1. Review. t 2. t 1. v w = vw cos( ) where is the angle between v and w. The above leads to the Schwarz inequality: v w vw.

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1 1. Review 1.1. The Geometry of Curves. AprmetriccurveinR 3 is mp R R 3 t (t) = (x(t),y(t),z(t)). We sy tht is di erentile if x, y, z re di erentile. We sy tht it is C 1 if, in ddition, the derivtives re continuous. We sy is C n if the first n derivtives exist nd re continuous. We sy tht is C or smooth if derivtives to ny order exist. The length of prmetric curve from t 1 to t 2 is the integrl where (t) = t 1 t 2 (t)dt ( dx dt )2 + ( dy dt )2 + ( dz dt )2 is the length of the velocity vector (t) = (x (t),y (t),z (t)). The length of the vector v = ( 1, 1,c 1 ) is v = c2 1 which is lso v v where is the sclr or dot product. We hve Also, v w = c 1 c 2. v w = vw cos( ) where is the ngle etween v nd w. The ove leds to the Schwrz inequlity: v w vw. Theorem 1.1. A line is the shortest curve etween two points Arclength prmetriztion. We sy tht curve is prmetrized y rclength s if the prmeter s is the length of the curve from certin point. Let s(t) = t (t) dt e the length of the curve from the point t =. A reprmetriztion of the curve is composition vrile. We sy curve is regulr if (t) is never 0. To reprmetrize curve y rclength, we must hve s(t) = t = t (t) = (u(t)) where u is function of rel (t) dt. Tking derivtives of oth sides, we otin (t) = 1ythefundmentltheoremofClculus. Conversely, if (t) = 1, then s(t) = t nd the curve is prmetrized y rclength.

2 So to prmetrize curve y rclength mens finding prmetriztion such tht the velocity vector lwys hs length 1. We hve Theorem 1.2. Every regulr curve cn e reprmetrized y rclength Frenet frmes. A Frenet frme cn e thought of s the pth long which you would wnt three fingers to gr curve so tht it hs no wy of escping! It is coordinte system tht moves on the curve. The first vector is the unit tngent vector. So it is most convenient to ssume tht the curve is prmetrized y rclength so tht the velocity vector utomticlly hs length 1. The other two vectors should e perpendiculr to the tngent vector ut how should we choose them? They should e determined y the curve. One importnt property of the curve is how it turns. So the second vector will e unit vector tht tells us how the tngent vector chnges. We define Assuming tht the curvture N(t) = T (t) T (t), pple(s) = T (s) is not zero. The curvture tells us how fst the tngent vector turns. Note tht y the product formul, ecuse the length of T is constnt, N is norml to T everywhere. Definition 1.3. The osculting plne t the point of prmeter s is the plne spnned y the vectors T (s) nd N(s). The third vector is the unique vector tht completes the system T,N into coordinte system. This vector is B = T N the cross-product of T nd N. Recll tht the cross-product of two vectors v = ( 1, 1,c 1 ) nd w = ( 2, 2,c 2 ) cn e defined s the determinnt i j k v w = 1 1 c c 2 where i, j, k re the unit vectors (1, 0, 0), (0, 1, 0), (0, 0, 1) respectively. The length of the crossproduct is T N sin( ). SoB hs length 1. The cross-product is iliner, non ssocitive: A (B C) = (A C)B (A B)C (A B) C,

