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(1)Sequences and couwgence. Def. Kulu. C. µ. HE. Rd. is called. tu. F N EIN. O. I. Cauchysequencey. a. N. m. f. i. n. Itu. i. e c. sequence. Rd xulu. 9. E. V E V NEN. E. called. is. In. I LE. It. i. n A point. Xm. if. hbupo.it of the. acc. if N. Xu. I. x. E. e. E. E. i. A. sequence. t E. Notation. 70. Kulu. a. converges n. N. Lim Xu. x. F N. n soo. V. i_. to. X G. i. ka. Rd e. Xu. l. if C. E. x. iT.

(2) Observations. sequece. a or. no. the sequence has. it is not. nee. Gulu. If. to. and the sequence is. Xu. is. n. then. X. T. on. O It does converge to. Assume. that has x. EIR EU is. are. we a. on. Miu. u. ell. more. n. i. n. E. supremum. converge on. Jo. r. 0,1T. ou. general. on. a. space. C 112. be. a. salad. of. U. ut. X. ofU if it is. if. or. E X. is. max. Colt of have a wax. To n doesnot. of U if. upp i. o n. Let U. a. called an. x is called. or. Vue U. and. tf. IR. point. i. total ordering called. is. point. ace. is the only ace. x. but does not. Cauchy. Mae sup. one. Cauchy. Au. Xu. just. Cauchy sequence. a. converges. E. points. acc. point. ace. if. even. leave many. can. 5 of. is an upper bound o A. the smallest upper bound A is the sup of Jo AE.

(3) lower bound. min. Analogously. infinum. liminf.hu For. Xu. sequence. a. we. hi. a. himhf. 42. C. u. define. xml. insta. f. Un. lineup u. For. him. Xu. u Sao. w. upper. bowed. lower bound. u. c CR. l. c CR. s. m. n. Hulu. bounded sequence. a. em. sup. th. i.e. there exists. Ku c IN. xn E. Que wi. en z. accumulation point the lineup is the largest. liminf e. Ei. then. the. liminf. b ESO F N. mint. and. u. l. of Gulu. smallest. the laxest ye. is. K. u. N. i. en T. CR such. y. E. that. an a.

(4) Continuity Y. A function f X 4 d X d. Defy. HE. For. o. fool.IE. continuous. V. f. diesel. f. X. s. Xu u C. X. X. f. 74. is. y c. X. Intuition. d. atto. called. cE. faro. dextro. c. if. d. it is. defeat. Liprchitzcoutinuous. if. far fly. if. if. have. called continuous. is. L with Lipschitz constant. bx. dffas feed. coat. foul. to 4. we. eX. of. called. 4 is. X for every to C X HE o For o feet. A function. d. L. fi. Xu. A function. coutinuous atto. out. sequence. X. called. is. two metric spaces. between. Hee X. o. Allunahikou for ewy. ai. EL. bounded derivative. If. x. y. fade.

(5) f. A function. HE. o. X. For. V xoc X V.ec Xi du role of. o. a. to. I. can choose. d far freaks wot. i. fed Gina E. if. 4 is called uniformlycoutinuouse. s. to. 5 that. unit. it. 0. J to bethe. works for all to. Cannot choose. luhuih.eu bounded derivative. Intuition unbounded. same. for. all. to. derivative. rcout.fxtius. lmporh.at Intermediate value theorem. if. cont. f. values. Ca b between. Ky. C. IR. continuous. is. then. f. attains all. fca and forb F. feat f Cbl. e E Ca b. f Cbl y fray. I. a. a. b. feel. y.

(6) with. fca with fool. a. find. b. then. you want to. If. Application. there. monotone. strictly. f. is. Additional note A. if. f. in. of. with fee. Caio. f. R. b. frat. R. D c. O. continuous. Then. f Cbl. continuous as. well. monotonicity. follows from. Continuity. function. x c. invertible and the inverse is Invertible. important. 0. D C a e. feet 0. with. e. 0. a. must airt. Inwhiblefunctions. find. the inwa follows directly from couh. off. forgot the following. the video I. of continuous functions between two metric spaces X all 14. characterization. f. if pre images of. and only B. c. 4 open. t. in 4. f. n. open sets B. are. is continuous. d. open. feet Ifni. c. B. open. t. in X.

