Implicit function theorem

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Easter Nicholson
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1 Jovo Jaric Implicit fuctio theorem The reader kows that the equatio of a curve i the x - plae ca be expressed F x, =., this does ot ecessaril represet a fuctio. Take, for example F x, = 2x x =. (1 either i a explicit form, such as = f ( x, or i a implicit form, such as However, if we are give a equatio of the form F( x, = 2 5 The equatio F( x, = does alwas represet a relatio, amel, that set of all pairs ( x, which satisf the equatio. The followig questio therefore presets itself quite aturall: Whe is the relatio defied b F( x, = also a fuctio? I other words, whe ca the equatio F( x, = be solved explicitl for i terms of x, ieldig a uique solutio. I the above example we ca solve the quadratic for ad obtai 2 ( 1 12 = f x = x± x + x ; (2 here a certai care is eeded, sice we reall have two explicit fuctios, correspodig to the positive ad egative sigs, from the oe implicit relatio. Accordigl we have to restrict attetio to the viciit of a particular poit. For example, the values x = 2, = 8 satisf the relatio F( x, = but ol oe of the explicit formulas. Geerall, what is =, which satisfies F x, f ( x = eeded is the assertio that there exists a fuctio f ( x ad for which f ( 2 = 8, eve though we could ot obtai the expressio 2 ( 1 12 f x = x+ x + x. (3 This is useful if we wat to treat x as a fuctio of, sa g(, for here o explicit formula exists. At the poit x = 1, = 1 ( or x =, = we ru ito trouble for here both formulas are valid. We shall obtai criteria for such poits. For that purpose we shall prove several geeral theorems that are required. The proof of the simplest theorem will be give i detail. The implicit fuctio theorem 1. Let x= x, = be a pair of values satisfig F( x, = ad let F ad its first derivatives be cotiuous i the eighborhood of this poit. The, if F does ot vaish at x= x, = there exists oe ad ol oe cotiuous fuctio = f ( x such that F( x, f ( x = ad f ( x =. (4

2 Proof. B the hpothesis, (, geeralit that F at x= x, = is positive, i.e. F ( x, >. ( If (, egative we could cosider G( x, F( x, x x positive. Moreover, sice F, F x, withi which (, F x. The we ca assume without loss of F x were = istead, ad G at =, = would be F are cotiuous we ca fid a box about poit ( x, x x δ, δ (5 F x, F ( x,, F ( x, are cotiuous ad F ( x, >. The x cosidered as a fuctio of, is a icreasig fuctio for δ. But F( x, =, b hpothesis, ad so it must be that (, ad that (, F( x, δ F x,, F x < for δ <, F x > for < + δ. I particular, F( x, δ + >. Further, b hpothesis, F( x, is cotiuous at poits ( x, δ ( x, + δ. Hece, F( x, δ < ad F( x, δ x, i.e. for ever < ad ad + > for ever x sufficietl close to x x l, where l > is sufficietl small umber. Let η be the smaller of two umbers l ad δ, i.e. η = mi ( l, δ. The, F ( x, >, F( x, δ <, F( x δ for ever x ad withi itervals, + > (6 x x η, δ. (7 Now, for a x from the iterval x x η fuctio F( x,, as a fuctio of, will be icreasig fuctio for a from the iterval δ, sice the F ( x, >. Thus, b the Itermediate value theorem for cotiuous fuctios, there must be a uique i the iterval δ for which F( x, =. Also whe x x the. To summarize, for each x i the iterval x x η we have show that there is oe ad ol oe i the iterval δ such that F( x, = ; this associatio of s with x s is a fuctio f ( x x x f ( x = with the domai x x η F x, f x =. Moreover, As = so that the fuctio is cotiuous at x. Note that we have ol gotte a elemet ad perhaps ol a small elemet of the fuctio, amel, the part withi x x η. Therefore, the implicit fuctio theorem deals the questio locall. However, we otice that the coditio of the theorem are satisfied for a poit x 1 withi this iterval. We ca therefore begi agai to costruct the fuctio x, ad hope to exted the iterval. I fact, we shall alwas be f ( x = from the poit 1 1 such that able to exted the iterval util we reach a poit where F = ad the there is o uique solutio to be foud. F x, is cotiuousl differetiable so is the Fiall, sice the fuctio fuctio = f ( x. I order to show this we start from the idetit

3 sice (, (, (, (, (, F x = F x F x + F x F x, (8 F x, =.The applig the first mea value theorem to each of these differeces, withi the itervals x x η ad δ, we have F( x, = ( x x Fx x + θ( x x, + ( F xo, + θ( But F x, f ( x = for = f ( x so that, θ 1. θ θ( = x x Fx x + x x, f x + f x F xo, + f x. If we write x = x + x, = + for a x ad i the above itervals, we get Fx( x + θ x, x+ F( x, + θ =. Now ad as x, because = f ( x is cotiuous ; also F ( x,. The the followig limit exists d ' Fx ( x, = f ( x = lim =. (9 dx x x F ( x, The theorem ma be exteded i a umber of was. For example the same techique ca be applied to solvig f( x,, z = for oe of variables i terms of the others. Now we proceed i solvig two simultaeous equatios ϕ xz ψ xz,, = (1 (,, =, Theorem 2. Let ϕ( x,, z = ad ψ ( x,, z =. Let ϕ( xz,, ad ψ ( xz differetiable i a eighborhood R of the poit ( x,, z ad let the Jacobia,, be ϕ ϕ ϕψ (, = (11 ( z, ψ ψ be osigular there. The, there is oe ad ol oe set of solutios = ( x, z = z( x which are cotiuous, satisf the equatios ϕ( xz,, =, ψ ( xz,, = ad for which = ( x, z = z( x. Furthermore, = ( x, z = z( x are differetiable. ` Proof. Sice The Jacobia (11 does ot vaish at ( x,, z the at least oe of partial derivatives ψ or ψ must ot vaish there. Let us assume that ψ does ot vaish at ( x,, z. The accordig to the Theorem 1 ψ ( xz,, = defies a uique differetiable fuctio z =φ ( x,. Now if we substitute this fuctio ito (11 we obtai F( x, = ϕ( x,, φ( x, =. Now, i order to prove the theorem 2 it is sufficiet to show that ϕ ϕ φ = + (12

