So the `chnge of vribles formul' for sphericl coordintes reds: W f(x; y; z) dv = R f(ρ cos sin ffi; ρ sin sin ffi; ρ cos ffi) ρ 2 sin ffi dρ d dffi So

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1 Mth 28 Topics for third exm Techniclly, everything covered on the first two exms, plus hpter 15: Multiple Integrls x4: Double integrls with polr coordintes Polr coordintes describe point in the plne by distnce nd direction, r nd. We cn trnslte from rectngulr to polr coordintes by (x; y) =(r cos ; r sin ) We cn use this new coordinte system to simplify some integrtion problems, in prt becuse circulr disk is polr rectngle, defined by» r» R nd»» 2ß. Similrly, circulr sectors cn be described s `polr rectngles'. But in so doing, we must interpret da in terms of dr nd d ; this is completely nlogous to wht we must do with u-substitution. If we hve smll circulr sector, mde between the circles of rdius r nd r + r, nd between the lines mking ngles nd +, it hs re pproximtely r r ; so nd so R f(x; y) da = D da = r dr d f(r cos ; r sin ) r dr d, where D is how we describe the region R in polr coordintes. For exmple, the integrl of the function f(x; y) =xy on the semicircle lying between the x-xis nd y= p 9 x 2 cn be computed s ß 3 (r cos )(r sin )r dr d x5: Triple integrls with sphericl nd cylindricl coordintes We cn in fct esily impose two new coordinte systems on 3-spce; ech cn sometimes be used to simplify n integrtion problem, usully by simplifying the region we integrte over. With cylindricl coordintes, we simply replce (x; y; z) with (r; ; z), i.e., use polr coordintes in the xy-plne. In the new coordinte system, dv = (r dr d ) dz, since tht will be the volume of smll `cylinder' of height dz lying over the smll sector in the xy-plne tht we use to compute da bove. Usully, we will ctully integrte in cylindricl coordintes in the order dz dr d, since this coordinte system is most useful when the cross-sections z=constnt of our region re disks (so the limits of integrtions for z will depend only on r). Sphericl coordintes re much like polr coordintes; we describe point (x; y; z) by distnce (which we cll ρ nd direction, except we need to use two ngles to completely specify the direction; first, the ngle tht (x; y; ) mkes with the x-xis in the xy-plne, nd then the ngle ffi tht the line through our point mkes with the (positive) z-xis (which we cn lwys ssume lies between nd ß). A little trigonometry leds us to the formuls x; y; z) =(ρ cos sin ffi; ρ sin sin ffi; ρ cos ffi) Agin, the ide is tht regions difficult to describe in rectngulr coordintes cn be fr esier to describe sphericlly; for exmple, sphere of rdius R cn be described s the rectngle» ρ» R, o»» 2ß, nd» ffi» ß. It is bit more trouble to work out wht dv is in sphericl coordintes; it turns out to be dv = ρ 2 sin ffi dρ d dffi.

