calcul diferential si integral 2

8/12/2019 Calcul Diferential Si Integral 2

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Differential and Integral Calculus

Volume 2

by R.Courant

6. 2reliminary Remar3s on Analytical )eometry and 7ector Analysis

66. 0unctions of Several 7ariales and their Derivatives

666. Development and Applications of the Differential 8alculus

67. "ultiple 6nte+rals

7. 6nte+ration over Re+ions in Several Dimensions

76. Differential uations

766. 8alculus of 7ariations

7666. 0unctions of a 8omple1 7ariale

Supplement: Real ;umers and the 8oncept of


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$.$.$ Co%ordinate Axes&6n order to fi1 a point in a plane or in space5 as is =ell 3no=n5one +enerally employs a rectan+ular co,ordinate system. 6n the plane5 =e ta3e t=operpendicular lines5 thex,a1is and they,a1is5 in space5 three mutually perpendicular lines5thex,a1is5 they,a1is and thez,a1is. *a3in+ the same unit len+th on each a1is5 =e assi+n to

each point of the plane anx,5 ay, and az,co,ordinate @0i+.% 8onversely5 therecorresponds to every set of values @15 y or @15 y5 )5 4ust one point of the plane or space? apoint is completely determined y its co,ordinates.

Bsin+ 2ytha+orasC theorem5 =e find that thedistanceet=een the t=o points@x%5y%5@x!5y! is +iven y

=hile the distance et=een the points =ith co,ordinates @x%5y%5z% and @x!5y!5z! is

6n settin+ up a system of rectan+ular a1es5 =e must pay attention to the orientation of theco,ordinate system.

6n'.!.%5 =e distin+uished et=een positive and ne+ative senses of rotation in the plane.*he rotation y 9#5 =hich rin+s the positivex,a1is of a plane co,ordinate system into theposition of the positivey,a1is in the shortest =ay defines a sense of rotation. Accordin+ to=hether this sense of rotation is postive or ne+ative5 =e say that the system of a1es isrig't%'anded or left%'anded @0i+s.!5E. 6t is impossile to chan+e a ri+ht,handed systeminto a left,handed one y a ri+id rotation5 confined to the plane. A similar distinctionoccurs =ith co,ordinate systems in space. 0or5 if one ima+ines oneself standin+ on thexy,plane =ith oneCs head in the direction of the positivez,a1is5 it is possile to distin+uisht=o types of co,ordinate system
http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat6/calc5.htm#5.2.1%20Orientation%20of%20Area:http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat6/calc5.htm#5.2.1%20Orientation%20of%20Area:http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat6/calc5.htm#5.2.1%20Orientation%20of%20Area:


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y means of the apparent orientation of the co,ordinate system in thexy,plane. 6f thissystem is ri+ht,handed5 the system in space is also said to e ri+ht,handed5 other=ise left,handed..@0i+s5 F/(. A ri+ht,handed system corresponds to an ordinary rig't%'anded

scre(? for if =e ma3e thexy,plane rotate aout thez,a1is @in the sense prescried y itsorientation and simultaneously translate it alon+ the positivez,a1is5 the comined motionis oviously that of a ri+ht,hand scre=. Similarly5 a left,handed system corresponds to aleft,handed scre=. ;o ri+id motion in three dimensions can transform a left,handedsystem into a ri+ht,handed system.

6n. =hat follo=s5 =e shall al=ays use ri+ht,handed systems of a1es.

-e may also assi+n an orientation to a system of three aritrary a1es passin+ throu+h onepoint5 provided these a1es do not all lie in one plane5 4ust as =e have done here for asystem of rectan+ular a1es.

$.$.2 Directions and Vectors. )ormulae for Transforming Axes&An oriented line linspace or in a plane5 i.e.5 a line traversed in a definite sense5 represents a direction? everyoriented line =hich can e made to coincide =ith the line lin position and sense ydisplacement parallel to itself represents the same direction. 6t is customary to specify adirection relative to a co,ordinate system y dra=in+ an oriented half,line in the +ivendirection5 startin+ from the ori+in of the co,ordinate system5 and on this half,line ta3in+

the point =ith co,ordinates 5 , 5 =hich is at unit distance from the ori+in. *he numers


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5 , are called the direction cosinesof the direction. *hey are the cosines of the three

an+les %5 !, E5 =hich the oriented line lma3es =ith the positivex,5 y, andz,a1es G @0i+.(? y the distance formula.5 they satisfy the relation

6f =e restrict ourselves to thexy,plane5 a direction can e

specified y the an+les %5 !=hich the oriented line l=iththis direction passin+ throu+h the ori+in forms =ith the

positivex, andy,a1es5 or y the direction cosines HH

cos%5 H cos!5 =hich satisfy the euation

A line,se+ment of +iven len+th and +iven direction =ill e called a#ector,morespecifically5 a bound #ector5 if the initial point is fi1ed in space5 and a free #ector5 if theposition of the initial point is immaterial. 6n the seuel5 and indeed throu+hout most of thisvolume5 =e shall omit the ad4ectives free and ound5 and if nothin+ is said to the contrary5=e shall al=ays assume vectors to e free. -e denote #ectorsy old type5 e.+.5 a, b, c,x,A.*=o free vectors are said to e eual if one of them can e made to coincide =ith theother y displacement parallel to itself. -e =ill at times call the lengt' of a #ectoritsasolute value and denote it y IaI.

**he an+le =hich one oriented line forms =ith another may al=ays e ta3en as ein+ et=een # and 5ecause in the seuel only the cosines of such an+les =ill e considered.

6f =e drop from the initial and final points of a vector vperpendiculars to an oriented linel5 =e otain an oriented se+ment on l correspondin+ to the vector. 6f the orientation of thisse+ment is the same as that of l5 =e call its len+th the com"onent of vin t'e direction ofl? if the orientations are opposite5 =e call the ne+ative value of the len+th of the se+mentthe com"onent of v in t'e direction of l.*he component of vin the direction of l =ill e

denoted y vl. 6f is the an+le et=een the direction of vand that of l@0i+.$5 =e al=ayshave

A vector vof len+th%iscalled a unit #ector. 6ts component in a direction l is eual to thecosine of the an+le et=een land v. *he components of a vector v in the directions of thethree a1es of a co,ordinate system are denoted y v%, v!, vE. 6f =e transfer the initial pointof vto the ori+in5 =e see that


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6f 5 , are the direction cosines of the direction of v5 then

A freevectoris

completely determined y its components v%, v!, vE.

