18. The Jacobian Formula - Probability Tutorials

1 Tutorial 18: The Jacobian Formula118. The Jacobian FormulaIn the following, 125We callK-normed space,anorderedpair(E, N),whereEis aK-vector space, andN:E R+is a norm definition (89)forvector space, and definition (95) , be an inner-product on aK-vector Show that = , is a norm Show that (H, )isaK-normed (E, )beaK-normed space:1. Show thatd(x, y)= x y defines a metric Show that for allx, y E,wehave| x y | x y . 18: The Jacobian Formula2 Definition 126 Let(E, )be aK-normed space, anddbe themetric defined byd(x, y)= x y .Wecallnorm topologyonE,denotedT , the topology onEassociated that this definition is consistent with definition (82) of the normtopology associated with an , Fbe twoK-normed spaces, andl:E Fbe alinear map. Show that the following are equivalent:(i)lis continuous ( to the norm topologies)(ii)lis continuous atx=0.

2 (iii) K R+, x E, l(x) K x (iv)sup{ l(x) :x E, x =1}<+ Definition 127 LetE,FbeK-normed spaces. TheK-vector spaceof allcontinuous linear mapsl:E Fis denotedLK(E, F). 18: The Jacobian Formula3 Exercise thatLK(E, F) is indeed aK-vector , FbeK-normed spaces. Givenl LK(E, F),let: l =sup{ l(x) :x E, x =1}<+ 1. Show that: l =sup{ l(x) :x E, x 1}2. Show that: l =sup l(x) x :x E, x =0 3. Show that l(x) l . x , for allx Show that l is the smallestK R+, such that: x E, l(x) K x 5. Show thatl l is a norm onLK(E, F). 18: The Jacobian Formula46. Show that (LK(E, F), )isaK-normed 128 LetE, FbeR-normed spaces andUbe an opensubset ofE. We say that a map :U Fisdifferentiableatsomea U,ifandonlyifthereexistsl LR(E, F)such that, for all >0,thereexists >0, such that for allh E: h a+h Uand (a+h) (a) l(h) h Exercise , Fbe twoR-normed spaces, andUbe open :U Fbe a map anda Suppose that :U Fis differentiable ata U,andthatl1,l2 LR(E, F) satisfy the requirement of definition (128).

3 Show that for all >0, there exists >0 such that: h E, h l1(h) l2(h) h 18: The Jacobian Formula52. Conclude that l1 l2 = 0 and finally thatl1= 129 LetE, FbeR-normed spaces andUbe an opensubset :U Fbe a map anda is differentiableata,wecalldifferentialof ata, the unique element ofLR(E, F),denotedd (a), satisfying the requirement of definition(128).If isdifferentiable at all points ofU,themapd :U LR(E, F)is alsocalled the differential of .Definition 130 LetE, FbeR-normed spaces andUbe an opensubset :U Fis said to be ofclassC1,ifandonlyifd (a)exists for alla U, and the differentiald :U LR(E, F)is a continuous , Fbe twoR-normed spaces andUbe open :U Fbe a map, anda 18: The Jacobian Formula61. Show that differentiable ata continuous If is of classC1, explain with respect to which topologies thedifferentiald :U LR(E, F) is said to be Show that if is of classC1,then is Suppose thatE=R.

4 Show that for alla U, is differentiableata U, if and only if the derivative: (a) = limt =0,t 0 (a+t) (a)texists inF,inwhichcased (a) LR(R,F)isgivenby: t R,d (a)(t)=t. (a)In particular, (a)=d (a)(1).Exercise , F, Gbe threeR-normed spaces. LetUbe openinEandVbe open :U Fand :V Gbe two 18: The Jacobian Formula7such that (U) V. We assume that is differentiable ata U,and we put:l1 =d (a) LR(E, F)We assume that is differentiable at (a) V, and we put:l2 =d ( (a)) LR(F, G)1. Explain why :U Gis a well-defined Given >0, show the existence of >0 such that: ( + l1 + l2 ) 3. Show the existence of 2>0 such that for allh2 Fwith h2 2,wehave (a)+h2 Vand: ( (a)+h2) (a) l2(h2) h2 4. Show that ifh2 Fand h2 2, then for allh E,wehave: ( (a)+h2) (a) l2 l1(h) h2 + l2 . h2 l1(h) 18: The Jacobian Formula85. Show the existence of >0 such that for allh Ewith h ,we havea+h Uand (a+h) (a) l1(h) h , togetherwith (a+h) (a) Show that ifh Eis such that h ,thena+h Uand: (a+h) (a) l2 l1(h) (a+h) (a) + l2.

5 H ( + l1 + l2 ) h h 7. Show thatl2 l1 LR(E, G)8. Conclude with the 18: The Jacobian Formula9 Theorem 110 LetE, F, Gbe threeR-normed spaces,Ube open inEandVbe open :U Fand :V Gbe two mapssuch that (U) ,if is differentiable ata U,and is differentiable at (a) V,then is differentiable ata U, and furthermore:d( )(a)=d ( (a)) d (a)Exercise ( ,T )and( ,T) be two topological spaces, andA P( ) be a set of subsets of generating the topologyT, thatT=T(A) as defined in (55). Letf: beamap,and define:U ={A :f 1(A) T }1. Show thatUis a topology on .2. Show thatf:( ,T ) ( ,T) is continuous, if and only if: A A,f 1(A) T 18: The Jacobian Formula10 Exercise ( ,T ) be a topological space, and ( i,Ti)i Ibea family of topological spaces, indexed by a non-empty be the Cartesian product = i I iandT= i ITibe the producttopology on . Let (fi)i Ibe a family of mapsfi: iindexedbyI,andletf: be the map defined byf( )=(fi( ))i Iforall.

6 Letpi: ibe the canonical projection Show thatpi:( ,T) ( i,Ti) is continuous for alli Show thatf:( ,T ) ( ,T) is continuous, if and only ifeach coordinate mappingfi:( ,T ) ( i,Ti) is , F, Gbe threeR-normed spaces. LetUbe openinEandVbe open :U Fand :V Gbe twomaps of classC1such that (U) For all (l1,l2) LR(F, G) LR(E, F), we define:N1(l1,l2) = l1 + l2 18: The Jacobian Formula11N2(l1,l2) = l1 2+ l2 2N (l1,l2) =max( l1 , l2 )Show thatN1,N2,N are all norms onLR(F, G) LR(E, F).2. Show they induce the product topology onLR(F, G) LR(E, F).3. We define the mapH:LR(F, G) LR(E, F) LR(E, G)by: (l1,l2) LR(F, G) LR(E, F),H(l1,l2) =l1 l2 Show that H(l1,l2) l1 . l2 , for alll1, Show thatHis We defineK:U LR(F, G) LR(E, F)by: a U,K(a) =(d ( (a)),d (a))Show thatKis 18: The Jacobian Formula126. Show that is differentiable Show thatd( )=H Conclude with the following:Theorem 111 LetE, F, Gbe threeR-normed spaces, U be open inEandVbe open :U Fand :V Gbe two mapsof classC1such that (U) , :U Gis of anR-normed space.

7 Leta, b R,a< :[a, b] Eandg:[a, b] Rbe two continuous maps whichare differentiable at every point of ]a, b[. We assume that: t ]a, b[, f (t) g (t)1. Given >0, we define :[a, b] Rby: (t) = f(t) f(a) g(t)+g(a) (t a) 18: The Jacobian Formula13for allt [a, b]. Show that is DefineE ={t [a, b]: (t) },andc=supE . Show that:c [a, b]and (c) 3. Show the existence ofh>0, such that: t [a, a+h[ [a, b], (t) 4. Show thatc ]a, b].5. Suppose thatc ]a, b[. Show the existence oft0 ]c, b] such that: f(t0) f(c)t0 c f (c) + /2andg (c) g(t0) g(c)t0 c+ /26. Show that f(t0) f(c) g(t0) g(c)+ (t0 c).7. Show that f(c) f(a) g(c) g(a)+ (c a)+ .8. Show that f(t0) f(a) g(t0) g(a)+ (t0 a)+ . 18: The Jacobian Formula149. Show thatc ]a, b[ leads to a Show that f(b) f(a) g(b) g(a)+ (b a)+ .11. Conclude with the following:Theorem 112 LetEbe anR-normed space. Leta, b R,a< :[a, b] Eandg:[a, b] Rbe two continuous maps whichare differentiable at every point of]a, b[, and such that: t ]a, b[, f (t) g (t)Then: f(b) f(a) g(b) g(a) 18: The Jacobian Formula15 Definition 131 Letn 1andUbe open :U Ebe a map, whereEis anR-normed space.

