Notation 1 Vectos As aleady noted, we neve wite vectos as pais o tiples of numbes; this notation is eseved fo coodinates, a quite diffeent concept. The symbols we use fo vectos have aows on them (to match what we wite by hand) as well as being bold-faced (to match the notation usually used in textbooks). The one exception to this ule is that we put hats on unit vectos, athe than aows. 1 2 Spheical Coodinates 2 2.1 The Poblem Nealy eveybody uses and θ to denote pola coodinates. Most Ameican calculus texts also utilie θ in spheical coodinates fo the angle in the equatoial plane (the aimuth o longitude), fo the angle fom the positive -axis (the enith o colatitude), and ρ fo the adial coodinate. Vitually all othe scientists and enginees as well as mathematicians in many othe counties evese the oles of θ and (and use some othe lette, such as R, fo the adial coodinate). Why is this a poblem? Afte all, the change in notation only affects students in paticula fields, such as physics o electical engineeing. Futhemoe, it s just a convention; suely these students have the matuity to 1 To ou dismay, some jounals seem to have difficulty typesetting hats above bold math italic Geek lettes, such as ˆθ. 2 This section is adapted fom [1]. 1
deal with it. Based on ou expeience tying to implement this change duing a second-yea couse in multivaiable calculus, we feel that such sentiments undeestimate the extent of the poblem. Students find the complete intechange of the oles of θ and to be teibly confusing and once confused, always confused. Using diffeent names fo the adial coodinate, on the othe hand, causes few poblems. The use of fo the spheical adial coodinate can be confused with the adial coodinate in pola o cylindical coodinates, but computations equiing both at the same time ae ae. While ρ is not available to the physicist, as it is used to epesent chage o mass density, students do not appea to be confused by the use of seveal diffeent names fo the spheical adial coodinate. Thee is howeve a much moe seious poblem. Seveal of the most commonly used calculus texts list spheical coodinates in the ode (ρ, θ, ); the est use (ρ,, θ). The fist of these is left-handed! An othogonal coodinate system is ight-handed if the coss poduct of the fist two coodinate diections points in the thid coodinate diection. This is immateial in the taditional mathematics teatment of vecto calculus, but cucial to the way physicists and enginees teat the same mateial. These scientists often intoduce basis vectos in the coodinate diections, analogous to {î,ĵ, ˆk} fo ectangula coodinates, and it is essential that these vectos fom a ighthanded system. This equies that the enith be listed befoe the aimuth; with the standad mathematics convention, this is (ρ,, θ). Books which use the standad mathematics definitions of the angles but wite (ρ, θ, ) ae doing thei students a majo dissevice, although we eiteate that this is only an issue fo mateial coveed in subsequent couses. 2.2 The Solution Thee is a unifom standad fo the use of spheical coodinates in applications, which is nowhee moe appaent than in the definition of spheical hamonics. These special functions on the sphee ae widely used, notably in the quantum mechanical desciption of electon obitals, which in tun undelies much of chemisty. It can not be stated too stongly that eveyone wites the spheical hamonics as Y lm (θ,), whee θ is the enith and the aimuth. Thee is simply no way to change this convention, which is embedded in geneations of standad efeence books. One objection to this is that it is confusing to use the same label, θ, fo
(, θ, ) θ (,,) θ y y x x Figue.1: Ou conventions fo spheical and cylindical coodinates. two diffeent angles in pola and spheical coodinates. This objection can be easily esolved, even if the esolution may not be popula: Change the conventions fo pola coodinates, that is, use athe than θ. We popose that these conventions be adopted by mathematicians, and we use them thoughout these mateials. Ou conventions fo both spheical and cylindical coodinates ae shown in Figue.1. 3 Integals Thee ae two common notations fo multiple integals, one being to use one integal sign fo each iteated integal which will ultimately be pefomed, the othe is to use a single integal sign fo each integal, since an integal just means add things up. We use the latte notation fo suface and volume integals, but iteated integals ae witten out in full. Fo instance, when
finding the flux of ˆk upwads though the unit disk, we wite ˆk d A = 2π 1 1 d d (1) 0 0 The in-between case in which the limits ae not given explicitly can be witten eithe way depending on the context (and pesonal pefeence), as in d d = d d (2)
Bibliogaphy [1] Tevian ay and Coinne A. Manogue, Spheical Coodinates, College Math. J. 34, 168 169 (2003). [2] Robet Osseman, Two-imensional Calculus, Hacout, Bace, and Wold, New Yok, 1968. [3] Tevian ay & Coinne A. Manogue, The Mude Mystey Method fo etemining Whethe a Vecto Field is Consevative, College Math. J. 34, 238 241 (2003). 11