(7) Sequences. Det Courider. fu. functions. We say that. of functions. the. Dc R. IR. D. fulue µ. convergespointwiuto if f R. D. b. sequence. x c. fax. face. D. fuel. Yu Yu. y. R. n. fu f i fo. Examplem. feel. y. fue. 1 feel Fn does. Det Kulu HE. n. otherwise. all. not imply that f to. o. O. x. pointwise. f. F N. E. 0. 0. f. IN V n. for continuous is continuous. uniformly N. this. V. if I fuel. fait. CE.

(8) i. I. if. c. such that all. with u N contained in. fu. are. tube. E. doa to 0 there will always be points x. i. not. yet in E tube Not uniformly cow. the fu f µ. D. f uniformly iff. fu. Alhuatiredefinite. Uniform. 7. o. convergence preserves. IR. yun µm. Dc. R. y.µ. continuity. all fee. f. fu fu. are. continuous. uµuu. o.

(9) Derivatives h. Def. IR. C. au. f. interval. diasleata. is called. f. c. h. we often write. R. feat. DI. f. de. t. ath. a. a. au. l function. for all. the function. a. thin. f. U. R. ha Uu. f. I. if it. n. f. cat. 2. O. is differentiable. continuously a. O. ath. a. f is called differential C U It is i. f. exists. h. so. i. the. the function. if. U. f Cath. him. ca. U. i. if. it is. is continuous. diff and.

(10) Highway we. can. derivatives repeat the process of taking. DI. f. de. Notation. n. f. DI. f. i. de. the. denotes. u. if. th derivative. aisha. Important theorem Let. f. Differentiable implies continuous. be differentiable at. such that. small ball around. a. on. f. f. e. E. l. a. fine a. theorem. a. constant ca. have. al a. for derivatives. that i.e functions on T.a.sc. b. Then there exist 3 E b. value. we. a. le. ca. is continuous at. Intermediate. theorem. t. fail. I feel In particular. Then there exists. a. we once. Cour differentiable. Ca. b. such that. fees. e. ca. 1.

(11) theorem fn. I. E. S. fu. him. u soo. counge. fu. R. Emb. i. fees s. Exchanging. face. uniformly. him and derivative E Ca b. C. exists. for. if. then. all. ee. linfu. ca. f. him. f. Ca b. and the derivatives. is coat differentiable. man. fl. the limit. If. ca. Cx. uniform coat is really important. rt Lfirst. and. motta. S we compute. compute. its. derivative. all. ful. then take limit these. of. derivatives. otherwise would be wrong.

(12) RiemanuintegralI Consider. a. f. function. a. that f is bounded Fl a c IR ke C Ca b Consider to a. er. In. l E. C. tu. n. feel. assume. Eu. with. xn. ko C en C Xz. R. b. Xu. b. let Nfl. en. 111. T. b. mm. Define mu. inf. f. Mu. sup. f. µ. exists because lower. Define the s. and the. sum. er. In. is bounded length of. n. f leo. f. In. en. E. her. Ik. M. l In t.mn. upper sum. Sff. fur. xu. Ital Mh. th. th n.

(13) Now define. I. i. sup. partitions. s. f partition. TT5IIhI.y. Jtieinfnn.u.lSffirarhiiouDf. we call. is anons. f. Rie. it. mean. tf. finer. is. ffHdtiy. ytYIghE i. i.

(14) Shortcomings. For eeauple. ugh integrable. are. Many feuchbus i. n. f't. e. IQ. otherwise. o. o. a. a. or. Beanie. Q. e. o. rational. number. a. Rid For any. interval. µn Mh. Thus. y. e. th 0. i. y. lb al. lb al O. with. liar. Hard to extend to. t. v. l. One cannot prove. In Ctu then. theorems. about. limffudt h soo. other spaces. n. exchanging. integral. flimfadt.

(15) Fundamenhaltheoreinulus TheoremI. IR. fi Ca b. i. Riemann. Let. continuous at 3 E Ca b. and. Then the function. Ca b. X. or. T. F CH. EE. c. is differentiable. E. Fant. c c. integrable. I. at 3 and. Cais. feel for all. fl Ce. Theorem. f Ctl Dt. F. i. aib. then. F. 3. F. e. GD. e. IR continuously differentiable Fca. F b. or. Feel. Ate. T. C. tayman The integral. is. operator. 9.1 Caio. with an. inverse. and. e e Ca b. b. F'Ctl dt. Edifmeuthiasle. isomorphism. E. C Carb. I. fc E. Ca b. linear. is the differential. X. fcc. bijective. operator. Cain O. and its. Iheu.