8 For alli=1,..,n,wesay that has anithpartial derivativeata U,ifandonlyifthelimit: xi(a) = limh =0,h 0 (a+hei) (a)hexists inE,where(e1,..,en)is the canonical basis 1andUbe open :U Ebe amap, whereEis anR-normed Suppose is differentiable ata U. Show that for alli Nn:limh =0,h 01 hei (a+hei) (a) d (a)(hei) =02. Show that for alli Nn, xi(a) exists, and: xi(a)=d (a)(ei) 18: The Jacobian Formula163. Conclude with the following:Theorem 113 Letn 1andUbe open :U Ebea map, whereEis anR-normed space. Then, if is differentiable ata U, for alli=1,..,n, xi(a)exists and we have: h =(h1,..,hn) Rn,d (a)(h)=n i=1 xi(a)hiExercise 1andUbe open :U Ebe amap, whereEis anR-normed Show that if is differentiable ata, b U, then for alli Nn: xi(b) xi(a) d (b) d (a) 18: The Jacobian Formula172. Conclude that if is of classC1onU,then xiexists and iscontinuous onU, for alli 1andUbe open :U Ebe amap, whereEis anR-normed space.

9 We assume that xiexists onU, and is continuous ata U, for alli Nn. We definel:Rn E: h =(h1,..,hn) Rn,l(h) =n i=1 xi(a)hi1. Show thatl LR(Rn,E).2. Given >0, show the existence of >0 such that for allh Rnwith h < ,wehavea+h U, and: i=1,..,n , xi(a+h) xi(a) 18: The Jacobian Formula183. Leth=(h1,..,hn) Rnbe such that h < .(e1,..,en)being the canonical basis ofRn, we definek0=aand fori Nn:ki =a+i j=1hjejShow thatk0,..,kn U, and that we have: (a+h) (a) l(h)=n i=1 (ki 1+hiei) (ki 1) hi xi(a) 4. Leti Nn. Assume thathi>0. We definefi:[0,hi] Eby: t [0,hi],fi(t) = (ki 1+tei) (ki 1) t xi(a)Showfiis well-defined,f i(t) exists for allt [0,hi], and: t [0,hi],f i(t)= xi(ki 1+tei) xi(a) 18: The Jacobian Formula195. Showfiis continuous on [0,hi], differentiable on ]0,hi[, with: t ]0,hi[, f i(t) 6. Show that: (ki 1+hiei) (ki 1) hi xi(a) |hi|7.

10 Show that the previous inequality still holds ifhi Conclude that for allh Rnwith h < ,wehave: (a+h) (a) l(h) n h 9. Prove the following:Theorem 114 Letn 1andUbe open :U Ebea map, whereEis anR-normed space. If, for alli Nn xiexistsonUand is continuous ata U,then is differentiable ata 18: The Jacobian Formula20 Exercise 1andUbe open :U Ebe amap, whereEis anR-normed space. We assume that for alli Nn, xiexists and is continuous Show that is differentiable Show that for alla, b Uandh Rn: (d (b) d (a))(h) n i=1 xi(b) xi(a) 2 1/2 h 3. Show that for alla, b U: d (b) d (a) n i=1 xi(b) xi(a) 2 1/24. Show thatd :U LR(Rn,E) is Prove the 18: The Jacobian Formula21 Theorem 115 Letn 1andUbe open :U Ebea map, whereEis anR-normed space. Then, is of classC1onU,if and only if for alli=1,..,n, xiexists and is continuous , Fbe twoR-normed spaces andl LR(E, F).