sin f sin sin f f V V V V sin sin sin sin V V V sin sin sin sin V V V cot V V sin All of the aboe ae undefined on the z-axis (sin ), whee thee is a coodinate singulaity. Howee, by taking the limit as sin, we may obtain well-defined alues fo some o all of the aboe expessions. The oigin ( ) poses a simila poblem.

21 ENGI 44 Non-Catesian Coodinates Page 7- Example 7.6 A ecto field has the equation, in cylindical pola coodinates,, z, k k F e n n Find the diegence of F and the alue of n fo which the diegence anishes fo all. In cylindical pola coodinates, n F k, F F z di F F F F F z z n k di n n F kn k n and clealy di F when n. F k is theefoe a souce-fee field eeywhee except on the z axis.

22 ENGI 44 Non-Catesian Coodinates Page 7- Example 7.7 In spheical pola coodinates, F,, f cot f g,, whee f is any diffeentiable function of only and g, is any diffeentiable function of and only. Find the diegence of F. cot,,, F f F f F g Fo spheical pola coodinates, F sin F F F sin sin F f cot sin f g, sin sin cot f sin f sin f cot f cot eeywhee (except possibly on the z axis, whee sin ).

23 ENGI 44 Non-Catesian Coodinates Page 7- Example 7.8 Find cul sin, whee, ae the two angula coodinates in the standad spheical pola coodinate system. Let sin F, then f F f sin sin sin sin sin f sin sin sin cos sin sin sin sin F cos sin sin d Note: Chain ule fo sin : d Let u sin, then d d du d u u u sin sin cos d du d d

24 ENGI 44 Non-Catesian Coodinates Page 7-4 Cental Foce Law If a potential function V x, y, z, (due solely to a point souce at the oigin) depends only on the distance fom the oigin, then the functional fom of the potential can be deduced. Using spheical pola coodinates:,, f V d df d f df V d d d d But, in any egions not containing any souces of the ecto field, the diegence of the ecto field F V (and theefoe the Laplacian of the associated potential function V ) must be zeo. Theefoe, fo all, d f d df d Sole this ODE by eduction of ode: Let y df then dy y dy y dy d d d d y dy d ln y ln C ln B y df y d B B f A V,, A B OR (a much faste solution!) d dv dv V B d d d dv d B V A B

25 ENGI 44 Non-Catesian Coodinates Page 7-5 Gaity is an example of a cental foce law, fo which the potential function must be of B the fom V,, A. The zeo point fo the potential is often set at infinity: B lim V lim A A The foce pe unit mass due to gaity fom a point mass M at the oigin is F V GM But, in spheical pola coodinates, V V V V dv B sin d GM B B GM Theefoe the gaitational potential function is V GM The electostatic potential function is simila, with a diffeent constant of popotionality.

26 ENGI 44 Non-Catesian Coodinates Page 7-6 [Space fo additional notes] [End of Chapte 7]