1. Generl Informtion Mth 223, Fll 2010 Review Informtion for Finl Exm Time, dte nd plce of finl exm: Mondy, ecember 13, 10:30 AM 1:00 PM, Wescoe 4051 (the usul clssroom). Pln to rrive 15 minutes erly so tht we cn strt on time. Bring pens or pencils, scrtch pper, nd clcultor. Check the btteries beforehnd two students my not shre clcultor, nd I will not hve extrs vilble. The exm is closed-book nd closed-notes; however, list of formuls will be provided to you (see next pge). You don t need to bring bluebook. You re responsible for the following topics in ddition to those covered on the two in-clss tests: Conservtive vector fields, pth-independence, nd the criterion involving curl ( 6.3, pp. 390 396) Finding sclr potentil functions ( 6.3, pp. 396 397) Generl theory of prmetrized surfces: coordinte curves, tngent nd norml vectors, tngent plnes, nd surfce re ( 7.1, pp. 405 417) Specific surfces you should know how to prmetrize: plne, disk, sphere, cylinder, cone (throughout chpter 7) Sclr nd vector surfce over orientble surfces ( 7.2, p. 419 435) Stokes Theorem nd Guss Theorem ( 7.3, p.439 444); you don t hve to know the proofs, but should understnd the interprettions discussed in clss I will hold my usul office hours on Thursdy 12/9 (1:00 2:15) s well s extr office hours on Fridy 12/10: 10:00 12:00 nd 1:00 3:00. I will lso try to check my e-mil regulrly over the weekend (jmrtin@mth.ku.edu).
2. Formuls tht will be given to you on the exm Projection: For ll vectors,b È R n, Tringle inequlity: For ll vectors,b È R n, ³² ³± proj b b Å Ð bð ÐÐ ÐbÐ Conversion between rectngulr nd sphericl coordintes: x ρ sin φ cos θ y ρ sin φ sin θ z ρ cos φ ³² ³± ρ 2 x 2 y 2 z 2 tn φ x 2 y 2 ßz tn θ yßx Product nd Quotient Rules: If f : R n R nd g : R n R re differentible t È R n, then ÔfgÕÔÕ gôõfôõ fôõgôõ ÔfßgÕÔÕ gôõfôõ fôõgôõ gôõ 2 Ôprovided tht gôõ 0Õ Chin Rule: If g : R n R m nd f : R m R k re functions such tht g is differentible t È R n, nd f is differentible t gôõ È R m, then Ôf gõôõ fôgôõõ gôõ Curvture: κ dt Ðv Ð ds ÐvÐ 3 Torsion: Line integrls: x f ds Green s Theorem: Surfce integrls: X f ds b db ds τn; fôxôtõõ ÐxÔtÕÐ dt ú M dx N dy fôxôs,tõõ ÐNÔs,tÕÐ ds dt Ôv Õ ½ τ Ðv Ð 2 x F ds b ÔN x M y Õ dx dy X F ds FÔxÔtÕÕ xôtõ dt FÔXÔs,tÕÕ NÔs,tÕ ds dt
Stokes Theorem: ivergence Theorem: S ù Ô FÕ ds Ô FÕ dv ú S û B B F ds F ds Two useful integrls (I my supply other integrl formuls s needed): sin 2 u du u sin ucos u 2 C cos 2 u du u sin ucos u 2 C
3. Review Exercises For ech topic, I ve listed some exercises from the textbook tht might resemble exm problems. Some of these exercises hve lredy ppered in homework ssignments. The problems on the finl exm will not necessrily look like these review exercises! You should not necessrily do ll the problems listed (which you won t hve time for nywy). One strtegy is to pick one topic t time nd work on the review exercises until they strt to become boring (which is sign tht you re more fmilir with the topic in question). Chpter 1: Find prmetric description of line in R 3, given two points on it, or point nd direction vector, or two plnes tht both contin the line, etc. ( 1.2: #13 20) Find the distnce between point nd line, or between two skew lines in R 3, or between point nd plne in R 3, etc. ( 1.5: #20 28) Prove simple