An euation

et=een t=o vectors is therefore euivalent to the three ordinary euations

*here are different reasons =hy the use of vectorsisnatural and advanta+eous.0irstly5

many +eometrical concepts and a still lar+er numer of physical concepts such as force5velocity5 acceleration5 etc.5 immediately reveal themselves as vectors independent of theparticular co,ordinate system. Secondly5 =e can set up simple rules for calculatin+ =ithvectors analo+ous to the rules for calculatin+ =ith ordinary numers? y means of thesemany ar+uments5 these can e developed in a simple =ay5 independently of the particularco,ordinate system chosen.

-e e+in y definin+ the sum of t'e t(o #ectorsaand b. 0or this purpose5 =e displacethe vector bparallel to itself until its initial point coincides =ith the final point of a. *henthe startin+ point of aand the end point of bdetermine a ne= vector c@0i+. &J the startin+point of =hich is the startin+ point of aand the end point of =hich is the end point of b.

-e call cthe sumof aand band =rite

0or this additive process5 there hold oviously the commutati#e la(


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and theassociati#e la(

as a +lance at 0i+s. & and 9 sho=s.

-e otain immediately from the definition of vector addition the"ro*ection t'eorem: *he component of the sum of t=o or morevectors in a direction lis the sum of the components of the individualvectors in that direction5 i.e.5

6n particular5 the components of abin the directions of the co,ordinate a1es are a%5 b%5a! b!5 aE bE.

>ence5 in order to form the sum of t(o #ectors5 =e have the simple rule: *hecomponents of the sum are eual to the sums of the correspondin+ summands.

very pointP=ith co,ordinates @x5y5z may e determined y the "osition #ector fromthe ori+in toP5 the components of =hich in the directions of the a1es are 4ust the co,ordinates of the pointP. -e ta3e three unit vectors in the directions of the three a1es5 e%inthex-direction5 e!in they-direction5 eEin thez-direction. 6f the vector vhas thecomponents v%5 v!5 vE5 then

-e call v%H v%e%5 v!%H v!e!5 vEH vEeE the #ector com"onentsof v.

Bsin+ the pro4ection theorem stated aove5 =e easily otain the transformationformulae=hich determine @xC5yC5zC5 the co,ordinates of a +iven point 2 =ith respect tothe a1es x!5 y!5 z!in terms of @x5y5z5 its co,ordinates =ith respect to another set G ofa1es x5 y5 z5 =hich has the same ori+in as the first set and may e otained from it yrotation. *he three ne= a1es form an+les =ith the three old a1es5 the cosines of =hich

may e e1pressed5 y the follo=in+ scheme5 =here5 for e1ample5 %is the cosine of thean+le et=een thexC,a1is and thez,a1is.

G ;ote that5 in accordance =ith the convention adopted5oth systems of a1es are to e ri+ht,handed.

-e drop fromPperpendiculars to the a1es x5 y5 zto their feetP%5P!5PE@0i+.%. *hevector from toP is then eual to the sum of the vectors from toP%5 from toP!and

from toPE. *he direction cosines of thexC,a1is relative to the a1es x5 y5 zare %5 %5

%5 those of theyC,a1is !5 !5 !and. those of thezC,a1is E5 E5 E. Ly the "ro*ection

t'eorem5 =e 3no= thatxC5 =hich is the component of the vector in the direction of the
http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#In.%20what%20follows,%20we%20shall%20always%20use%20right-handed%20systems%20of%20axes%23In.%20what%20follows,%20we%20shall%20always%20use%20right-handed%20systems%20of%20axeshttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#In.%20what%20follows,%20we%20shall%20always%20use%20right-handed%20systems%20of%20axes%23In.%20what%20follows,%20we%20shall%20always%20use%20right-handed%20systems%20of%20axeshttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#Co-ordinate%20Axes%23Co-ordinate%20Axeshttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#In.%20what%20follows,%20we%20shall%20always%20use%20right-handed%20systems%20of%20axes%23In.%20what%20follows,%20we%20shall%20always%20use%20right-handed%20systems%20of%20axeshttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#Co-ordinate%20Axes%23Co-ordinate%20Axes


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xC,a1is5 must e eual to the sum of the components of in the direction ofthexC,a1is5 =hence

ecause %xis the component ofxin the direction of thexC,a1is5 etc. 8arryin+ out similarar+uments foryC andzC5 =e otain the transformation formulae

Since the components of a. ound vector v in the directions of the a1es are e1pressed ythe formulae

in =hich @x%5y%5z% are the co,ordinates of the startin+ point and @x!5y!5z! the co,ordinatesof the end point of v5 it follo=s that the same transformation formulae hold for thecomponents of the vector as for the co,ordinates:

$.$.+ ,calar -ulti"lication of Vectors&0ollo=in+ conventions similar to those for theaddition of vectors5 =e no= define the product of a vector vy a numer c: 6f vhas thecomponents v%5 v!5 vE5 then cv is the vector =ith the components cv%5 cv!5 cvE.*hisdefinition a+rees =ith that of vector addition5 ecause vvH !v5 vv v H Ev5 etc. 6f cM#5 cvhas the same direction as vand the len+th cIvI? if cN #5 the direction of cvis oppositeto the direction of vand its len+th is @,c IvI. 6f cH #5 =e see that cv is the ero #ector =iththe components #5#5#.

-e can also define the product of t=o vectors wandv5 =here this multi"licationof#ectorssatisfies rules of calculation =hich are partly similar to those of ordinarymultiplication. *here are t=o different 3inds of vector multiplication. -e e+in =ith

scalar multi"lication=hich is simpler and the more important for our purposes.

*he scalar "roductG uvof the vectors uand vis the product of their asolute values and

the cosine of the an+le et=een their directions:


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>ence5 the scalar product is simply the component of one of the vectors in the direction ofthe other multiplied y the len+th of the second vector.

G 6t is sometimes also called the inner "roduct.

