[PDF] comment calculer le prix de revient d'un produit f
[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
Differential Calculus of Vector Valued Functions
Functions of Several Variables
We are going to consider scalar valued and vector valued functions of several real variables. For example, z ?f?x,y?,w?F?x,y,z?,y?G?x1,x2,...,xn?
V??v1?x,y,z?i??v2?x,y,z?j??v3?x,y,z?k?
Here?x,y?,?x,y,z?or?x1,x2,...,xn?denote the independent variables for the functionsf,F andG,andz,w,yandV?are referred to as dependent variables. A real valued function like z ?f?x,y?,assigns a unique value to each point?x,y?of a setDof the plane called the domain of the function. The set of values forf ?x,y?as?x,y?ranges over all points in the domain is called the range of the functionf.For each ?x,y?inDthe functionfassumes a scalar value (i.e., the value is a real number) andfis therefore called a scalar valued function or scalar field. The functionV ??v1?x,y,z?i??v2?x,y,z?j??v3?x,y,z?k?assigns to each point ?x,y,z?in its domain a unique value?v1,v2,v3?in3?spaceand since this value may be interpreted as a vector, this function is referred to as a vector valued function or vector field defined over its domainD.
Continuity
LetFdenote a real or vector valued function of n real variables defined over domainD.We say thatFis continuous at the pointPinDif, for eachQthat is "close toP",the value of F ?Q?is "close to" the valueF?P?.The precise definition of this vague statement is the following;Fis continuous at the pointPinDif, for every ??0,there exists a??0such that ?F?P??F?Q????whenever,?P?Q???.IfFis continuous at each pointPin its domain
D,we sayFis continuous onD.
Variation of a scalar field
Variations in the values of a real valued function of one variable are described in terms of its derivative. For a function of more than a singe variable there are several analogues of the derivative of a function of one variable. For example, letu ??pi??qj??rk?denote a unit vector inR
3and letf?f?x,y,z?denote a scalar function defined on domainDwithP??a,b,c?a
point ofD.Then if the following limit exists, it is defined to be the directional derivative offat
Pin the directionu
?u?f?a,b,c??limh?0 f?P?hu???f?P? h?limh?0 f?a?ph,b?qh,c?rh??f?a,b,c? h Clearly the directional derivative can be defined for functions ofnvariables fornother than
3.In the special case that the unit vector is in the direction ofone of the coordinate axes,
we refer to the derivative as the "partial derivative off"with respect to that variable along whose axis the unit vector is directed. That is, ifu ???1,0,0??u?f??f ?x??xf?the partial derivative offwith respect tox 1 ifu???0,1,0??u?f??f ?y??yf?the partial derivative offwith respect toy ifu ???0,0,1??u?f??f ?z??zf?the partial derivative offwith respect toz
We will use the notations
?f ?x??xf?fxinterchangeably to denote the partial derivative off with respect tox,with similar notations for the partial derivatives with respect to other independent variables. A function that is continuous onD,together with all its partial derivatives will be said to be a smooth function onD.
The Del operator
It is convenient for what follows to define the vector differential operator ???i?? ?x?j?? ?y?k?? ?z and to refer to this as the "del" operator. Then the followingoperations are defined for smooth scalar fieldsf ?x,y,z?or smooth vector fieldsV??x,y,z?: a) ??f?i??f ?x?j??f ?y?k??f ?z?"gradient off" b) ???V????x,?y,?z???v1,v2,v3???v1?x??v2?y??v3?z?"divergence ofV?" c) ???V?? i?j?k? x?y?z v1v2v3 ?i???yv3? ?zv2??j???xv3? ?zv1??k???xv2? ?yv1? "curl ofV?" Clearly this definition of the del operator as an operator onfunctions of three variables can be generalized to an operator on functions of n variables forevery n.
Example 1
1. Forf
?x,y,z??x2?y2?z2,we have??f?2xi??2yj??2zk?and forg?x,y,z??x2?y2?z2we find ?xg?1
2?x2?y2?z2??1/22x?x
x
2?y2?z2etc
Then ??g?1 x
2?y2?z2xi??yj??zk?
Note that if we letr?denote the radial vector,r??xi??yj??zk?,then?r???x2?y2?z2equals the length ofr ?and both of the functionsfandgare functions of the scalarR??r??;i.e. 2 f?R??R2andg?R??R.Then ??f?2r??2Rr?
Rand??g?r?
R.
More generally, forF
?F?R?,a scalar function ofR,we have??F?F??R?u?rwhereu?r?r? Ris a unit vector in the direction ofr
2. ForV
??x,y,z??xi??yj??zk??r?,we computedivV?????V???x?x???y?y???z?z??3. ForW ??xi??yj??zk? x2?y2?z2?1
Rr?,we compute
?xx x
2?y2?z2?x2?y2?z2?x2?x2?y2?z2??1/2
x2?y2?z2?1R?x2 R3 ?yy x
2?y2?z2?1R?y2
R3?zz x
2?y2?z2?1R?z2
R3
ThendivW??3R?x2?y2?z2
R3?2R
3. ForV??x,y,z??xi??yj??zk??r?,
???V??curlV?? i?j?k? x?y?z x y z ?0
Note also thatdiv V
??1?1?1?3.HereV?is an example of a "radial field"; i.e., the vector flow emanates out of the origin. ForW ?? ???r?where????w1,w2,w3?is a constant vector, W ?? ???r?? i?j?k? w1w2w3 x y z ?i??w2z?w3y??j??w3x?w1z??k??w1y?w2x? The vector fieldW?describes the velocity field for a rigid body rotation aboutthe axis??with angular speed equal to ???w12?w22?w32.In this case, we computecurlW??2??and div W ??0.
Each of the quantities
??f?grad f,???V??div V?,and???V??curl V?has physical meaning. The meanings for the divergence and curl must wait until necessary vector integration results have been derived. However, the meaning of the gradient is contained in the 3 following theorem.Theorem 1- Letf ?f?x,y,z?denote a smooth scalar field defined overDinR3.Then a) For a smooth curveC: x?x?t? y?y?t? z?z?t? a?t?b,the rate of change offalongCis given by df dt dsdt?V??x??t?2?y??t?2?z??t?2 andT??1
V?V??unit tangent vector toC
b) For every unit vectoru ?,the directional derivative offin the directionu?is given by ?uf???f?u? c) At each point P inD,fincreases most rapidly in the direction of??fand decreases most rapidly in the opposite direction, ???f. d) At each interior point ofDwherefhas a relative max or min, we have ??f?P??0? e) LetSdenote a level surface forf(i.e.S??x,y,z?|f?x,y,z??constant?Then at each pointPofS,the vector ??f?P?is normal to the surface.
Proof- (a) Supposew
?t??f?x?t?,y?t?,z?t??.Then the chain rule implies dw dt ??f ?x dxdt??f ?y dy dt??f ?z dzdt???f?V?whereV???x??t?,y??t?,z??t?? SinceV?is known to be tangent toCat each point ofC, V ??1
V?V?V??T?dsdt
?b?By definition?u?f?a,b,c??limh?0 f?P?hu???f?P? h?limh?0 f?a?u1h,b?u2h,c?u3h??f?a,b,c? h The mean value theorem for derivatives asserts that for?1,?2,?3such that0??j?h, 4
Thenlim
h?0 f?P?hu???f?P? i.e.,?u?f?a,b,c????f?a,b,c??u? (c) Since?u?f?a,b,c????f?a,b,c??u????f?a,b,c??u??cos? it is clear that???f?a,b,c??|?u?f?a,b,c?|???f?a,b,c?; i.e. ??f?a,b,c?when???i.e.,u?????f?a,b,c? Then??f?a,b,c?is in the direction of most rapid increase forfatP??a,b,c?,while???f?a,b,c? is in the direction thatfis most rapidly decreasing. (d) Supposef ?x,y,z?has an interior extreme point atP??a,b,c?.Theng1?x??f?x,b,c?has an interior extreme point atx ?a,which meansg1??x???xf?x,b,c??0atx?a.Similarly, ?yf?a,y,c??0aty?band?zf?a,b,z??0atz?c.But then (e) LetS??x,y,z?|f?x,y,z??AforAa constant and forPa point ofS,letCdenote a curve inSpassing throughP.ThenV ???x??t?,y??t?,z??t??is tangent toC,which is to say,V? lies in the plane that is tangent toSatP.Sincefis constant onS,fis constant onCand it follows from (a) that df dt ???f?V??0i.e.??f?V?at P Since this holds for anyClying inSand passing throughP, ??fmust, in fact, be normal to the plane that is tangent toSatP,which is the same as being normal toS.
Identities
Just as there are rules for the derivative of sums and products of differentiable functions of one variable, there are similar rules for applying the del operator to sums and various products of scalar and vector fields. Theorem 2- Letfandgdenote smooth scalar fields on domainDinR
3and letV?andW?
denote smooth vector fields onD. a) ???f?g????f???g;i.e.,grad?f?g??grad f?grad g b) 5 d)???fg??g??f?f??g; e) ??F?f?x,y,z???F??f???fforFa smooth function of one variable f) ???f V??grad f?V??f div V? g)???f V??grad f?V??f curl V? These rules can be seen to hold by using the definitions of theoperations together with the product or chain rules for differentiation. In addition to the rules for the del operator acting on sums and products, there are rules for combining the various operations with the del operator. Theorem 3-Letfdenote a sufficiently smooth scalar field on domainDinR
3and letV?
denote a similarly smooth vector field onD. Thengrad fandcurl V?are vector fields so the following operations are defined: div Similarly,div V?is a scalar field sograddiv V???????V?is a defined operation. We have the following identities a) ???grad f?div?grad f???2f??xxf??yyf??zzf b)curl curl V??graddiv V???????V? c)divcurl V??0 d)curl grad f?0?
Example 2-
1. Considerf
?x,y,z??1 x
2?y2?z2?1R,whereR??r??for r??xi??yj??zk?.
Then from problem 1.1, we have
??f??1 R 2r?
R??xi??yj??zk?
R3 and?xx R
3?R3?3x2R
R 3,?yy R
3?R3?3y2R
R 3,?zz R
3?R3?3z2R
R 3
Therefore,
div ?grad f???R3?3?x2?y2?z2?R R 3?0 6 A smooth functionfthat satisfiesdiv?grad f???2f?0at each point of a domainDis said to be harmonic inD.
2. For smooth vector fieldV
??v1?x,y,z?i??v2?x,y,z?j??v3?x,y,z?k?, ???V??curlV?? i?j?k? x?y?z v1v2v3 ?i???yv3? ?zv2??j???zv1? ?xv3??k???xv2? ?yv1?
Then??????V??
i?j?k? x?y?z ?yv3? ?zv2?zv1? ?xv3?xv2? ?yv1 i???y??xv2? ?yv1?? ?z??zv1? ?xv3???j???x??xv2? ?yv1?? ?z??yv3? ?zv2?? k???x??zv1? ?xv3?? ?y??yv3? ?zv2?? i???x??xv1??yv2??zv3?? ?2v1??j???y??xv1??yv2??zv3?? ?2v2? k???z??xv1??yv2??zv3?? ?2v3? i?? ?x?j?? ?y?k?? ?z?divV?? ?2v1i??v2j??v3k??graddivV???div grad?V?
3. For smooth vector fieldV??v1?x,y,z?i??v2?x,y,z?j??v3?x,y,z?k?,
???V??i???yv3? ?zv2??j???zv1? ?xv3??k???xv2? ?yv1? Thendivcurl V???x??yv3? ?zv2???y??zv1? ?xv3???z??xv2? ?yv1? xyv3? ?yxv3??zxv2? ?xzv2??yzv1? ?zyv1 Sincev1,v2andv3are all smooth functions, the mixed partial derivatives areequal and it follows thatdiv curl V??0.Thus ifW??curl V?for some smooth vector fieldV?,then the divergence ofW ?must vanish. The converse result can also be shown to be true.That is, if div W??0,thenW??curl V?for some smooth vector fieldV?.
4. Letf
?f?x,y,z?denote a smooth scalar field defined overDinR3.Then ?????f?curl??f? i?j?k? x?y?z ?xf?yf?zf ?i???yzf? ?zyf??j???xzf? ?zxf??k???xyf? ?yxf? Forfa smooth scalar field, the mixed partial are all equal andcurl??f?0?.Thus ifW?is a so 7 called gradient field (i.e., ifW????ffor some smooth scalar fieldf?thencurl W?vanishes. The converse of the result is also true. Ifcurl W??0?,thenW?must be the gradient of some smooth scalar field. A vector field whose curl vanishes is said to be a conservative field or irrotational field. Then every conservative field is a gradient field. 8quotesdbs_dbs8.pdfusesText_14
Introduction to differential calculus - University of Sydney