vector identities involving dot nd/or cross products, projections, nd norms: ( 1.3: #17 19; 1.4: #20; 1.6: #9 12) Chpter 2: Sketch the grph nd/or level curves of function f : X R 2 R ( 2.1: #10 19), or the level surfces of function f : X R 3 R ( 2.1: #28 32) Evlute limit of multivrible function, or prove tht it does not exist ( 2.2: #7 22, 28 33) Understnd wht it mens for function to be continuous t point in its domin ( 2.2: #34 42) Clculte the derivtive mtrix of function ( 2.3: #20 25) Find the tngent plne to the grph of function f : X R 2 R ( 2.2: #29 33) or to level surfce of function f : X R 3 R ( 2.6: #16 23) Understnd wht it mens for function to be of clss C k, nd wht tht implies bout its prtil derivtives ( 2.4: #9 18) Apply the Chin Rule in its mtrix form ( 2.5: #15 21) Clculte directionl derivtives nd the grdient, nd determine the direction of gretest increse or decrese of function ( 2.6: #2 15) Chpter 3: Sketch prmetrized curve in R 2 or R 3 ( 3.1: #1 6) etermine the tngent vector (or tngent line), ccelertion vector, nd speed of prmetrized curve t point ( 3.1: #7 10, 15 19) Set up n integrl for the rclength of prmetrized curve, nd evlute it if possible ( 3.2: #1 10) Sketch vector field in R 2 or R 3 ( 3.3: #1 12) Given vector field F, verify tht given prmetrized curve is flow line of F ( 3.3: #17 19) or set up (nd if possible solve) system of differentil equtions to find flow lines of F ( 3.3: #20 22) Clculte the divergence nd curl of vector field ( 3.4: #1 11) Prove identities involving divergence nd curl ( 3.4: Theorem 4.3 nd Theorem 4.4 (p. 218), nd #14,15,21 24) Chpter 6:
Set up nd evlute sclr or vector line integrl of function f : R 2 R long prmetrized curve in R 2 ( 6.1: #1 11, 13, 14, 17 21) Understnd wht hppens to line integrl when the curve is reprmetrized (in prticulr, know wht is ment by n orienttion of curve) ( 6.1: #15) Use Green s Theorem to convert line integrl to double integrl, or vice vers ( 6.2: #5 8,11) Use Green s Theorem to find the re of bounded region ( 6.2: #9,10,12 16) Use Green s Theorem to deduce fcts bout line integrls of conservtive vector fields ( 6.2: #19 21, 6.3: #1,2) etermine whether vector field is conservtive, nd if so find sclr potentil function ( 6.3: #3 16) Use the sclr potentil function of conservtive vector field to evlute line integrls ( 6.3: #17 24) Chpter 7: For given prmetrized surfce X : R 2 R 3, find coordinte curves, tngent vectors, norml vectors nd tngent plnes, nd express the underlying surfce s the grph of function f : R 2 R or s level surfce of function f : R 3 R ( 7.1: #1 4,6 11,14 17; 7.6: #1 8) Find the surfce re of prmetrized surfce ( 7.1: #18 21) Set up nd evlute sclr or vector surfce integrl long prmetrized surfce in R 3 ( 7.2: #1 3,5,6,9 23; 7.6: #9 12) Know wht hppens to surfce integrl when the surfce is reprmetrized ( 7.2: #4) Verify Stokes Theorem for given surfce nd vector field ( 7.3: #1 4) Verify the ivergence Theorem for given closed surfce nd vector field ( 7.3: #6 9 Use Stokes Theorem or the ivergence Theorem to replce difficult integrl with esier ones ( 7.3: #5,11,12,14,16) Other pplictions of Stokes Theorem nd the ivergence Theorem ( 7.6: #21,22,26,28,38) Understnd the geometric interprettions of Stokes Theorem nd the ivergence Theorem ( 7.3: #28)