Ly the the pro4ection theorem5 the distributi#e la( for multi"licationis

=hence follo=s at once the commutati#e la(

On the other hand5 there is an essential difference et=een the scalar "roduct of t(o#ectorsand the ordinary "roduct of t(o numbers5ecause the product can vanishalthou+h neither factor vanishes.

6f the len+th of uand vare not ero5 the product uvvanishes if5 and only if the t=o vectorsuand vare perpendicular to each other.

6n order to e1press the scalar product in terms of the components of the t=o vectors5 =eta3e oth the vectors uand v=ith their startin+ points at the ori+in. -e denote their vectorcomponents y u%5 u!5 uEand v%5 v!5 vE5 respectively5 so that uHu%+u!uEand vHv%+v!vE. 6nthe euation uv H @u%+u!uE@v%+v!vE5 =e can e1pand the product on the ri+ht hand side

in accordance =ith the rules of calculation5 =hich =e have 4ust estalished? if =e note thatthe products u%v!5 u%vE, u!v%5 u!vE5uEv%5 uEv!vanish5 ecause the factors are perpendicularto each other5 =e otain uvH u%v% u!v!+uEvE. ;o= the factors on the ri+ht have thesame direction5 so that5 y definition5 u%v% H v%u%5 etc.5 =here u%5 u!5 uEand v%5 v!5 vEare thecomponents of uand v5 respectively. >ence

*his euation could have een ta3en as the definition of the scalar "roductand is animportant rule for calculatin+ the scalar product of t=o vectors +iven in terms of theircomponents. 6n particular5 if =e ta3e uand vas unit #ector =ith direction cosines

%, !, Eand%, !,E5 respectively5 the scalar product is eual to the cosine of the an+leet=een uand v5 =hich is accordin+ly +iven y the formula

*he "'ysical meaning of t'e scalar "roductis e1emplified y the fact5 proved inelementary physics5 that a forcef =hich moves a particle of unit mass throu+h the directed


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distance vdoes=or3amountin+ tofv.

$.$./ T'e

0uations of

t'e ,traig't

1ine and of

t'e Plane&


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straig't line@or plane y multiplyin+ y an aritrary5 non,vanishin+ factor. 8onversely5an aritrary linear euation

represents a strai+ht line @or plane provided not all the coefficients&,'@or&,',()areero.G 0or e1ample5 in the second of these euations5 =e may divide y

and set

6n this =ay5 =e otain an euation =hich is seen to represent a plane at a distancepfrom

the ori+in5 the normal of =hich has the direction cosines , , .8orrespondin+ remar3shold for the euation of the strai+ht line.

G 6f&H' # @or& '(H #5must also e ero5 and any point of the plane @or of space satisfies the

euation.

A strai+ht line in space may e determined y any t=o planes passin+ throu+h the line.*hus5 =e otain for a line in space t=o linear euations

=hich are satisfied y @x5y,z5 the co,ordinates of any point on the line. Since an infinitenumer of planes pass throu+h a +iven line5 this representation of a line in space is notuni0ue.

0reuently5 it is more convenient to represent a line analytically in "arametric formymeans of a parameter t.6f =e consider three linear functions of t


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=here the biare not all ero5 then5 as ttraverses the numer a1is5 the point @x,y,z)descries a strai+ht line. -e see this at once y eliminatin+ tet=een each pair ofeuations5 =herey =e otain t=o linear euations forx,y,x.

*he direction cosines , , of the line in its parametric form are proportional to thecoefficients b%5 b!, bE,ecause these direction cosines are proportional @0i+.%% tox%- x!, y%

- y!, z%- z! the differences of the co,ordinates of t=o pointsP%, P!=ith the co,ordinates

and

>ence


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=here denotes the len+th of the se+mentP%P!. >ence

Since the sum of the suares of the direction cosines is unity5 it follo=s that

=here the doule si+ns of the suare root correspond to the fact that =e can choose eitherof the t=o possile senses on the line. Ly means of the direction cosines5 =e can easily

rin+ the parametric representation of the line into the form

=here @x#5y#5z# is a fi1ed point on the line? the ne= parameter is connected =ith theprevious parameter ty the euation

0rom the fact that " H l follo=s that

>ence the asolute value of is the distance et=een @x#5y#, z# and @x5y5z.*he si+n of indicates =hether the direction of the line is from the point @x#5y#, z# to the point @x5y5z

or vice versa? in the first case5 is positive5 in the second case ne+ative.

0rom this result =e otain a useful e1pression for @x5y5z,the co,ordinates of a pointPonthe se+ment 4oinin+ the pointsP#@x#5y#, z# andP%@x%5y%, z%5 namely

=here#and%are positive and #" % %.6fand +denote the distances fromPof

the pointP andP+, respectively5 =e find that +,ecause if =e calculate, say,


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fromx% x# %5 and sustitute this value5 H @x%- x#/#5 in the euationx x# %5=e otain the aove e1pression.

ence5 the euation of the plane throu+h @x#5y#, z# perpendicular to the line =ith direction

cosines 5 5 is

6n the same =ay5 the euation of a strai+ht line in thexy,plane =hich passes throu+h the

point @x#5y# and is perpendicular to the line =ith direction cosines 5 is


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6n thexy,plane5 =e have

xercises $.$

$.2rove that the uantities 5 5 PPP 5 [email protected] =hich define a rotation of axes5 satisfy therelations

2. 6f aand bare t=o vectors =ith startin+ point and end points&and'5 then the vector

=ith O as startin+ point and the point dividin+&'in the ratio :% , as final point is+iven y

+.*he centre of mass of the vertices of a tetrahedronP/0may e defined as the point

dividin+10in the ratio % : E5 =here1is the centre of mass of the trian+leP/.Sho=that this definition is independent of the order in =hich the vertices are ta3en and that ita+rees =ith the +eneral definition of the 8entre of mass.

/.6f in the tetrahedronP/0the centres of the ed+esP,/0,P/,0,P0,/aredenoted y&,&!,','!,(5 (!,respectively5 then all the lines&&!,''!,((!pass throu+hthe centre of mass and isect one another at that point

3.


