Class #16 Monday, March 20, 2017

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1 D. Pogo Class #16 Monday, Mach 0, 017 D Non-Catesian Coodinate Systems A point in space can be specified by thee numbes:, y, and z. O, it can be specified by 3 diffeent numbes:,, and z, whee = cos, y = sin, and z = z. Thee ae in fact many options. y d Looking at the y plane, you can see that the aea of a small element measuing d dy is just da = d dy. In the pola/cylindical coodinates, a little ectangle of aea is still base height, but tuns out to be da = d d. d d Similaly, if you want to know the total distance between two neaby points, you use the Pythagoean theoem: ds = d + dy in D Catesian coodinates. In D cylindical coodinates, this is not much hade: d ds d d d 1 d ( ) = + = + If you ae integating ds to find the peimete of a cicle (assuming you hadn t aleady memoized it), this would be easie in cylindical coodinates than in Catesian coodinates. See Mathematica demo fo eamples of use. 3D Cuilinea Coodinate Systems (See Page 60) ds d dy Thee ae thee common ways of ceating coodinate systems that we use to descibe motion, positions, and the like. We know about Catesian coodinates (, y, z). They ae named afte Renee Descates. The net most common is cylindical. Cylindical coodinates ae just a 3D esion of pola coodinates. To conet fom cylindical to Catesian, we use the following elationships: CYLINDRICAL z = cos y = sin z = z You can easily eese these by using the Pythagoean theoem and actangent to find o, fom and y as usual. Fo each coodinate system, it is useful to addess ahead of time some common questions. Fo eample, in E&M, you might be asked to compute the total chage of a cylindical wie, which would equie a olume integal. So, the following might be useful: d d d dz y dv = d dy dz = d d dz units ae indeed olume! Note that in this sketch, dv is a cube (like d dy dz) as d, d, and dz 0. Page 1 of

2 D. Pogo Class #16 Monday, Mach 0, 017 O, you might need to integate oe some cued line in cylindical coodinates. Along a cue s, the position diffeential ds is gien by the Pythagoean theoem: ds = d + dy + dz = d + d + dz units ae indeed all!! We can edo all of this in spheical coodinates as well: SPHERICAL = sin cosφ y = sin sinφ z = cos dv = sin d d dφ ds = d + d + sin dφ z d sindφ d d Note that, ey unfotunately, what is called in the fist pictue is called φ in the second pictue. Stangely, in math classes, they ll do this moe sensibly, and use fo the same thing in each definition. Not so in physics. Velocity φ dφ y Velocity is a ecto, usually descibed as ds ˆ V i y ˆ = = + j + z kˆ. So, if you know (t), etc., t t t you can easily find the elocity. How would we descibe it in a spheical coodinate system? We can do it, poided that we hae an appopiate definition of ds. Fom aboe, ds = d + d + sin dφ gies us a magnitude fo ds. But in ecto fom, this is ds = ( d) ˆ + ( d ) ˆ + ( sin d ) φ ˆ φ. So, ds V ˆ ˆ φ = = + + sin ˆ φ. t t t Of couse, you need to know (t), etc., to sole a poblem like this. dv Similaly, once you hae elocity, you can always find acceleation, since a =. These poblems ae almost staggeingly had to wok out, as you ll see on the homewok. Page of

3 D. Pogo Class #16 Monday, Mach 0, 017 Jacobians The peious eamples ae the most common coodinate systems, but in some poblems (especially those inoling gaitation, whee space-time is cued) it is desiable to come up with othe, stange, coodinate aes. When you ae tying to elate one coodinate system to anothe, the Jacobian is a useful tool that elates olumes in each to the othe. It is the deteminant of a mati, and is theefoe just a scala epession. The Jacobian descibing the elationship between an (, y, z) coodinate system and an abitay (m, n, p) coodinate system is defined as this deteminant: m n p y y y J = m n p z z z m n p deˆ 11 deˆ 1 deˆ 13 The useful equation fo Jacobians is J =, whee each e 1i is a unit ecto in the deˆ ˆ ˆ 1 de de3 fist coodinate system, and each e i is a unit ecto in the second coodinate system. Eample: The Jacobian elating (, y, z) to (,, φ) is: J φ y y y =. φ z z z φ Note that the elements of the mati ae not typically dimensionally homogeneous. Since we know the definitions = sin cosφ, y = sin sinφ, and z = cos, this becomes: sin cosφ cos cosφ sin sinφ J = sin sinφ cos sinφ sin cosφ cos sin 0 We can multiply out the deteminant in ou usual way, and we discoe that this paticula Jacobian educes to J = sin. d dy dz Theefoe, using ou second useful Jacobian equation, sin =. This allows us to d d dφ conclude that d dy dz = sin d d dφ, which we aleady saw a few minutes ago. Page 3 of