(16) Proof I. Need to prove. i. A. Consider. F. h. that F Sth. diff at 3. is. f 5. h. fetid. fetid. f 5th. Want to pwn. far df. In. as. h. S. courage. f I. Iu. II. f Formally. aid. fit t'idt. eTia. Sth. 5. Yadier which we. hwan. fan. yes df. Intuition small. given E. fat fat Then. if. Human. E. 0. we. due to continuity of can. K t a. find h 5,3 4. g. geongottderd. o. ft. Chl. f. O. Want to pwn A. to. 0. fat. 5. such that. fol w. mm intral.

(17) f. that. fat. E. Edt. f. fail. I felt. f. E. that. E. I. f. at E. h. E DM. P. theoremI. I know that. by theoremI. thus. F continuous. F CH dt. Glx. the function. and. differentiable. is. a. Ceil. Consider ii. By Hence. we. by def of. 0. Gea. Cil. F. G ca. x. know that. we. It is. a. know that. by theorem Il. Ca b. ou. Girl. Fil. It Cx. constant function It. Consider. Fca. x. Fla. Gca. Ffa. a. Fca. Has. G'cxl 0 for all. F ai. H ai. im. 0. Ciii. G. means. i. constant. b Heb. If. f Cbl. Gcb. b. Fal. f. a. F CH at. I't. this.

(18) b. F'Cfldt a. F. Cbl. Fla. Dan. Th I.

(19) Powerseries D. Def. the form. A series of called. E. au X. n. is. n 0. power series. a. Radius. Theorem. of. convergence o. For every power series constant. poet. in. E. pox. neo. there exists. ane. a. called the. O E r E. radiusofcouvergencesuch r. that. absolutely. The series counges. 1 1. L. r. that the. meaning. as. sequence. Io. E. h O. of partial. au 1 1. can. radius. 1. of. be computed r. r. converges. sums. in the usual. converges. It is unclear what happens. The. 1 1. with. seven. a. N. If 1. an. N. e. pN. that. meaning. for all. L r. the series. coungence. various. f. where. 1. r. converges. only depends. by. tie I. even. for 1. on. uniformly. the Cantu and. formulas L. ling p. I. lanl. if exists.

(20) E. n M. E. pal. x. n. in. 0. n. limII. lima. u.in. I. r. c. some constant c. for. An. an. Case. n. c. lim. c. nIa. ha. x. 1 radius. cour. r. 1. no. general. eine For. a. For. t. 0. i. both. it. 1. s. it diverges. 71. E. Esp Ext. e. E. for 1 1. r. er. D. n n. e. has. n. has. n. neo. r. an. t.nu. beam. neo. x. diners. is. E. E. E. x. n. and. A. x. converges. r. 0. a. IITi Au. lau.nl. sa. un. µ. n. In. o.

(21) From. power. I. fan. power series. Given. Obarvation. to Taylor. series. Let's take its derivative. f. C. ca. al. x. ao e ar. t. an D. E. n. f. n. i. an. u. ex a. series. Io. azce. a. Zaz Ct. a. au. x. a. tag Ce t. al't. I. Ia Ce al t. n. cel i i. f. k. p. E. cel n. an. k. Cu. n. al Ca 4. n. k int. n. x. k. a. omiaaiI M. Theorin Ix al. e r. we. feel Intuition we. E anti. Let feel. a. n. us 0. with. 0. r. then. for. with. have. f. E n. 0. start with. a. r. Cai. I power. VI x. a. series that converges. Then. the coeff in terms of that upresses hail the heat formula. derivation.

(22) Question. Does. it work the. given any function. i. possibly. L. That is way round with nice assumptions. other. m. ai. fax.