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%.!.% T'e Area of a Triangle&6n order to calculate the area of a trian+le m thexy,plane5=e ima+ine it moved parallel to itself until one of its vertices is at the ori+in? let the othert=o vertices eP%@x%5y% andP!@x!5y! @0i+.%!. -rite do=n the euation of the line4oinin+P%to the ori+in in its canonical form

hence one has for the distance %of the pointP!from this line @e1cept possily for thesi+n the e1pression

Since the len+th of the se+ment Pis ,=e find that t=ice the area of thetrian+le,=hich is the product of the baseP%and the 'eig't%is +iven @e1cept possilyfor the si+n y

*his e1pression can e either positive or ne+ative? it chan+es si+n if =e interchan+eP%and 2!. -e no= ma3e the follo=in+ assertion: *he e1pression&has a positive or ne+ativevalue accordin+ to =hether the sense5 in =hich the vertices P%P!are traversed5 is thesame as the sense of the rotation associated =ith the co,ordinate a1es or not.6nstead of


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provin+ the fact y a more detailed investi+ation of the ar+ument +iven aove5 =hich isuite feasile5 =e prefer to prove it in the follo=in+ =ay. -e rotate the trian+le P%P!aout the ori+in untilP%lies on the positivex,a1is. @*he case in =hich ,P%,P!lie onone line5 so that&Q@x%y!,x!y% H # can e omitted. *his rotation does not chan+e the

value of A. After the rotation5P%has theco,ordinatesx%C M #5y%C H #5 and the co,ordinates of the ne=P!arex!C andy!!.*he area of the trian+le is no=

=hence it has the same si+n asy!!.>o=ever5 the si+n ofy!!is the same as

the si+n of the sense in =hich thevertices P%P!are traversed @0i+.%E5and thus the statement is proved.

0or the e1pressionx%y!,x!y,=hich+ives t=ice the area =ith its propersi+n5 it is customary to introduce thesymolic notation

=hich =e call a t(o%ro( determinant or a second order determinant.

6f no verte1 of the trian+le is at the ori+in of the co,ordinate system5 for e1ample5 if thethree vertices are @x#5y#5 @x%,y%,@x!5y!5 =e otain y movin+ the a1es parallel tothemselves for the area& of the trian+le

$.2.2 Vector -ulti"lication of t(o Vectors&. Leside the scalar "roductof t=o vector5

=e have the important conce"t of t'e #ector "roduct. *he vector product ab or abofthe vectors aandbis defined as follo=s @0i+.%F:

-e lay off aand bfrom a point .*hen aand bare t=o sides of a parallelo+ram in space.*he vector product ab cis a vector the len+th of =hich is numerically eual to thearea of the parallelo+ram and the direction of =hich is perpendicular to the plane of the


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parallelo+ram5 the sense of direction ein+ such that the rotation from ato band cabis ri+ht,handed5 i.e.5 if =e loo3 at the plane from the end point of the vector c5 =e see theshortest rotation from the direction of ato that of bas a positive direction. 6f a and blie inthe same strai+ht line5 =e must have ab HH #5 since the area of the parallelo+ram is ero.

Rules of Calculation of t'e Vector Product&

7$86f a band b#5 then ab # if5 and only if5aandbhave the same direction oropposite directions5 ecause then and only then the area of the parallelo+ram =ith sides aand beuals ero.

@! *here holds the euation

*his follo=s at once from the definition of ab.

@E 6f aand bare real numers5 then

ecause the parallelo+ram =ith sides aaand bbhas an area abtimes that of theparallelo+ram =ith aides aand band lies in the same plane as the latter.

@F *he distributi#e la( holds:

-e shall prove the first of these formulae? the second follo=s from it =hen Rule @! isapplied.

-e shall no= +ive a geometrical construction for t'e #ector "roductab =hich =illdemonstrate directly the truth of the distributi#e la(.


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ence, 4e can form ab in the follo=in+ =ay: 2ro4ect bortho+onally onto the plane2,len+then it in the ratio IaI : % and rotate it positively throu+h 9#T aout the vector a.

6n order to prove that a@bc H ab ac5. =e proceed as follo=s: band care thesides ',(of a parallelo+ram '(,the dia+onal of =hich is the sum b c.-eno= perform the three operations of pro4ection5 len+thenin+ and rotation on the =holeparallelo+ram '(instead of on the individual vectors b,c,bc? =e thus otain aparallelo+ram '%%(%the sides '%,(%of =hich are the vectors ab and ac and thedia+onal of =hich is the product a@bc. >ence follo=s the euation ab ac H a@b c @0i+.%(.


19/735

@'


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=hich5 y Rules @% and @E5 may e re=ritten

;o=5 y the definition of the #ector "roduct5

=hence

*he components of the vector product ab H care therefore

6n 2hysics5 =e use the vector product of t=o vectors to represent a moment.A forcefactin+ at the end point of the position vectorxhas the moment fx aout the ori+in.


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$.2.+ T'e Volume of a Tetra'edron&8onsider a tetrahedron @0i+.%$ the vertices of=hich are the ori+in and three other points 2%5P!PE=ith the co,ordinates 7x%5y%5z%5 7x!5y!5z!5 7xE5yE5zE5 respectively. 6n order to e1press the volume of this tetrahedron in termsof the co,ordinates of its vertices5 proceed as follo=s: *he vectorsx%P%andx!P!are the sides of a trian+le the area of =hich is half the len+th of the vector product x%x!5*his vector product has the direction of the perpendicular from 2Eto the plane of thetrian+le P%P!5 hence the len+th %of this perpendicular @the hei+ht of the tetrahedron is+iven y the scalar "roduct of the vectorxEPEand the unit vector in the direction ofx%x!5 ecause %is eual to the component of PEin the direction of x%x!5Since the

asolute value of x%x! is t=ice the area&of the trian+le P%P!and the volume 6of thetetrahedron is eual to&%/E,=e have

Or5 since the components of x%x! are +iven y


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=e can =rite

*his also holds for the case in =hich ,P%5P!lie on a strai+ht line? in this case5 it is true5the direction of x%x! is indeterminate5 so that %can no lon+er e re+arded as thecomponent of Pin the direction of x%x!5 ut nevertheless so that 6#5 and thisfollo=s also from the aove e1pression for 65 since in this case all the components of

x%x! vanish.

>ere a+ain the volume of the tetrahedron is +iven =ith a definite si+n as the area of thetrian+le =as earlier? and =e can sho= that the si+n is positive5 if the three a1es P%,P!,PEta3en in that order form a system of the same type @ri+ht,handed or left,handed5 asthe case may e as the co,ordinate a1es5 and ne+ative if the t=o systems are of the

opposite type. 6n fact5 in the first case5 the an+leet=een x%x! andxElies in the interval

# /! and5 in the second case5 in the interval /!5 as follo=s immediately fromthe definition of x%x! and 6is eual to

*he e1pression

in our formulae may e e1pressed more riefly y the symol

called a t'ree%ro(ed determinant or determinant of t'ird order.On e1pandin+ thet=o,ro=ed determinants5 =e see that
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Vust as in the case of the trian+le5 =e find that the volume of the tetrahedron =ith vertical@x#5y#5z#5 @x%5y%5z%5 @x!5y!5z! is

xercises $.2 7more difficult exercises are indicated by an 98

%. -hat is the distance of the pointP@x#5y#5z# from the strai+ht line l+iven y

!G. 0ind the shortest distance et=een t=o strai+ht lines land l!in space5 +iven y theeuations

E. Sho= that the plane throu+h the three points @x%5y%, z%5 @x!5y!, z!5 @xE5yE, zE is +iven y

F. 6n a uniform rotation5 let @5 5 e the direction cosines of the a1is of rotation5 =hich

passes throu+h the ori+in5 and the an+ular velocity. 0ind the velocity of the point @x,y,z.

'. 2rove 1agrange:s identity


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(. *he area of a conve1 poly+on =ith the verticesP%@x%5y%,P!@x!5y!5 PPP 5Pn@xn5yn is+iven y half the asolute value of

>ints and Ans=ers

$.+ ,I-P1 TH4R-, 45 DTR-I5A5T, 4) TH ,C45D A5D THIRD

4RDR

$.+.$ 1a(s of )ormation and Princi"al Pro"erties&*he determinants of the second andthird order occurrin+ in the calculation of the area of a trian+le and the volume of atetrahedron5 to+ether =ith their +eneraliation5 the determinant of order n,or n%ro(eddeterminant,are very important in that they enale formal calculations in all ranches ofmathematics to e e1pressed m a compact form. -e shall develop no= the properties ofdeterminants of the second and third order? those of hi+her order =e shall rarely need. 6tmay5 ho=ever5 e pointed out that all the principal theorems may e +eneralied at once todeterminants =ith any numer of ro=s. 0or their theory5 =e must refer the reader to oo3son al+era and determinants.GLy their definitions in!.%and !.E5 the determinants

are e1pressions formed in a definite =ay from their elements a,b5 c5 d and a5 b5 c5 d,e,f,g,%,75 respectively. *he horiontal lines of elements @such asd,e5f in our e1ample arecalled ro(sand the vertical lines @such as c5f5 7 are called columns.

G 0or e1ample5 >. -. *urnull5 8%e8%eoryofDeterminants51atricesand9nvariants@Llac3ie W Son5


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symmetrical =ay in =hich the determinant is formed:

-e repeat the first t=o columns after the third and then form the product of each triad ofnumers in the dia+onal lines5 multiply the products associated =ith lines slantin+do=n=ards and to the ri+ht y %5 the others y ,%5 and add them. 6n this =ay5 =e otain

-e shall no= prove several t'eorems on determinants:

@% 6f the ro=s and columns of a determinant are interchan+ed5 its value is unaltered5 i.e.5

*his follo=s immediately from the aove e1pressions for the determinants.

@! 6f t=o ro=s Xor t=o column.s of a determinant are interchan+ed5 its si+n is altered5


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i.e.5 the determinant is multiplied y ,%.

Ly virtue of @%5 this need only e proved for the columns5 and it can e verified at oncey the la= of formation of the determinant +iven aove.

@E 6n!.E5 =e have introduced three,ro=ed determinants y the euations

Bsin+ @!5 =e =rite this in the form

then in the determinants on the ri+ht hand side the elements are in the same order as onthe left hand side. 6f =e interchan+e the last t=o ro=s and then =rite do=n the sameeuation5 usin+ @!5 =e otain

and similarly

-e call these three euations the ex"ansion in terms of t'e elements of t'e t'ird ro(t'e second ro( and t'e first ro(5 respectively. Ly interchan+in+ columns and ro=s5=hich accordin+ to @% does not alter the value of the determinant5 =e otain theex"ansion by columns5
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An immediate conseuence of this is the theorem:

@FJ6f all the elements of one ro= @or column are multiplied y a numer 5 the value of

the determinant is multiplied y.

0rom @! and @F5 =e deduce the theorem:

@' %f the elements of t=o ro=s @or t=o columns are proportional5 i.e.5 if every element of

one ro= @or column is the product of the correspondin+ element in the other ro= @orcolumn and the same factor5 t%ent%edeter$inantise:ualtozero.

6n fact5 accordin+ to @F5 =e can ta3e the factor outside the determinant. 6f =e theninterchan+e the eual ro=s5 the value of the determinant is unchan+ed5 ut y @! it shouldchan+e si+n. >ence its value is ero.

6n particular5 a determinant5 in =hich one ro= or column consists entirely of eros5 has thevalue ero5 as also follo=s from the definition of a determinant.

@( *he sum of t=o determinants5 havin+ the same numer of ro=s5 =hich differ only in

the elements of one ro= @or column is eual to the determinant =hich coincides =iththem in the ro=s @or columns common to the t=o determinants and in the one remainin+ro= @or column has the sums of the correspondin+ elements of the t=o non,identicalro=s @or columns.

0or e1ample5


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6n fact5 if =e e1pand in terms of the ro=s @or columns in uestion5 =hich in our e1ampleconsist of the elements b,e,%and $,n,p,respectively5 and add5 =e otain the e1pression

=hich clearly is 4ust the e1pansion of the determinant

in terms of the column m5 e n5 h p. *his proves the statement.

Similar statements hold for t=o,ro=ed determinants.

@$6f =e add to each element of a ro= @or column of a determinant the same multiple5 ofthe correspondin+ element of another ro= @or column5 the value of the determinant isunchan+ed.

Ly @(5 the ne= determinant is the sum of the ori+inal determinant and a determinant=hich has t=o proportional ro=s @or columns? y @'5 this second determinant is ero.G

G *he rule for an e1pansion in terms of ro=s or columns may e e1tended to define determinants of thefourth and hi+her order. )iven a system of si1teen numers5 for e1ample5

=e define a determinant of the fourth order y the e1pression


29/735

and similarly =e can introduce determinants of the fifth5 si1th5 .. .5 nth order in succession. 6t turns out that

in all essential properties these a+ree =ith the determinants of t=o or three ro=s. >o=ever5 determinants of

more than three ro=s cannot e e1panded y the diagonal rule. -e shall not consider further detailshere.

*he follo=in+ e1amples illustrate ho= the aove theorems are applied to the evaluation ofdeterminants. -e have

as =e can prove y the diagonal rule. A determinant in =hich only the elements in the so,called "rinci"al diagonal differfro$ zero is e:ual to t%e product of t%ese ele$ents.

#aluation of a determinant:

>ence


30/735


31/735

=hich can e verified y sustitution. >o=ever5 if the determinant vanishes5 theeuations

=ould lead to a contradiction if either the determinant and =ere differentfrom ero. >o=ever5 if

our formulae tell us nothin+ aout the solution.

>ence5 =e otain the fact5 =hich is particularly important for our purposes5 that a systemof euations of the aove form5 the determinant of =hich is different from ero5 al=ayshas a uniue solution.

6f our system of euations is 'omogeneous,i.e.,if&H'#,our calculations lead to the

solutionx#,y#,provided that #.

0or the three euations =ith three un3no=ns

a similar discussion leads to a similar conclusion. -e multiply the first euation y

5 the second y the third y and add to otain


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>o=ever5 y our formulae for the e1pansion of a determinant in terms of the elements of acolumn5 this euation can e =ritten in the form

Ly Rule @F5 the coefficients ofyandzvanish5 so that

6n the same =ay5 =e derive the euations

6f the determinant


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is not ero5 the last three euations yield the value of the un3no=ns. 2rovided that thisdeterminant is not ero5 the euations can e solved uniuely forx5y,. 6f the determinantis ero5 it follo=s that the ri+ht hand sides of the aove euations must also e ero5=hence the euations cannot e solved unless&,',(satisfy the special conditions =hichare e1pressed y the vanishin+ of every determinant on the ri+ht hand side.

6n particular5 if the system of euations is homo+eneous5 so that&'(#5 and if itsdeterminant is different from ero5 it a+ain follo=s thatx H y HHzH #. 6n addition to thecases aove5 in =hich the numer of euations is eual to the numer of un3no=ns5 =e

shall occasionally encounter systems of t(o 7'omogeneous8 e0uations (it' t'reeun!no(ns5 for e1ample

6f not all of the three determinants

are ero5 if5 for e1ample5E #5 our euations can first e solved forxandy;

or

*his has the +eometrical meanin+: -e are +iven t=o vector = and v=ith the componentsa,b,cand d.e,f5 respectively. -e see3 a vectorx=hich is perpendicular to uand v,i.e.5=hich satisfies the euations


34/735

*hus5xis in the direction of uv.

xercises $.+

%. Sho= that the determinant

can al=ays e reduced to the form

merely y repeated application of the follo=in+ processes:

@% 6nterchan+in+ t=o ro=s or t=o columns5 @! addin+ a multiple of one ro= @or columnto another ro= @or column.

!. 6f all the three determinants

vanish5 then the necessary and sufficient condition for the e1istence of a solution of thethree euations

is


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E. State the condition that the t=o strai+ht lines

either intersect or are parallel

FG. 2rove 2roperties @% to @$5 +iven inE.%for determinants of the fourth order.

'. 2rove that the volume of a tetrahedron =ith vertices @x%5y%5z%5 @x!5y!5z!5 @xE5yE5zE,@xF5yF5zF is +iven y

>ints and Ans=ers

$./ A))I5 TRA5,)4R-ATI45, A5D TH -61TIP1ICATI45 4)

DTR-I5A5T,

-e shall conclude these preliminary remar3s y discussin+ the simplest facts relatin+ tothe so,called affine transformationsandat the same time otain an important theorem ondeterminants.

$./.$. Affine Transformations of t'e Plane and ,"ace&-e mean y a ma""ingortransformationof a portion of space @or of a plane a la= y =hich each point hasassi+ned to it another point of a space @or a plane as image "oint? =e call the point itselfthe original "oint,or sometimes the model@in contrast to the image.-e otain aphysical e1pression of the concept of mappin+ y ima+inin+ that the portion of space @orplane in uestion is occupied y some deformale sustance and that our transformationrepresents a deformation in =hich every point of the sustance moves from its ori+inal
http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#3.1%20Laws%20of%20Formation%20and%20Principal%20Properties:%233.1%20Laws%20of%20Formation%20and%20Principal%20Properties:http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#3.1%20Laws%20of%20Formation%20and%20Principal%20Properties:%233.1%20Laws%20of%20Formation%20and%20Principal%20Properties:http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#*%20The%20rule%23*%20The%20rulehttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/anshints.htm#1.3http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#3.1%20Laws%20of%20Formation%20and%20Principal%20Properties:%233.1%20Laws%20of%20Formation%20and%20Principal%20Properties:http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#*%20The%20rule%23*%20The%20rulehttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/anshints.htm#1.3


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position to a certain final position.

Bsin+ a rectan+ular system of co,ordinates5 =e ta3e @x5y, as the co,ordinates of theori+inal point and @xC5y!,z! as those of the correspondin+ ima+e point.

*he transformations =hich are not only the simplest and most easily understood ones5 utare also of fundamental importance for the +eneral case5 are the affine transformations.An affine transformation is one in =hich the co,ordinates @xC5y!,z! @or in the plane @x!,y!of the ima+e point are e1pressed linearly in terms of those of the ori+inal point. Such atransformation is therefore +iven y the three euations

or in the plane y the t=o euations

=ith constant coefficients a5 b5 PPP. *hese assi+n animage "ointto every point of space @orplane. *he uestion at once arises =hether =e can interchan+e the relationship et=eenima+e and ori+inal point5 i.e.5 =hether every point of space @or of the plane has anori+inal point correspondin+ to it. *he necessary and sufficient condition for this is thatthe euations

shall e capale of ein+ solved for the un3no=nsx,y,z@orx,y),no matter =hat thevalues ofx!,y!,z!are. Ly %.E.!5 an affine transformation has an inverse and5 in fact5 a

uni0ue in#[email protected] every ima+e point has one and only one ori+inal point5provided

that its determinant
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is different from ero. -e shall confine our attention to affine transformations of this type

and shall not discuss =hat happens =hen #.Ly introducin+ an intermediate point @xY5yY5zY5 =e can decompose the +eneral affine

transformation into the transformations

and

>ere @x,y,z is mapped first onto @x3,y3,z3)and then


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*he characteristic +eometrical properties of affine transformations are stated in thet'eorems.

@%6n space5 the ima+e of a plane is a plane? in the plane5 the ima+e of a strai+ht line is a

strai+ht line.

6n fact5 y%.%.F5 =e can =rite the euation of the plane @or the line in the form

*he numers&,',(@or&,')are not all ero. *he co,ordinates of the ima+e points of theplane @or of the line satisfy the euation

>ence the ima+e points themselves lie on a plane @or a line5 ecause the coefficients

of the co,ordinatesx!,y!,zC @orxC,y!)cannot all e ero? other=ise the euations

=ould hold5 and these =e may re+ard as euations in the un3no=ns&,',( @or&,').Lut

=e have sho=n aove that it follo=s from these euations that& ' ( @or& ' #.

@! *he ima+e of a strai+ht line in space is a strai+ht line.

*his follo=s immediately from the fact that a strai+ht line may e re+arded as theintersection of t=o planes? y @%5 its ima+e is also the intersection of t=o planes and is
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therefore a strai+ht line.

@E *he ima+es of t=o parallel planes of space Xor of t=o parallel lines of the plane areparallel.

6n fact5 if the ima+es had points of intersection5 the ori+inals =ould have to intersect at theori+inal points of these intersections.

@F *he ima+es of t=o parallel lines in space are t=o parallel lines.

6n fact5 as the t=o lines lie in a plane and do not intersect one another5 y @% and @!5 thesame is true for their ima+es5. *he ima+es are therefore parallel.

*he ima+e of a vector vis of course a vector v!leadin+ from the ima+e of the startin+point of v to the ima+e of the end point of v.Since the components of the vector are the

differences of the correspondin+ co,ordinates of the startin+ and end points5 under themost +eneral affine transformation5 they are transformed accordin+ to the euations

$./.2 T'e Combination of Affine Transformations and t'e Resolution of t'e General

Affine Transformation. 6f =e map a point @x5y5z onto an ima+e point #x!,y!,z' ymeans of the transformation

and then map #x!,y!,z!)onto a point @x3,y3,z3)y means of a second affinetransformation

=e readily see that @x5y,z)and #x3,y3,z3)are also related y an affine transformation. 6n


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fact5

=here the coefficients are +iven y the euations

-e say that this last transformation is the combination orresultantof the first t=otransformations. 6f the determinants of the first t=o transformations are different fromero5 their inverses can e formed5 =hence the compound transformation also has aninverse. *he coefficients of the compound transformation are otained from those of theori+inal transformations y multiplyin+ correspondin+ elements of a column of the firsttransformation and of a ro= of the second transformation5 addin+ the three products thusotained5 and usin+ this "roductof column and ro= as the coefficient =hich stands in thecolumn =ith the same numer as the column used and in the ro= =ith the same numer asthe ro= used.

6n the same =ay5 a comination of the transformations

yields the ne= transformation

-e mean y a "rimiti#e transformationone in =hich t=o @or one of the three @or t=oco,ordinates of the ima+e are the same as the correspondin+ co,ordinates of the ori+inalpoints. 6n physical terms5 =e may thin3 of a primitive transformation as one in =hich thespace @or plane under+oes stretchin+ in one direction only @the stretchin+5 of course5varyin+ from point to point so that all the points are simply moved alon+ a family ofparallel lines. A primitive affine transformation in =hich the motion ta3es place parallel to


41/735

thex,a1is is analytically represented y formulae of the type

*he +eneral affine transformation in the plane

=ith a non,vanishin+ determinant5 can e otained y a comination of primitivetransformations.

6n the proof =e may assume G that a#. -e introduce an intermediate point @5 y theprimitive transformation

the determinant of =hich does not vanish. 0rom 5 , =e otainxC5yC y a second primitivetransformation

=ith the determinant

*his yields the reuired resolution into primitive transformations.

G 6f aH #5 then b#5 and =e can return to the case a#. Such an interchan+e. represented y thetransformation=y, >x5 is itself effected y the three successive primitive transformations

1=x - y, 2= 1,= 2+ 2=y

1=y, 2= 1+ 1=x, >= 2 z.


42/735

6n a similar =ay5 theaffine transformation in s"ace

=ith a non,vanishin+ determinant5 can e resolved into primitive transformations.

At least one of the three determinants

must e different from ero? other=ise5 as the e1pansion in terms of the elements of thelast ro= sho=s5 =e should have

As in the precedin+ case5 =e can them assume =ithout loss of +enerality that @%

and @! that a#. *he first intermediate point @5 is +iven y theeuations

*he determinant of this primitive transformation is a,=hich is not ero. 0or the second

transformation to !5 C5 C5 =e =ish to set ! 5 ! 5 and also to have C HyC. Oneprimitive transformation then remains. 6f =e introduce in the euation

the uantities 5 5 instead ofx,y,z,=e otain the second primitive transformation in


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the form

*he determinant of this transformation is *he third transformation mustthen e

ne1t+o to start of chapter

$./.+ T'e Geometrical -eaning of t'e Determinant of Transformation and t'e

-ulti"lication T'eorem&0rom the considerations of the last section5 =e can deduce the+eometrical meanin+ of the determinant of an affine transformation and5 at the same time5an al+eraic t'eorem on t'e multi"lication of determinants.