4 D. Pogo Class #16 Monday, Mach 0, 017 So, when you e faced with eally weid coodinate systems, the Jacobian can help you quickly set up and pefom olume integals. Notation Aboe, I used a lowecase e as a geneic unit ecto. That s petty common. In this notation, e ˆ is the same as î, e ˆy is the same as ĵ, e ˆ is the same as ˆ, and so on. Howee, it also common to use numbes to epesent diections. e 1 is the same as e, e = e y, and e 3 =e z. We would obiously like to be able to elate diffeentials to each othe. Fo eample, how ae the unit ectos in cylindical e =1 coodinates elated to the unit ectos in Catesian e =1 cos coodinates? sin sin Fom looking at the pictues hee: cos eˆ cos ˆ sin ˆ = i + j, and eˆ sin iˆ cos ˆ = + j, and eˆ = 1kˆ z It pobably goes without saying that e is pependicula to e and e z, etc. We could do simila things with spheical coodinates. Fo eample: eˆ = sin cosφ i ˆ + sin sinφ ˆ j + cos k ( ) ( ) ( ) ˆ = ( ) ˆ + ( ) ˆ + ( ) ˆ and ˆ ( sin ) ˆ ( cos ) eˆ cos cosφ i cos sinφ j sin k Gadients and Opeatos in Non-Catesian Coodinates e i ˆ φ = φ + φ j. You emembe the gadient, ight? If U is some scala field, then U ˆ U i U ˆ = + j + U kˆ. At y z least, that s what it is in Catesian coodinates. What about in cylindical coodinates? In the -diection, looking at ou oiginal olume sketch, U ds = d, so U =. OK, that s one component. 1 U In the -diection, ds = d, so U =. U In the z-diection, ds z = dz, so zu =. z U 1 U U Putting togethe all thee pats: U = eˆ + eˆ + eˆ z. CYLINDRICAL z Page 4 of

5 D. Pogo Class #16 Monday, Mach 0, 017 Scale Factos If you look at the diffeential length in any coodinate system, you ll see that it usually follows a paticula fom. Compae these: ds = ( ) eˆ ( ) ˆ ( ) ˆ + y ey + z ez ds = ( d ) eˆ ( ) ˆ ( ) ˆ + d e + dz ez ds = d eˆ + d eˆ + sindφ eˆ ( ) ( ) ( ) In any coodinate system, this can be epessed geneally as: ds = h eˆ + h eˆ + h eˆ ( ) ( ) ( ) So, in Catesian coodinates, h = 1. Also, h y = 1, h z = 1. Fo the cylindical coodinates: h = 1, h =, and h z = 1. Fo the spheical coodinates: h = 1, h =, and h φ = sin. These h tems ae called scale factos. Also, some people like to use fo eey coodinate ais, whethe it is, y, z,,, o φ. When you do this, you use numeical subscipts to distinguish the diffeent component diections. The book does this sometimes, too. So, in Catesian coodinates: 1 =, = y, 3 = z. So, in cylindical coodinates: 1 =, =, 3 = z. So, in spheical coodinates: 1 =, =, 3 =φ. In this notation, if you know the scale factos fo a paticula coodinate system, you can compute all kinds of deiaties: 1 U 1 U 1 U U = eˆ + eˆ + eˆ h h h φ 1 V = h h V + h hv + h h V h h h ( ) ( ) ( ) hh3 U h3h1 U h1h U U = + + h 1 h h 3 1 h 1 1 h 3 h 3 3 h1e ˆ1 h ˆ ˆ e h3e3 1 V = h h h hv h V h V Page 5 of

6 D. Pogo Class #16 Monday, Mach 0, 017 Eample Gien the position s = ( 3ˆ i + ˆ) j + ( 1ˆ i + 3 ˆj ) t Detemine elocity in Catesian and pola coodinates Obiously, (t) = 3 t (the î component of s ). y(t) = + 3t (the ĵ component of s ). ds Theefoe, = = ( 1ˆ i + 3 ˆj ) In othe wods, = 1, and y = +3 (both ae constants, and the speed is 10 ). Let s look at cylindical = + y = ( 3 t) + ( + 3t ) = Taking the deiatie: d = + 3 y + 3t Also, ( t) = atan = atan. 3 t Taking the deiatie: d = 11 So we e almost done: d + 3 = = and d 11 = = It s woth noting that in pola coodinates, the elocities ae not constant (but the speed still is 10, o about 3.16, in each coodinate system). Page 6 of