4 at ( x,. To do this we elimiate φ from this expressio b makig use of the idetit ψ( x,, φ( x, =.The ψ ψ φ + =. (13 which ca be solved for φ i the eighborhood of the poit x z,,. I fact, from (12 ad (13 we have ϕψ (, ψ =. (14 (, z ϕψ (, But ad ψ at ( x,, z do ot vaish b hpothesis ad our assumptio, ( z, respectivel. Therefore, the same holds for F. Hece (12 defies a uique fuctio = ( x. The, substitutig this fuctio ito z =φ ( x, we obtai z =φ ( x, ( x, i.e. z = z( x. A more geeral theorem follows. Theorem 3. Let Fi( x1,..., xm, 1,...,, i= 1,..., be differetiable i a eighborhood of the poit ( x, =( x1,..., xm, 1,...,,. Further, let F i x, = ad let the Jacobia be osigular at ( x, set of solutios of the equatios Furthermore, f i are differetiable. ( F1,..., F (,..., = ( The there exists eighborhood R of ( x, ad a uique f x,..., x (16 i = i 1 m F x,..., x,,..., =. (17 i 1 m 1 Proof. If we assume the theorem true for ( 1 equatios ad prove its truth for equatios, the b iductio we ca rise to the geeral case from = 12, (alread proved b

5 the theorem 1 ad theorem 2. Sice the Jacobia does ot vaish at least oe of cofactors of its last row must ot vaish ad for coveiece we ma take this to be ( F1,..., F 1. ( 1,..., 1 1, we ca solve the first The sice the theorem is assumed for the case ( 1 equatios i the form = ϕ x1,..., xm;, = 1,..., 1, ad the ϕ are differetiable. The substitutig i the last equatio we have Φ( x1,..., xm; = F( x1,..., xm, ϕ1,..., ϕ 2; =. If the derivative Φ =ϕ x x 1,..., ad m the substitutio gives However, The ϕ From this does ot vaish, the this ca be solved to give ( 1 m ( 1 m (,..., (,..., = f x,..., x =ϕ x,..., x ; ϕ, = 1,..., 1, = f x x ϕ x x. (18 1 m 1 m ϕ Φ = +. are calculated from the first ( 1 of (17: ϕ β ϕ β + =, β, = 1,..., 1. ( F1,..,.., F 1 ( F1,..., F 1 ( 1,.., 1,, + 1,..., 1 ( 1,..., 1 ( F1,..,.., F 1 ( F1,..., F 1 ( 1 (,..,,,...,, (,..., = = = so that 1 Φ ( F,..,.., F (,...,,..., F F F F = ( 1 +,..,,,...,,,...,,..., = F ( F1,..,.., F 1 ( 1,..., F F 1 ( 1. (,..,,,...,, (,..., = = Fiall, ( ( F,..,.., F F,..., F,...,,..., Φ = (

6 Φ ad sice either term o the right-had side is zero,. The particular form i which the theorem is most eeded is the iversio of a fuctioal trasformatio. If m= ad F i has the form Fi = g (,..., i 1 x, i where g i are cotiuous differetiable fuctios the the theorem 3 takes the followig form. Theorem 4. If xi = g (,..., i 1, i = 1,...,, (2 are cotiuous fuctios of the variables 1,..., with cotiuous first partial derivatives, ad if the Jacobia ( x1,..., x J = (21 ( 1,..., does ot vaish, the the trasformatio from to x ca be uiquel iverted to give i = fi( x1,..., x. (22 Proof. We simpl appl the teorem 3. But ow the Jacobia (15 reads ( g1,..., g J = ( 1,..., because of the form of fuctios F i. The (21 follows from (2. Refereces [1] Tom M. Apostol, Mathematical Aalsis, Secod editio, Addiso-Wesle Publishig Compa, Readig, Massachusetts, Amsterdam, Lodo, Maila, Sigapore, Sde, Toko, pp. 373, 1974 [2] Howard Ato, Calculus, Secod editio, Joh Wile & Sos& Sos, New York, Chichester, Brisbae, Toroto, Sigapore, pp. 177, 1984 [3] Abraham Schwartz, Calculus ad Aaltic Geometr, Third editio, pp [4] Rutherford Aris, Vectors, Tesors ad the Basic Equatios of Fluid Mechaics, Dover Publicatio. Ic. New York, 1989

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial