[PDF] comment calculer le prix de revient d'un produit fini
[PDF] résultats affelnet 2017 toulouse
[PDF] guide de lecture biblique quotidienne 2017
[PDF] la bible en 6 ans 2017
[PDF] guide de lecture biblique quotidienne 2017 pdf
[PDF] guide de lecture biblique quotidienne 2016
[PDF] plan de lecture biblique
[PDF] lecture biblique du jour protestant 2017
[PDF] calendrier biblique 2018
[PDF] passage biblique du jour
[PDF] comment calculer le cycle d'une fille
[PDF] calcul d un cycle menstruel irrégulier
[PDF] programme national d'immunisation maroc 2017
[PDF] programme national d'immunisation 2016
[PDF] programme national d'immunisation 2017
![[PDF] Differential Calculus](https://pdfprof.com/Listes/17/25100-172ch6a.pdf.pdf.jpg "[PDF] Differential Calculus")
Chapter 6Differential CalculusIn this chapter, it is assumed that all linear spaces and flat spaces under
consideration are finite-dimensional.
61 Differentiation of Processes
LetEbe a flat space with translation spaceV. A mappingp:I→ Efrom some intervalI?SubRtoEwill be called aprocess. It is useful to think of the valuep(t)? Eas describing thestateof some physical system at time t. In special cases, the mappingpdescribes the motion of a particle and p(t) is theplaceof the particle at timet. The concept of differentiability for real-valued functions (see Sect.08) extends without difficulty to processes as follows: Definition 1:The processp:I→ Eis said to bedifferentiable at t?Iif the limit tp:= lims→01 s(p(t+s)-p(t)) (61.1) exists. Its value∂tp? Vis then called thederivative ofpatt. We say thatpisdifferentiableif it is differentiable at allt?I. In that case, the mapping∂p:I→ Vdefined by(∂p)(t) :=∂tpfor allt?Iis called the derivativeofp. Givenn?N×, we say thatpisntimes differentiableif∂np:I→ V can be defined by the recursion
1p:=∂p, ∂k+1p:=∂(∂kp) for allk?(n-1)].(61.2)
We say thatpisof class Cnif it isntimes differentiable and∂npis continuous. We say thatpisof class C∞if it is of classCnfor alln?N×. 209
210CHAPTER 6. DIFFERENTIAL CALCULUS
As for a real-valued function, it is easily seen that a processpis contin- uous att?Dompif it is differentiable att. Hencepis continuous if it is differentiable, but it may also be continuous without being differentiable. In analogy to (08.34) and (08.35), we also use the notation p (k):=∂kpfor allk?(n-1)](61.3) whenpis ann-times differentiable process, and we use p :=p(1)=∂p, p:=p(2)=∂2p, p:=p(3)=∂3p,(61.4) if meaningful. We use the term "process" also for a mappingp:I→ Dfrom some intervalIinto asubsetDof the flat spaceE. In that case, we use poetic license and ascribe topany of the properties defined above ifp|Ehas that property. Also, we write∂pinstead of∂(p|E) ifpis differentiable, etc. (IfD is included in some flatF, then one can take the direction space ofFrather than all ofVas the codomain of∂p. This ambiguity will usually not cause any difficulty.) The following facts are immediate consequences of Def.1, and Prop.5 of
Sect.56 and Prop.6 of Sect.57.
Proposition 1:The processp:I→ Eis differentiable att?Iif and only if, for eachλin some basis ofV?, the functionλ(p-p(t)) :I→Ris differentiable att. The processpis differentiable if and only if, for every flat functiona?
FlfE, the functiona◦pis differentiable.
Proposition 2:LetE,E?be flat spaces andα:E → E?a flat mapping. Ifp:I→ Eis a process that is differentiable att?I, thenα◦p:I→ E? is also differentiable att?Iand t(α◦p) = (?α)(∂tp).(61.5) Ifpis differentiable then(61.5)holds for allt?Iand we get ∂(α◦p) = (?α)∂p.(61.6) Letp:I→ Eandq:I→ E?be processes having the same domainI. Then (p,q) :I→ E ×E?, defined by value-wise pair formation, (see (04.13)) is another process. It is easily seen thatpandqare both differentiable at t?Iif and only if (p,q) is differentiable att. If this is the case we have t(p,q) = (∂tq,∂tq).(61.7)
61. DIFFERENTIATION OF PROCESSES211
Bothpandqare differentiable if and only if (p,q) is, and, in that case, ∂(p,q) = (∂p,∂q).(61.8) Letpandqbe processes having the same domainIand the same codomainE. Since the point-difference (x,y)?→x-yis a flat mapping fromE × EintoVwhose gradient is the vector-difference (u,v)?→u-v fromV × VintoVwe can apply Prop.2 to obtain Proposition 3:Ifp:I→ Eandq:I→ Eare both differentiable at t?I, so is the value-wise differencep-q:I→ V, and t(p-q) = (∂tp)-(∂tq).(61.9) Ifpandqare both differentiable, then(61.9)holds for allt?Iand we get ∂(p-q) =∂p-∂q.(61.10) The following result generalizes the Difference-Quotient Theorem stated in Sect.08. Difference-Quotient Theorem:Letp:I→ Ebe a process and let t
1,t2?Iwitht1< t2. Ifp|[t1,t2]is continuous and ifpis differentiable at
eacht?]t1,t2[then p(t2)-p(t1) t2-t1?CloCxh{∂tp|t?]t1,t2[}.(61.11) Proof:Leta?FlfEbe given. Then (a◦p)|[t1,t2]is continuous and, by Prop.1,a◦pis differentiable at eacht?]t1,t2[. By the elementary
Difference-Quotient Theorem (see Sect.08) we have
(a◦p)(t2)-(a◦p)(t2) t2-t1? {∂t(a◦p)|t?]t1,t2[}.
Using (61.5) and (33.4), we obtain
?a?p(t2)-p(t1) t2-t1? ?(?a)>(S),(61.12) where
S:={∂tp|t?]t1,t2[}.
Since (61.12) holds for alla?FlfEwe can conclude thatb(p(t2)-p(t1) t2-t1)≥0 holds for all thoseb?FlfVthat satisfyb>(S)?P. Using the Half-Space Intersection Theorem of Sect.54, we obtain the desired result (61.11).
Notes 61
(1) See Note (8) to Sect.08 concerning notations such as∂tp,∂p,∂np,p·,p(n), etc.
212CHAPTER 6. DIFFERENTIAL CALCULUS
62 Small and Confined Mappings
LetVandV?be linear spaces of strictly positive dimension. Consider a mappingnfrom a neighborhood of zero inVto a neighborhood of zero in V ?. Ifn(0) =0and ifnis continuous at0, then we can say, intuitively, that n(v) approaches0inV?asvapproaches0inV. We wish to make precise the idea thatnissmallnear0? Vin the sense thatn(v) approaches0? V? fasterthanvapproaches0? V. Definition 1:We say that a mappingnfrom a neighborhood of0inV to a neighborhood of0inV?issmall near 0ifn(0) =0and, for all norms
νandν?onVandV?, respectively, we have
lim u→0ν ?(n(u))
ν(u)= 0.(62.1)
The set of all such small mappings will be denoted bySmall(V,V?). Proposition 1:Letnbe a mapping from a neighborhood of0inVto a neighborhood of0inV?. Then the following conditions are equivalent: (i)n?Small(V,V?). (ii)n(0) =0and the limit-relation(62.1)holds for some normνonV and some normν?onV?. (iii)For every bounded subsetSofVand everyN??Nhd0(V?)there is a
δ?P×such that
n(sv)?sN?for alls?]-δ,δ[ (62.2) and allv? Ssuch thatsv?Domn.
Proof: (i)?(ii):This implication is trivial.
(ii)?(iii):Assume that (ii) is valid. LetN??Nhd0(V?) and a bounded subsetSofVbe given. By Cor.1 to the Cell-Inclusion Theorem of Sect.52, we can chooseb?P×such that By Prop.3 of Sect.53 we can chooseε?P×such that
εbCe(ν?)? N?.(62.4)
Applying Prop.4 of Sect.57 to the assumption (ii) we obtainδ?P×such that, for allu?Domn, ?(n(u))< εν(u) if 0< ν(u)< δb.(62.5)
62. SMALL AND CONFINED MAPPINGS213
such thatsv?Domn. Therefore, by (62.5), we have ifsv?=0, and hence n(sv)?sεbCe(ν?) for alls?]-δ,δ[ such thatsv?Domn. The desired conclusion (62.2) now follows from (62.4). (iii)?(i):Assume that (iii) is valid. Let a normνonV, a norm ?onV?, andε?P×be given. We apply (iii) to the choicesS:= Ce(ν), N ?:=εCe(ν?) and determineδ?P×such that (62.2) holds. If we put s:= 0 in (62.2) we obtainn(0) =0. Now letu?Domnbe given such that
0< ν(u)< δ. If we apply (62.2) with the choicess:=ν(u) andv:=1
su, we see thatn(u)?ν(u)εCe(ν?), which yields ?(n(u))
ν(u)< ε.
The assertion follows by applying Prop.4 of Sect.57.
The condition (iii) of Prop.1 states that
lim s→01 sn(sv) = 0 (62.6) for allv? Vand, roughly, that the limit is approached uniformly asvvaries in an arbitrary bounded set. We also wish to make precise the intuitive idea that a mappinghfrom a neighborhood of0inVto a neighborhood of0inV?isconfinednear zero in the sense thath(v) approaches0? V?not more slowlythanvapproaches 0? V. Definition 2:A mappinghfrom a neighborhood of0inVto a neigh- borhood of0inV?is said to beconfined near 0if for every normνonV and every normν?onV?there isN ?Nhd0(V)andκ?P×such that The set of all such confined mappings will be denoted byConf(V,V?). Proposition 2:Lethbe a mapping from a neighborhood of0inVto a neighborhood of0inV?. Then the following are equivalent: (i)h?Conf(V,V?).
214CHAPTER 6. DIFFERENTIAL CALCULUS
(ii)There exists a normνonV, a normν?onV?, a neighborhoodNof0 inV, andκ?P×such that(62.7)holds. (iii)For every bounded subsetSofVthere isδ?P×and a bounded subset S ?ofV?such that h(sv)?sS?for alls?]-δ,δ[ (62.8) and allv? Ssuch thatsv?Domh.
Proof: (i)?(ii):This is trivial.
(ii)?(iii):Assume that (ii) holds. Let a bounded subsetSofVbe given. By Cor.1 to the Cell-Inclusion Theorem of Sect.52, wecan choose b?P×such that By Prop.3 of Sect.53, we can determineδ?P×such thatδbCe(ν)? N.
Hence, by (62.7), we have for allu?Domh
such thatsv?Domh. Therefore, by (62.9) we have and hence h(sv)?sκbCe(ν?) for alls?]-δ,δ[ such thatsv?Domh. If we putS?:=κbCe(ν?), we obtain the desired conclusion (62.8). (iii)?(i):Assume that (iii) is valid. Let a normνonVand a normν? onV?be given. We apply (iii) to the choiceS:= Ce(ν) and determineS?and δ?P×such that (62.8) holds. SinceS?is bounded, we can apply the Cell- Inclusion Theorem of Sect.52 and determineκ?P×such thatS??κCe(ν?). We putN:=δCe(ν)∩Domh, which belongs to Nhd0(V). If we puts:= 0 in (62.8) we obtainh(0) =0, which shows that (62.7) holds foru:=0. Now letu? N×be given, so that 0< ν(u)< δ. If we apply (62.8) with the choicess:=ν(u) andv:=1 su, we see that h(u)?ν(u)S??ν(u)κCe(ν?), which yields the assertion (62.7). The following results are immediate consequences of the definition and of the properties on linear mappings discussed in Sect.52:
62. SMALL AND CONFINED MAPPINGS215
(I) Value-wise sums and value-wise scalar multiples of mappings that are small [confined] near zero are again small [confined] near zero. (II) Every mapping that is small near zero is also confined near zero, i.e.
Small(V,V?)?Conf(V,V?).
(III) Ifh?Conf(V,V?), thenh(0) =0andhis continuous at0. (IV) Every linear mapping is confined near zero, i.e.
Lin(V,V?)?Conf(V,V?).
(V) The only linear mapping that is small near zero is the zero-mapping, i.e.
Lin(V,V?)∩Small(V,V?) ={0}.
Proposition 3:LetV,V?,V??be linear spaces and leth? Conf(V,V?)andk?Conf(V?,V??)be such thatDomk= Codh. Then k◦h?Conf(V,V??). Moreover, if one ofkorhis small near zero so is k◦h. Proof:Let normsν,ν?,ν??onV,V?,V??, respectively, be given. Sinceh andkare confined we can findκ,κ??P×andN ?Nhd0(V),N??Nhd0(V?) such that for allu? N ∩Domhsuch thath(u)? N?∩Domk, i.e. for all u? N ∩h<(N?∩Domk). Sincehis continuous at0? V, we have h <(N?∩Domk)?Nhd0(V) and henceN ∩h<(N?∩Domk)?Nhd0(V). Thus, (62.7) remains satisfied when we replaceh,κandNbyk◦h,κ?κ, and N ∩h<(N?∩Domk), respectively, which shows thatk◦h?Conf(V,V??). Assume now, that one ofkandh, sayh, is small. Letε?P×be given. u? N ∩Domhwithκ:=ε for allu? N ∩h<(N?∩Domk). Sinceε?P×was arbitrary this proves that lim u→0ν ??((k◦h)(u))
ν(u)= 0,
i.e. thatk◦his small near zero. Now letEandE?be flat spaces with translation spacesVandV?, respec- tively.
216CHAPTER 6. DIFFERENTIAL CALCULUS
Definition 3:Letx? Ebe given. We say that a mappingσfrom a neighborhood ofx? Eto a neighborhood of0? V?issmall nearxif the mappingv?→σ(x+v)from(Domσ)-xtoCodσis small near0. The set of all such small mappings will be denoted bySmallx(E,V?). We say that a mapping?from a neighborhood ofx? Eto a neighborhood of?(x)? E?isconfined nearxif the mappingv?→(?(x+v)-?(x))from (Dom?)-xto(Cod?)-?(x)is confined near zero.
The following characterization is immediate.
Proposition 4:The mapping?is confined nearx? Eif and only if for every normνonVand every normν?onV?there isN ?Nhdx(E)and
κ?P×such that
We now state a few facts that are direct consequences of the definitions, the results (I)-(V) stated above, and Prop.3: (VI) Value-wise sums and differences of mappings that are small [confined] nearxare again small [confined] nearx. Here, "sum" can mean either the sum of two vectors or sum of a point and a vector, while "differ- ence" can mean either the difference of two vectors or the difference of two points. (VII) Everyσ?Smallx(E,V?) is confined nearx. (VIII) If a mapping is confined nearxit is continuous atx. (IX) A flat mappingα:E → E?is confined near everyx? E. (X) The only flat mappingβ:E → V?that is small near somex? Eis the constant0E→V?. (XI) If?is confined nearx? Eand ifψis a mapping with Domψ= Cod? that is confined near?(x) thenψ◦?is confined nearx. (XII) Ifσ?Smallx(E,V?) andh?Conf(V?,V??) with Codσ= Domh, then h◦σ?Smallx(E,V??). (XIII) If?is confined nearx? Eand ifσis a mapping with Domσ= Cod? that is small near?(x) thenσ◦??Smallx(E,V??), whereV??is the linear space for which Codσ?Nhd0(V??). (XIV) An adjustment of a mapping that is small [confined] nearxis again small [confined] nearx, provided only that the concept small [confined] nearxremains meaningful after the adjustment.
63. GRADIENTS, CHAIN RULE217
Notes 62
(1) In the conventional treatments, the normsνandν?in Defs.1 and 2 are assumed to be prescribed and fixed. The notationn=o(ν), and the phrase "nis small oh ofν", are often used to express the assertion thatn?Small(V,V?). The notation h=O(ν), and the phrase "his big oh ofν", are often used to express the assertion thath?Conf(V,V?). I am introducing the terms "small" and "confined" here for the first time because I believe that the conventional terminology is intolerably awkward and involves a misuse of the = sign.
63 Gradients, Chain Rule
LetIbe an open interval inR. One learns in elementary calculus that if a functionf:I→Ris differentiable at a pointt?I, then the graph offhas a tangent at (t,f(t)). This tangent is the graph of a flat functiona?Flf(R). Using poetic license, we refer to this function itself as thetangent tofat t?I. In this sense, the tangentais given bya(r) :=f(t)+ (∂tf)(r-t) for allr?R. If we putσ:=f-a|I, thenσ(r) =f(r)-f(t)-(∂tf)(r-t) for allr?I.
We have lim
s→0σ(t+s) s= 0, from which it follows thatσ?Smallt(R,R). One can use the existence of a tangent todefinedifferentiability att. Such a definition generalizes directly to mappings involving flatspaces. LetE,E?be flat spaces with translation spacesV,V?, respectively. We consider a mapping?:D → D?from an open subsetDofEinto an open subsetD?ofE?. Proposition 1:Givenx? D, there can be at most one flat mapping α:E → E?such that the value-wise difference?-α|D:D → V?is small nearx. Proof:If the flat mappingsα1,α2both have this property, then the value-wise difference (α2-α1)|D= (?-α1|D)-(?-α2|D) is small near
218CHAPTER 6. DIFFERENTIAL CALCULUS
x? E. Sinceα2-α1is flat, it follows from (X) and (XIV) of Sect.62 that
2-α1is the zero mapping and hence thatα1=α2.
Definition 1:The mapping?:D → D?is said to bedifferentiable atx? Dif there is a flat mappingα:E → E?such that ?-α|D?Smallx(E,V?).(63.1) This (unique) flat mappingαis then called thetangent to?atx. The gradient of?atxis defined to be the gradient ofαand is denoted by x?:=?α.(63.2) We say that?isdifferentiableif it is differentiable at allx? D. If this is the case, the mapping ??:D →Lin(V,V?) (63.3) defined by (??)(x) :=?x?for allx? D(63.4) is called thegradientof?. We say that?isof class C1if it is differentiable and if its gradient??is continuous. We say that?istwice differentiable if it is differentiable and if its gradient??is also differentiable. The gradient of??is then called thesecond gradientof?and is denoted by (2)?:=?(??) :D →Lin(V,Lin(V,V?))≂=Lin2(V2,V?).(63.5) We say that?isof class C2if it is twice differentiable and if?(2)?is continuous. If the subsetsDandD?are arbitrary, not necessarily open, and ifx? IntD, we say that?:D → D?isdifferentiable atxif?|EIntDis differentiable atxand we write?x?for?x(?|EIntD). The differentiability properties of a mapping?remain unchanged if the codomain of?is changed to any open subset ofE?that includes Rng?. The gradient of?remains unaltered. If Rng?is included in some flatF?inE?, one may change the codomain to a subset that is open inF?. in that case, the gradient of?at a pointx? Dmust be replaced by the adjustment x?|U?of?x?, whereU?is the direction space ofF?. The differentiability and the gradient of a mapping at a pointdependquotesdbs_dbs29.pdfusesText_35
[PDF] Introduction to differential calculus - The University of Sydney