3 nti-commuttive. If oth v nd w hve length 1, their cross-product is the unit vector perpendiculr to oth of them, oriented with the thum-screw rule. The set (T,N,B) is the Frenet frme long.wesytht(t,n,b) is n orthonorml sis of R 3.ThevritionofT,N nd B will tell us how twists nd turns through spce. Their vritions re determined y their derivtives T,N nd B. We lredy know T = pplen y the definition of N. Thetorsion of the curve is: = N B = B N. We hve the Frenet reltions for curve of unit speed: T = pplen N = pplet + B B = N So the torsion is the component of the derivtive of the norml vector N on the vector B. Ittells us how the curve twists out of its osculting plne or how twisted the curve is! Torsion is the noun ssocited to the French ver tordre which mens to twist. Recll tht pple = T is lwys 0. The curvture of circle is the inverse of its rdius. Theorem 1.4. Let e unit speed curve. Then (1) pple = 0 if nd only if is line. (2) for pple > 0, = 0 if nd only if is plne curve. Theorem 1.5. A curve constnt. is prt of circle if nd only if it is plne curve ( = 0) nd pple > 0 is Definition 1.6. For generl plne curve, the curve defined y is the evolute of (s) = (s) + 1 pple(s) N(s) : the locus of centers of its osculting circles Non unit speed curves. As we sw, it is not lwys esy to reprmetrize curve y rclength so we need to modify our Frenet formuls so tht we cn lso compute with ritrry prmetriztions. For this we hve to recll (s) = (t).

4 The Frenet frme is defined in terms of unit speed prmetriztion. So we theoreticlly switch to unit speed nd then use the Chin rule to compute with n ritrry prmetriztion. T (s) = T (t) is lwys the unit tngent vector. So T (t) = (t) (t) = (s). Recll tht the curvture is pple(s) = dt ds. If we don t hve n explicit rclength prmetriztion, we cn tke derivtives with respect to t. Recll the formuls Using the chin rule, we otin dt ds = pplen dn ds = pplet + B db ds = N. dt dt = dt ds ds dt = pple ds dt N dn dt = dn ds ds dt = pple ds dt T db dt = db ds ds dt = ds dt N. ds + dt B First note tht Computing (t), (t) nd (t) we otin: nd we lso otin N s pple = dt ds = 1 (t) dt dt. pple = 3, B = N = B T nd s or since pple 3 =. = N B = ( ) = ( ) pple 2 6 2

5 Definition 1.7. The involute of plne curve (t) is given y nd tht the evolute of plne curve is given y I(t) = (t) s(t) (t) = (t) s(t)t (t) (t) E(t) = (t) + N(t) pple(t). These two opertions re inverses to ech other The Fundmentl theorem (1.5.17). Theorem 1.8. Given smooth functions pple(s) > 0 nd (s) on n intervl I R, there exists regulr rclength prmetrized curve I R such tht pple is its curvture nd its torsion. Such curve is unique up to rigid motion (i.e. comintion of rottion nd trnsltion) of R Green s theorem nd the isoperimetric inequlity. Asmoothcurve [, ] R 2 is closed if (n) () = (n) () for ll n 0. Closed curves often represents periodic orits of physicl systems. A closed curve is simple if it does not cross itself, in other words, is one-to-one on [, ). Theorem 1.9. (The isoperimetric inequlity) Let C e simple closed curve of length L, ounding region of re A. Then L 2 4 A with equlity exctly when C is circle. For the proof we use the formul A = x(t)y (t)dt = y(t)x (t)dt. If the curve cn e divided into the union of the grphs of two functions f nd g, then A = x 1 x 2 g(x)dx x 1 x 2 f(x)dx = t 2 t 1 y(t)x (t)dt Integrtion y prts yields the second formul: Or one cn use y(t)x (t)dt = [ y(t)x(t)] + t 1 y(t)x (t)dt t 2 x(t)y (t)dt. y(t)x (t)dt = y(t)x (t)dt.

6 Theorem (Green s theorem) Let P nd Q e smooth functions on simply connected region (i.e., region without holes) R of the plne with oundry simple closed curve C. @x dxdy = C (Pdx Qdy) Surfces in R 3. Surfces in R 3 re unions of imges of C (prtils with respect to the two vriles to ny order exist) mps ' D R 3 where D is n open (mening D does not contin its oundry) domin in R 2, usully n open rectngle ], [ ]c, d[ or n open disc (x, y) P < r. Ifu nd v re the coordintes on D, wewrite '(u, v) = (x(u, v),y(u, v),z(u, v)) nd define ' @u, ' @v. Amp' is clled regulr if the derivtives ' u,' v re linerly independent everywhere on D, i.e., the cross-product ' u ' v is everywhere nonzero. The vector ' u ' v is norml to the surfce nd we cn otin unit norml y dividing y its length. We cn usully mke D smller so tht the mp ' is one-to-one nd regulr. Such mp ' is then clled coordinte chrt or ptch for the surfce. For such mp ' there exists n inverse (defined on the imge of '). If we hve two coordinte chrts ' nd we cn form the composition 1 ' D R 2. Asurfceisclledsmooth if for ll coordinte chrts ',, thecomposition 1 ' is smooth. The composition 1 ' is chnge of coordintes. AsetofcoordintechrtscoveringS is usully clled n tls Grphs (or Monge ptches). '(u, v) = (u, v, f(u, v)) for function of two vriles f. Alwys coordinte ptch if f is smooth The implicit function theorem. When is n implicit eqution f(x, y, z) = c surfce? If we could solve for z, wewouldeletowritez = g(x, y). Thenonthesurfce, nd @f dx + dz = 0 @f dx

7 Theorem (The implicit function theorem) Ner ny point (,, c) stisfying f(x, y, z) = c 0, z cn e written s smooth function of x nd y whose prtils @y = c is clled regulr vlue of f, ifnopointstisfiesf(x, y, z) @y = Spheres. x 2 + y 2 + z 2 = 1. Upper, lower, left, right hemispheres... z = x 2 + y 2 etc. Sphericl coordintes:!(, ) = R(sin cos,sin sin,cos ). Geogrphicl coordintes: u =, v = 2. Compute the derivtives nd the unit norml. Wht points on the sphere do we miss? Are these coordinte ptches? How should we define the domins D? Surfces of revolution. Regulr curve (u) = (g(u),h(u), 0). Rotteoutthex-xis: How should we @f '(u, v) = (g(u),h(u) cos v,h(u) sin v) Ruled surfces. Asurfceisruledifithsprmetriztion where nd w(u, v) = (u) + v (u) re two curves. This my not e coordinte ptch, we might hve to remove some points to get coordinte ptches. Ech time we fix u, wegetlineins. Exmples: Cones: w = P + v Sddles: z = xy (douly ruled). where P is fixed (ll the lines go through one point). Hyperoloid of one sheet: x 2 + y 2 z 2 = 1 (douly ruled, see exercises in the ook) The geometry of surfces. Curves hve tngent lines, surfces hve tngent plnes. There re 3 equivlent wys of looking t the tngent spces of surfce: (1) The tngent plne to S t the point (,, c) is the union of ll the tngent lines to curves in S through (,, c). (2) The tngent plne is the plne through (,, c) spnned y the vectors ' u,' v in coordinte chrt ' ner (,, c). (3) The tngent plne is the plne through (,, c) with norml vector ' u ' v..

8 The first one is not so esy to compute with useful for geometric rguments nd it is nice ecuse it shows tht the tngent plne is independent of the choice of chrt. AcurvedefinedonnintervlI = [, ] lies on surfce S if the mp I R 3 fctors through S. In other words, I S R 3. The dtum of curve in S is equivlent to the dt of two smooth mps u(t),v(t) I R such tht (t) = '(u(t),v(t)). (SeeLemm2.1.3) Recll tht the eqution of plne cn e written using the dot product: N (Q P ) = 0 where N is norml vector to the plne, P is fixed point on the plne nd Q is vrile point on the plne. We define the unit norml vector U = ' u ' v ' u ' v. Definition A surfce is orientle if it hs consistent norml vector defined everywhere on it. Single coordinte ptches re lwys orientle. Prolems might occur when we hve to use more thn one ptch: we need to e le to glue the normls from one ptch to the other. Essentilly, the surfce is orientle if, when we re moving on ny loop (simple closed curve), then the norml t the strting point is equl to the norml t the ending point. Exmple:: The Möius strip is not orientle. First ptch '(u 1,v 1 ) = 2 v 1 sin u 1 2 sin u 1, 2 v 1 sin u 1 2 cos u 1,v 1 cos u 1 2 where u 1 ]0, 2 [,v 1 ] 1, 1[. Secondptch (u 2,v 2 ) = 2 v 2 sin 4 + u 2 2 sin u 2, 2 v 2 sin 4 + u 2 2 cos u 2,v 2 cos 4 + u 2 2 where u 2 ]0, 2 [,v 2 ] 1, 1[. From now on we ssume tht our surfces re orientle. Definition The shpe opertor S for surfce M is defined s S p (V ) = V U t point p M where V U is the directionl derivtive of U in the direction of V. In other words, if U = (f,g,h) nd V = (,, c), then V U = ( V f, V g, V @y c@h.

9 Theorem (Lemm ) S p is liner trnsformtion from T p M to itself. Note tht if M is contined in plne, then U is constnt nd ll its derivtives re 0. We sy tht S is pth connected, if for ny two points p, q S, thereexistscurve [0, 1] S such tht (0) = p nd (1) = q. Conversely Theorem Assume M is pth connected. If S p = 0 t every point of M, then M is contined in plne. Exmple: The sphere of rdius R: '(u, v) = R(cos u cos v,sin u cos v,sin v). S p (V ) = V R for ll V The liner lger of surfces. The computtion of the shpe opertor doesn t lwys give us geometric informtion out the surfce, i.e., its shpe (see the exmple of the sddle, homework ). However, there re other things we cn do with the shpe opertor tht do give us informtion out the shpe of the surfce. Review few notions from liner lger: Mtrix of liner trnsformtion on sis. Eigenvlues nd eigenvectors. Mtrix on sis of eigenvectors nd digonliztion. Determinnt nd Trce in terms of eigenvlues. Definition We sy tht liner opertor T is symmetric if, for ny vectors v nd w, T (v) w = v T (w). Definition (reminder) A sis is orthonorml if it consists of orthogonl (perpendiculr) unit vectors. Exercise 2.3.4: With respect to n orthonorml sis, the mtrix of symmetric opertor is symmetric. Theorem (2.3.5) We hve S(' u ) ' u = ' uu U, S(' u ) ' v = ' uv U = S(' v ) ' u, S(' v ) ' v = ' vv U. In prticulr, the shpe opertor is symmetric. The shpe opertor S p contins informtion out the ccelertion/curvture of curves in M through p.

10 Lemm (2.4.1) For ny curve (t) in M, U = S( ). The sclr product U is the prt of the ccelertion due to the ending of M. The ove formul shows tht it only depends on the tngent vector to the curve. If the curve hs unit speed, it only depends on the tngent direction to the curve. If hs unit speed nd is n eigenvector of S p,then S p ( ) = nd = = S p ( ) = U. Definition Given unit vector u in the tngent spce T p M, the (norml) curvture of M in the direction of u is k(u) = S p (u) u. This is the curvture of the curve P M where P is the plne through p nd prllel to U(p) nd u (see elow nd in the ook). The (rel) eigenvlues k 1,k 2 of S p re the principl curvtures of M t p. The corresponding unit (orthogonl) eigenvectors u 1, u 2 re the principl directions. The determinnt K = k 1 k 2 is the Gussin curvture t p. Hlf the trce H = 1 2 (k 1 + k 2 ) is the men curvture t p. A point where k 1 = k 2 is umilic: S p = k 1 id nd every direction is principl. Note tht S, k 1,k 2,H chnge sign under chnge of orienttion ut not K. The norml curvture is the norml component of ccelertion: y the Frenet formuls, for unit speed curve k( ) = S p ( ) = U = pplen U = pple cos where is the ngle etween the surfce norml nd the curve norml (drw picture). Proposition (2.4.3) Let u e unit vector nd P the plne through p nd prllel to U(p) nd u. Let e the unit speed curve formed y P M with (0) = p. Then k(u) = ±pple (0).