(23) Tayherseries. s.TCIRopeuiuhrvalifiT.sk f Theoreur e. E. Cais. T. e e. a. Define. i. Then. Proof. feel Tn ca. need to. i. fee Inductive ship Consider. kn Ce. a. Fundamental theorem. follows from. B. t. al. f. ca. f. t. by. inductile. on n. prove. Ctl. E Fandom. Dt. theorem. or. u runner. F CH. tt n. int. f. finer. t. Take its derivative theorem fundamental and uploit Integrate Date.

(24) with Lagrange remainder Taylor. Theorem. f. e. i i 1001. airts. there. Then. J. a e E. J. e. i. some. i. Let 3 Ca b G E two functions F. Consider. e. Ca 43. Assume. bJ. tha Now. Ct. O. m. Gca. G Cbl Assume. F b. F C. G b. G. Fla. F b. in. S. E. F Cbl. g. a Cbl. g. f. We. ca ca. in. 3 for some. t_. f. F G Celie. f. can. iterate. 0. We would obtain. For all k. some 5eca.bg. 3. that F and G also satisfy ED. Now chose. for. x. OE k En. Tut 0. a. and. E. Ca b. Ru Ce. a. a. we. have so we. by. construction Hut. in particular. have. Glk. a. O.

(25) For. ne n. f. f. ce. E. By Fai. Race. f. them. u. then. al. For x. example C. 1 f. 0 cu. this. Cf. is. 51 DMD. I. E. Tuen. such. I. x a c. fax. Gentil g. f. I. Refine. f. n O. soo. we have. urn. ce. F. ace. htt. T. lim. T Celis. E. a. Ken e. G. ex. obtain. we. x. e. have. now. we. Tai. if. the. case. n. ca. I. Ca al. ku Ce al. if. there. exist constants. that E. x. C. KEE T. the Lagrangian remainder from Follows directly. o. Ku c IN.

(26) E. es. Exponential series. apt. u. E. x. trio. pour series. with. p. coincides with its Taylor eep always series. n. slope f cat. anti. t. ftp. series. fr. e a. f. fair feat. fax. Convergence. can prove. outside. For make. sense. feel Has the Consider. at. exp. of J. r. is. rep. does not. all. Ale 2. to. it. e o. o. funny. Taylor sein. 1C. t. o. a. of Taylor series. radius. al. Y. Taylor series about. Att. Log. x. ca. property. the Taylor. that. series. knew. f. cu. derived about. a. o. O. O.

(27) All tours will be 0. but. b. x. of t. cause. 0. f. so. Ku. is not 50. Tu self. ful. O. Tu Ce so. we. get. r. e.

(28) G. AlgebratffEFEEt a. Tik. vol. luwuals. eh then. Xuan Xu. affair 2h Def X non. F. c. is. F of. Det a. G. closed. F. are. over. called. of X. such. a. that. F. e. counhable curious G. X. is. taking complements. under. measwa blespace. algebra F. sets in. subsets. XIA. F. Aa Az. A. stage. A. is closed under. A ii. collection. empty. F. Cil. empty set. non. rollin f. D. U. F. consist. i n. of. Notation bLe. a. Ai. E. F. set X and X. F. the. mid.

(29) Imporhautebras Trival. o. useless. 0. we. can. define two. pretty. X. Act. PK. F lef. Given k the. X. 6 algebras. Fa e. Gira. algebras. g. subsets be any collection of. G. o. such that. G. c. can. be proved easily. construction or. g. N X d. E. f. g. Notation. F. Existence. Newark. is the smallest r algebra. E is. a o. by. an. explicit. Algebra that containsG. let 5 be the collection of and metric Coincides a space is defined r Then the Borel algebrae subsets of X open as. 5. g.

(30) Measuresy Given. Def is a. F. map p. i. 01. p. Cii. measurable space. a. e. o. F. X. a. measure. such that. a. O. countable collection of disjoint with Si E F we have subsets Si lien For. a. Niven i. measure. a. si. w. function on. F. not. Exams Disenhureasur X. x. Given. i. in ez. the set of all. milien. subsets. finite. I. Example. mi. µ. F. space algebra F to be. Iz IR. E. is. mi. finite. Proceed as. define. F you. For all other strs due to countability measure. A. X. fr countable. mi. Xi. on. define the r of X Gurieli a sequence. such that. Want to define. µ. a. si. p. follows. you. can. by. now. all. elementary sets. deduce. the.

(31) E. A. µ. ai C A. E. 3. pi't. mi. X cA. A funny measure on IR Example Want to define r Habra on R Borel the be f Lef numbers rational to mass that assigns measure just be all rational numbers Let q Ok Couaidu. mi. aid. µ. c µ. mi. Fz. mi. E. Csis. µ. before. as. aie Cai's. IR. plain. sum up on. E. mi. ai cCais. Q. i. iif. all the. rational. interval. points in the infinitely man points. Exauple X. A. i. IR. a. more. useful class. F Bord. o. of. measures. Let. algebra. be monotonically increasing. th a. Fi. 112. on. 3112. continuous. b. IR.

(32) a measure. Define. S. re. ou. IR F. Ffa. SC. yup. Fcb. inf. by. setting. Taj. i. R. b. s. w. w. w. L. Fast Flan. by. Cover 5. intervals. To each interval. we. elementary volume F Cbl. assign. Fay. Take best cowing. this is. Need to prove. Def. is called. A. subat. We. say. a. X F. measure space. N E. that. F is called a. it holds for all in probability. property. a. holds. we. say. a. measure. pl. nail if µ N O HU admosthere if. xeX except. theory. endowed with. F. X. A measurable space. measure. a. for. in. a. null set N. l. µ.

(33) 2. igu.eu want to form. Can b. volume. t. given. az. by. Cz. First. approaches. Want that rectangles of the. measure on 112. construct a. belt. II. x. au. have the. but. natural. bi ai. vol CRI. Jordan Riemann. E ang Cz. attempted the following i. in. i. a. Innerapproximation t. i. A would be called. measurable. inner approximation. courage. A. C. if. outer and.

(34) Now. generalization. this approach. of. Allow for counhable coverings Replace inner approximation. of. by. me complement. ouhr approx. ace. DODD DOE D. 499 outer approx. E of. ELAM LEIA El. ICAI Need a algebra as. underlying. µ. p. A. pl E. E. Al. structure. Outrlebesguemeasureset rectangles natural volume. the. an.br. R. IIa. IRI. Ac. a. A is. Car.bz. Can by. t. C 42. bi ai. outer Lebesgue measure. Definition of Let. t. of. 112. We. be arbitrary. inf. En. I Ri. I. I. define A C. YI. Ri. Ri. rectangle.

(35) by. We cow A. a. I t.tl. observe. countable union. R. E. Want to make this into. PCR Need to. for. t. algebra. we run. E. a. E. a. if. Problem. measure. a. we say that. inf. we. use. into contradictions a. smaller or algebra. set A. c 112. is measurably. c 112. ALENA. ACETATE a. tha oh. Ken hale. rectangles. fo. v. unknit ourselves to. Beth if. r. as. of. by L all. measurable subsets. ae4. n. i i T. nm. i. Example i. E3. O. X. IR. a. E. of R. e me.

(36) A. Ql is. measurable. I E. define for all ai. For E 70. the interval. ti. CA. such that. yi. Fi in. Ai. ki. 0. Q. and has 1. Proofshet. In particular. 1 A1 0. The. countable. air. E. t. ai. Yi. EI in. zu in. R ti. U C DA i. Yi. ii tic. n. a. Eric il. h. E. E. i ca. i n. E. ion. Taking the. X C Al. inf. E. je ah. ow. wings. Hh shows. that. O. Geupmiag.ofebugwmeaswabenlwiktheBorl. alge8reD r. 18c2io.o intervals pea any. of. open. are. afFin. intervals open. measurable 112. can. A. thus in. be written c. 00. b isn. Ii. L as. a. countable union. Ii. open intervals.

(37) L. Him exist For every Lebesgue measurable set L A N O such that E L with N and B E B a set. L Summary. B. u. N. L. E. B. up to. sets. of. measure. 0.

(38) Consider on. O. 0,1C. Define. l. x. y. I. II. t. Q. y. E. Iz. au. I. Iff. t. would be equivalent. the equivalence classes. I. t. Q. f. t. Q. EE. we pick. equivalence relation. an. as follows. x. consider. set. measurable. Anon. a. Iq c. a. Q. Q. t. representative. f't. of. each. of the. and denote. classes. byNtheretofallsacGrepresentati N is not Lebesgue measurable Want to prove. 1. Intuition irrational. 1 o. nx.i. e.e. x. x B. x.ee. V2 2. blacl points. blue. crosses. rz x t. Q. Q. r.