2 So the `chnge of vribles formul' for sphericl coordintes reds: W f(x; y; z) dv = R f(ρ cos sin ffi; ρ sin sin ffi; ρ cos ffi) ρ 2 sin ffi dρ d dffi So, for exmple, the integrl of the function f(x; y; z) =xz over the top hlf of sphere of rdius 5 could be computed s 5 2ß 2 ß=2 (ρ cos sin ffi)(ρ cos ffi) (ρ 2 sin ffi) dρ d dffi hpter 16: Prmetrized curves x1: Prmetrized curves So fr, we hve tlked bout functions of severl vribles; functions which need severl inputs in order to get single output. Our next topic is prmetrized curves; functions which hve one input but severl outputs. We will focus on functions of the form ~r(t) =(x(t);y(t)) or ~r(t) = (x(t);y(t);z(t)) i.e., curves in the plne or 3-spce. If we think of t s time, then wht ~r does is give us point in the plne or 3-spce t ech moment of time. Thinking of ~r s the position of prticle, the prticle sweeps out pth or curve,, in the plne or 3-spce s time psses; we think of ~r s prmetrizing this curve. We therefore mke distinction between curve (= collection of point lid out in string) nd prmetrized curve (= function which trces out curve). A single curve cn hve mny different prmetriztions; for exmple, ~r 1 (t) = (cos t; sin t),» t» 2ß ~r 1 (t) = (cos 2t; sin 2t),» t» ß ~r 1 (t) = (sin t; cos t),» t» 2ß ~r 1 (t) = (cos t 2 ; sin t 2 ),» t» p 2ß ll prmetrize the (unit) circle in the plne. Their diffences with the first re tht they go twice s fst, or trvel in the opposite direction, or strts slowly nd then moves fster nd fster, respectively. Of specil interest re lines; they cn be described s hving strting plce nd direction they trvel, nd so cn be prmetrized by ~r(t) =P + t~v, where P is the strting point nd ~v is the direction (for exmple, the differencve of two points lying long the line). As with ordinry functions, we cn build new prmetrized curves from old ones by, for exmple, dding constnts to ech coordinte (which trnsltes the curve by those mounts), or multiplying coordintes by constnts (which streches the curve in those directions). x2: Velocity nd ccelertion When we think of t s time, we cn imgine ourselves s trvelling long the prmetrized curve ~r(t), nd so t ech pointwe cn mke sense of both velocity nd ccelertion. Velocity, which is the instntneous rte of chnge of position, cn be esily clculted from our prmetriztion ~r(t) =x(t);y(t);z(t) s~v(t) =~r (t) =x (t);y (t);z (t) Similrly, ccelertion cn be computed s ~(t) = ~r (t) = x (t);y (t);z (t) On useful fct: if the length of the velocity (i.e., its speed), jj~v(t)jj is constnt, then ~(t) is lwys perpendiculr to ~v(t) And speking of length, we cn compute the length of prmetrized curve cn be computed by integrting its speed: the length of the prmetrized curve ~r(t),» t» b, is hpter 17: Vector fields Length = b jj~v(t)jj dt

3 x1: Vector fields A vector field is field of vectors, i.e., choice of vector F (x; y) (or F (x; y; z)) in the plne for every point in some prt of the plne (the domin of F ), nd similrly in 3-spce. We cn think of F s F (x; y) = (F 1 (x; y);f 2 (x; y)) ; ech coordinte of F is function of severl vribles. We cn represent vector field pictorilly by plce the vector F (x; y) in the plne with its til t the point (x; y). A vector field is therefore choice of direction (nd mgmitude) t ech point in the plne (or 3-spce...). Such objects nturlly occur in mny disciplines, e.g., vector field my represent the wind velocity t ech point in the plne, or the direction nd mgnitude of the current in river. One of the most importnt clss of vector fields tht we will encounter re the grdient vector fields. If we hve n (ordinry) function f(x; y; z) ofseverl vribles, then for ech point (x; y; z), r(f) cn be thought of s vector, which we hve in fct lredy tken to drwing with its til t the point (x; y; z) (so tht, for exmple, we cn use it s norml vector for the tngent plne to the grph of f). Mny vector fields re grdient vector fields, e.g., (y; x) =r(xy) ; one of the question we will need to nswer is `How doyou tell when vector field is grdient vector field?'. We shll see severl nswers to this question in the next chpter. hpter 18: Line Integrls x1: The bsic ide We introduced vector fields F (x; y) in the previous chpter in lrge prt becuse these re the objects tht we cn most nturlly integrte over (prmetrized) curve. The reson for this is tht long curve wehve the notion of velocity vector ~v t ech point, nd we cn compre these two vectors, by tking their dot product. This tells us the extent to which F points in the direction of ~v. Integrtion is ll bout tking verges, nd so we cn think if the integrl of F over the curved s mesuring the verge extent to which F points in the sme direction s. We cn set this up s we hve ll other integrls, s limit of sums. Picking points ~c i strung long the curve, we cn dd together the dot products F (~c i ) ffl ( c i+1 ~ ~c i ), nd then tke limit s the lengths of the vectors c i+1 ~ ~c i between consecutive points long the curve goes to. We denote this number by F ffl d~r Such quntity cn be interpreted in severl wys; we will mostly focus on the notion of work. If we interpret F s mesuring the mount of force being pplied to n object t ech point (e.g., the pull due to grvity), then F ffl d~r mesures the mount ofwork done by F s we move long. In other words, it mesures the mount tht the force field F helped us move long (since moving in the sme direction, it helps push us long, while when moving opposite to it, it would slow us down). In the cse tht F mesures the current in river or lke or ocen, nd is closed curve (mening it begins nd ends t the sme point), we interpret the integrl of F long s the circultion round, since it mesures the extent to which the current would push you round the curve. x2: omputing using prmetrized curves Of course, s usul, we would never wnt to compute line integrl by tking limit! But if we use prmetriztion of, we cn interpret F ffl d~r s n `ordinry' integrl. The ide is tht if we use prmetriztion ~r(t) for then F (~c i ) ffl ( c i+1 ~ ~c i ) becomes

4 F (~r(t i )) ffl (~r(t i+1 ) ~r(t i )) But using tngent lines, we cn pproximte ~r(t i+1 ) ~r(t i )by ~r (t i )(t i+1 t i )=~r (t i ) y. so we cn insted compute our line integrl s b F ffl d~r = where ~r prmetrizes with» t» b. F (~r(t)) ffl ~r (t) dt Some nottion tht we will occsionlly use: If the vector field F = (P; Q; R) nd ~r(t) = (x(t);y(t);z(t)), then d~r = (dx; dy; dz), so F ffl d~r = Pdx+ Qdy + Rdz. So we cn write b F ffl d~r = Pdx+ Qdy + Rdz x3: Grdient fields nd pth independence In generl, the computtion of line integrl cn be quite cumbersome, in prt becuse we need to evlute the vector field F t the point ~r(t), while cn yield quite complicted formuls. But there is one clss of vector fields tht re relly quite esy to integrte: grdient vector fields. This is becuse we cn compute: so if F = r(f), then F (~r(t)) ffl ~r (t) b F ffl d~r = F (~r(t)) ffl ~r (t) dt b d dx dy dz = d (f(~r(t))) dt dt dt (f(~r(t))) dt = f(~r(b)) f(~r()). We cll this the Fundmentl Theorem of Line Integrls. We sy tht vector field f is pth-independent (or conservtive) ifthe vlue of line integrl over curve depends only on wht the endpoints P; Q of re, i.e., the integrl would be the sme of ny other curve running from P to Q. Our result right bove cn then be interpreted s sying tht grdient vector fields re conservtive. Wht is mzing is tht it turns out tht every conservtive vector field F is the grdient vector field for some function f. We cn lctully write down the function, too (by steling n ide from the Fundmentl Theorem of lculus...), s f(x; y) = F ffl d~r, where is ny curve from (,) to (x; y). x4: Green's Theorem All of which is very nice, but fr too theoreticl for prcticl purposes. Wht we need re simple wys to tell tht vector field is conservtive, nd to build the function f when it is. Luckily, this is not too hrd! R First, slight reinterprettion: vector field F is pth-independent if F ffl d~r= for every closed curve. If F is conservtive, then F = (F 1 ;F 2 ) = (f x ;f y ) for some function f. But then by using the equlity of mixed prtils for f, we cn then conclude tht we must hve (F 1 ) y = (F 2 ) x. In fct, this is enough to gurntee tht F is conservtive; this is becuse of Green's Theorem: defining the curl of F to be (F 2 ) x (F 1 ) y, we hve If R is region in the plne, nd is the boundry of R, prmetrized so tht we trvel counterclockwise round R, then R F ffl d~r = R R curl(f ) da In prticulr, if the curl is, then the integrl of F long is lwys for every closed curve, so F is conservtive. We cn ctully use this result to evlute line integrls or double integrls, whichever we wish. For exmple, we cn compute the re of region R s line integrl, byintegrting

5 the functionb 1 over R, nd then using vector field round the boundry whose curl is 1, such s (;y)or( x; ) or (x; 2y) or... This llows us to spot conservtive vector fields quite esily, but doesn't tell us how to compute the function it is the grdient of (clled its potentil function). But in prctice this is not hrd; we simply write down function =F 1 (e.g., f(x; y) = R F 1 (x; y)dx). This is ctully fmily of functions, becuse we hve the constnt of integrtion to worry bout, which we should relly think of s function of y (becuse we integrted function of two vribles, dx). To figure out which function of y, simply of your nd compre with F 2 ; just djust the constnt of integrtion Finlly, there is similr result for vector fields in dimension 3; for F = (F 1 ;F 2 ;F 3 ), we cn define curel(f ) = r F " = ((F 3 ) y (F 2 ) z ; ((F 3 ) x (F 1 ) z ); (F 2 ) x (F 1 ) y ) Then F = rf exctly when curl(f ) = (,,) ; nd we cn ctully construct f using procedure nlogous to the one we cme up with for vector fields with two vribles. hpter 19: Flux Integrls x1: The bsic ide The bsic ide is tht we cn lso integrte vector fields (in 3-spce) over surfce. The interprettion we will use is tht we re mesuring the mount of fluid flowing through surfce (e.g., cell membrne) immersed in the fluid. We cn think of wire-frme surfce sitting in river; we would like to compute the mount of wter flowing (ech second, perhps) flowing through the surfce. (Or, you cn think of computing the mount of rin flling on the surfce of your body...) Our input is (velocity) vector field F, nd surfce S, described in some fshion (e.g., s the grph of function of two virbles). The ide is tht piece of surfce which is tilted with respect to the vector field will not contribute much to the totl. In other words, the mount flowing through the surfce is relted to the extent to which the (unti) norml vector for the surfce is pointing in the sme direction s F. We mesure this with the dot product, F ffl ~n. This mount islso clerly proportionl to the size of the surfce; twice s much surfce will give twice s much flow. This leds us to believe tht wht we need to dd up in order to compute the flow through the surfce is F ffl ~n da (to tke into ccount tilt nd size). So we define the flux integrl of vector field F over surfce S to be RS ~ F ffl d ~ A = R S ( ~ F ffl ~n)da Now t every point of the surfce S, we ctully hve two choices of unit norml vector ~n; we will see in the next section how to mke more or less `obvious' consistent choice of norml, the outwrd pointing norml. For exmple, if S is sphere of rdius R, centered t (,,), the outwrd unit norml t (x; y; z) is just (x=r; y=r; z=r). If we choose F to be this sme vector, then it is esy to see tht F ffl ~n = 1, nd so our flux integrl will just compute the re of the surfce S. x2: omputing using grphs, cylindricl, nd sphericl coordintes Of course, still don't wnt to compute flux integrls s limits of sums, either! Wht we need is some pproches to clculting ~nda. We study three cses: Swuppose S is the grph of function f, hving domin R in the plne. Wht we would relly like to do is to compute the flux integrl s the integrl of function over R. To do this, we note tht the vector v = ( f x ; f y ; 1) is norml to the grph of f; it's the norml vector we used to express the tngent plne to the grph of f. It just so hppens tht v =

6 (1; ;f x ) (; 1;f y ), nd so its length is equl to the re of the prllelogrm tht these two vectors spn. But!, these re exctly the prlleogrms we would use to pproximte the grph, i.e., this length is lso da. So, ~nda = ( f x ; f y ; 1), nd so RS F ffl ~nda = R R F (x; y; f(x; y)) ffl ( f x; f y ; 1) dx dy dz We cn lso use cylindricl nd sphericl coordintes, in specil cses. If S is piece of cylinder cylinder, given by r = r, for nd z in some rnge of vlues R, then the outwrd norml t r ; ;z is (cos ; sin ; ), while da = r d dz, so RS F ffl ~nda = R R F (r cos ; r sin ; z) ffl (cos ; sin ; )r d dz If S is piece of sphere, given by ρ = ρ for nd ffi in some rnge R of vlues, then the outwrd norml is (cos sin ffi; sin sin ffi; cos ffi) while da is ρ 2 sin ffi d dffi, so F ffl ~nda = RS RR F (ρ cos sin ffi; ρ sin sin ffi; ρ cos ffi) ffl (cos sin ffi; sin sin ffi; cos ffi) ρ 2 sin ffi d dffi

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