(39) Proof. by contradiction. i. We. N is measurable. Assume. whi. following. Na. N. gt. now. Ce 1C. For g. e. v. g n. n. N. 9. na. g. rat. N is measurable. 1. U. 1C. o. measurable g t N is. Meu i. Nq. 9. irrational. 91 N. q n 1N. and. arc. E. 7. if. construct the. kg C Co if. N. Ng. q CCannon. Ng. n. Np. consequently o. f Of. U Ng. Np Ng is. disjoint. additivity. ily. t. a. 9. IT.

(40) Could be that. 1 Ng. Could be that t Cng. I. ly. O. actual. 1. But then. 0. Nq as. 0. But then.

(41) thelebesgueinkgralou.IR Intuition Riemann. then. Lebesgue. t. too. f. n. G. Je. between two measurable CY F X function A Defy f called measurably if pre images of measurable sets is spaces. g. are. measurable i. f a. e. g. i. f. G ee. E. X. F. l feel c. G.

(42) R is called a IR IO Det Sc c 42 there exist measurable sets. I. Si. For such. a. 4. I. is n. ai t. n. such. that. as. Lebesgue integral. Csi 1 E. Co. IR. ft. function. define its. can. function we. Q d 1. a. cR. ai. in. For. ai. Assi. ai. ion. siuplefuuch.ie. we. define. its. lo Ef. a simple. Lebesgue integral. ft del might be For. a. into when. and. fed. f. di. o. general function positive. fol. sup. n. IR. f. neg. s 112. park. f. we. fax. if fees. o. our win. split. ft. the function. f. 20. ft Wsh. ft f. ar. measurable. if f is. measurable.

(43) if. ft. both. then. we. call. and. f. If. more. At. di. old. a. ff. A. Twoiuportan.tn monotone. Theorem. that is pointwin i. fun CH. lim face. ff. c. e. thing. de. fuori de. c. di. inheral. t CQ. e. O. eeus. functions. of. 112. o. decreasing i. non. I. fu. fue't. i. Eii. it feel. Then. fax. di. convergence. Couridu a sequence. b xi. di. than Riemann powerful notion. Example. k ra e IR. If. and define. integrable. f di Much. satisfy. f. lim. k soo. f fue. de.

(44) f. ie. B. integrable. fee. twice s. R. I. face I. E. ga. Assume that the pointwise. liar u. 5. fax de. facet him U SD. Then. fue de. ou. limit. B aids. g CH. is. Axe B.

(45) Pantialderivative Consider. IR. is called. partially differentiable 112 if point 3 C. variable Xj x. g. f. IR. f. Det f. IR. g. at x. i. f. 4,32. a. Tj. IR 3112. Notation. at 3. er. If. Lim. J xj. h so. f C3. fi R2. n. titter R. with resp to the function. Kj th. is diperentiable. feet. iej.hr. Sj. i. r. Sn. rector. f. c. 3. h. derivatim exist then the rector of all partial if called the gradient partial derivatives is au. grad CHCH. Tfc. Ijf. C R. i.

(46) Rm. If f IR. we. decompose. Jacobian matrix. in. We define the. ftL. f. component functions. its. into. f. iii. Even. if. all partial derivatives exist. know whether. f. is continuous at 5. Exampley. f. R. I. fogy. at 5. we. do not. IR. it. yr. if. o. Caylt e. y. Coco. o. For Cecil f fo o. grad. fans. gradflo 01. ly 0. because. it fee o f Co yl. but. f. is not continuous at 0. ya O. ke. 0. Ky.

(47) Totalderivative f f. IR. IR. 9. at 3 if. dig. is. such. that for h. E. 5th. fcs. f. theorem. f. L h. f then. f. is. 7112. IR. µy. R. h. Intuition. Rm. h. tr. Chl. him. with. exists. there. IR. linear mapping L. a. i. U. E. I. so. h. O. 1. locally linear at. differentiable. 3. is continuous at. The linear functional. f Geht fest. L. coincides. II. II. with the gradient. hj. 151. grad f Gl h. f f for. lR. IR. fur. it is. are differentiable. L h. e. differentiable. if. all. their all partial. Jacobi matrix. h. h. r. e. r. i. h. coordinate luncheons derivatives. aid and.

(48) and If all partial derivatives exist. theorem. then. f. are. all continuous. is differentiable. derivatives exist but are not continuous partial If need to be differentiable. then. f. doesn't.

(49) Directional derivatives R. u. Det. Assume. fi. R. is. cont differentiable. The directional derivative. is. theorem. defined. fi. er. as. f. 31. 112. R. directional derivatives. eye. t.tv. fist. lim. with Urkel. R. at 3 in direction of. f. of. v e. direction. fol. t. t so. differentiable in 3. Then all the. and. compute them de. aid. we can. npartial. rfls.ie. gradflt.y. vi. II. Csl. tint the. largest value. directional of all. attained in direction. grad f Gl. v 11. grad. f. 5111. derivatives is. by.

(50) Higherorderderivat IR. IR. all partial. derivatives. Courier so. f. is differentiable. assume. 7. can. we. it is. hate its derivative. a. These are called second order. 7. f. i2x Example. we. partial. 2xj. of. derivatives. ki. XIN2. fax. X't y. i. grade. Have. derivatives. cannot change the order. t. If this. IR. IR. Iii In general. differentiable. IIe. O y. I. y. I. for all y. function.

(51) If. Ct. J. 01. k all. 0. x. I Defy. if. R. R. f. we say that. we say that. is. f. derivatives. Schwartz. Then. are. continuous. if. differentiable. all partial. again continuously. or. 22. Notation. partial. and. twice continuously. differentiable. bcouh.dimhaole. Analogously.sk. differentiable. 2ti. exist. continuously differentiable and. f is is. theorem. Zf. all partial derivatives. continuously differentiable. is. Ek. IR. Rm. e. IR. Rm. Assume that. f. f. IR. or. ft. R. R. is twice. exchange the order. we can. I l. k times wah. dik a. other coat. diff. continuously. in which. we. take. derivatives. If. z2f 2ti. Analogously. 2x. 2x. k times coat. 2x. diff. can exchange order. k. partial. of first. derivatives.

(52) Caution. f. dimensions. IR. 112. Pf. IR. Itf. R. function. first derivative. 112. a. partial deriv. If. ati. second derivative. 112. n. i. partial derivatives 2f. Fei. Bet. Hessian. f. IR. Itf. 2tj. matrix 112. then. cry. we. define the. 2xi2xj. Hessian. CH. at point. off i. j. 1. u. by.

(53) Minimalmatin Def. f. then. f. IR. IR call. we. has. If Of exi. differentiable a. criticalpoi. local minimum at. a. V. such that. c. x. Be. there. if. xo. f. i. exists. local min. f. He. f. has. c. Be. m. If. f. a. choral. diff. and. win. t. c. resp. BeCeo. to. 16cal. in a. innit. o. h. ah to. feel. Xo i. a. He. is. iuimum. stria.tw. a. E. fool. I. x. to has. so. foto. strict local wax. foxtel. feed. critical point that is neither we. saddUpoint. call it. De.

(54) f. has. global minimum at Xo. a. Vx. if. Vocal. 5. fat. free. minimum. global minimum. How. we. can. see. which. Whuition. type. we. have. IR. in. n. u local win f. f cry. f. theorem it If. is. ko. Hf ai. i. critical point. a. If. exo. a. then. f. e. E. Pf. is. positive semi positive. strict local min do in. a. ex. IR. definite. wax. saddle point. ca. that eo. O O. is. then. maximum. definite. f cat. f. co. Assume. 0. to. local minimum. Hfceo is. is a. e. f. so. R. IR. f'also. O. Ct. t.li. t. then the Hessian. neg semi def. neg definite. If. Hf Cr. then is. to. indefinite.

(55) Matrix vector derivatives linear least Mum. Example. f. IR. IR. fincool. Ily e. Awh. of. model. Willi para. Want to minimize. parameters. e. we. want to find. W. Need. few. tf. to look at. 42. 112. t. d write functieu. ft. a. predicted output. weight vector. pndicham. q. y. input data. trueoutputvalues. ho good is. Y. explicitly. II. L. IT. It. Aw. I5. E. f g it 24. Einainwn Aw. u. 2. E j. n. aji. y. fat. y. Au Aw. i.

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