8onsider a plane trian+le =ith vertices @#5 #5 @x%5y%5 @x!5y! =ith the area @cf. %.!.%

-e shall investi+ate the relationship et=een&and the area&!of its ima+e otained ymeans of a "rimiti#e affine transformation
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*he vertices of the ima+e trian+le have the co,ordinates @#5 #5 @ax% by%,y%,@ax! by!5

y!5 =hence

>o=ever5 this determinant can e transformed y the theorems of %.E.%in the follo=in+=ay:

i.e.5

6f =e had ta3en the "rimiti#e transformation

=e should have found in the same =ay that

-e thus see that a "rimiti#e affine transformation has the effect of multiplyin+ the areaof a trian+le y a constant independent of the trian+le.6f the verte1 of the trian+le lies atthe ori+in5 the same fact applies5 y virtue of the +eneral formulafor the area. Since the+eneral affine transformation can e formed y cominin+ primitive transformations5 thestatement remains true for any affine transformation.6n the case of an affinetransformation5 the ratio of the area of an ima+e trian+le to the area of the ori+inaltrian+le is constant and independent of the choice of trian+le5 dependin+ only on thecoefficients of the transformation.6n order to find this constant ratio5 =e consider5 inparticular5 the trian+le =ith vertices @#5#5 @%5# and @#5%5 =ith area Q .Since the ima+eof this trian+le5 accordin+ to the transformation
http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#1.3.1%20Laws%20of%20Formation%20and%20Principal%20Propertieshttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01a.htm#2.1%20The%20Area%20of%20a%20Triangle%232.1%20The%20Area%20of%20a%20Trianglehttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#1.3.1%20Laws%20of%20Formation%20and%20Principal%20Propertieshttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01a.htm#2.1%20The%20Area%20of%20a%20Triangle%232.1%20The%20Area%20of%20a%20Triangle


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has the vertices @#5 #5 @a5 c,@b,d5 its area is

and =e thus see that the constant ratio of area AC/A. for an affine transformation is thedeterminant of the transformation.

-e can proceed in e1actly the same =ay for transformations in space. 6f =e consider thetetra'edron=ith the vertices @#5#5#5 @x%5y%,z%5 @x!5y!,z!5 @xE5yE,zE and su4ect it to theprimitive transformation

the image tetra'edronhas the vertices @#5 #5 #5 @ax% by% cz%5y%,z%5 @ax! by! cz!5y!,z!5 @axE byE czE5yE, zE5 so that its volume is

>ence

=here 6is the volume of the ori+inal tetrahedron. 0or the volume of the ima+e5 +iven ythe primitive transformation


46/735

=e find in a similar manner that

and for the primitive transformation

that

>ence an aritrary affine transformation has the effect of multiplyin+ the volume of atetrahedron y a constant. 6f no verte1 of the tetrahedron coincides =ith the ori+in5 thistheorem follo=s from the +eneral formulafor the #olume of a tetra'edron. 6n order tofind this constant for the transformation

=e consider the tetrahedron =ith the vertices @#5 #5 #5 @%5 #5 #5 @#5 %5 #5 @#5 #5 %5 theima+e of =hich has the vertices @#5#5#5 #a,d,g),@b5e5%),@c5f57).0or the volumes 6!and 6of the ima+e and the ori+inal =e thus have

=hence the determinant is the constant sou+ht.
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*he si+n of the determinant also has a +eometrical meanin+. 6n fact5 from =hat =e haveseen in !.Ere+ardin+ the connection et=een the sense of rotation and the volume of thetetrahedron or area of the trian+le5 it follo=s at once that a transformation =ith a positivedeterminant preserves the sense of rotation5 =hile a transformation =ith a ne+ativedeterminant reverses it.

-e no= consider the combination of t(o transformations

As =e pass fromx5y5ztoxC5yC5zC5 the volume of a tetrahedron is multiplied y

as =e pass fromxC5yC5ztoxY5yY5zY5 y

and y direct chan+e fromx,y,ztox3,y3,Y5 it is multiplied y

*his oservation yields the follo=in+ relation5 3no=n as the t'eorem for t'emulti"lication of determinants:
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As efore5 =e call the elements of the determinant on the ri+ht hand side the "roducts of

t'e ro(s of and the columns of ? at the intersection of the i,th

ro= and7,th column of the product of the determinants5 there stands the e1pression

formed from thei,th ro= of and the 7-th column of . Sincero=s and columns are interchan+eale5 the product of the determinants can also eotained y cominin+ columns and. ro=s5 columns and columns5 or ro=s and. ro=s.

0or t=o,ro=ed determinants5 naturally5 the correspondin+ theorem @cominin+ ro=s andcolumns holds:

xercises $./

%.valuate the determinants

!. 0ind the relation =hich must e1ist et=een a,b,cin order that the system of euations


49/735

may have a solution.

EG. @a 2rove the ineuality

@ -hen does the euality si+n holdU

F. -hat conditions must e satisfied in order that the affine transformation

may leave the distance et=een any t=o points unchan+edU

'. 2rove that in an affine transformation the ima+e of a 0uadratic

is a+ain a uadratic.

(.G 2rove that the affine transformation

leaves at least one direction unaltered.

$. )ive the formulae for a rotation y the an+le aout the a1isx:y:zH % : # : ,% suchthat the rotation of the planexHHzis positive =hen loo3ed at from the point @,%5 #5 %.

&. 2rove that an affine transformation transforms the centre of mass of a system ofparticles into the centre of mass of the ima+e particles.


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9. 6f15 PPP 5 3denote the uantities at the end of%.!5 definin+ a rotation of a1es5 then

>ints and Ans=ers

lastne1t

CHAPTR 2

)unctions of ,e#eral Variables and t'eir Deri#ati#es

-e have already ecome acuainted =ith functions of several varialesand have learnedthere enou+h to appreciate their importance and usefulness. -e are no= aout to enter ona more thorou+h study of these functions5 discussin+ properties =hich =ere notmentioned efore. and provin+ theorems =hich =ere made merely plausile. ;o proof inthis volume =ill involve previous 3no=led+e of any proof developed in 8hapter Z. [etthe student should read a+ain that chapter5 as the intuitive discussion +iven there =illassist in formin+ mental ima+es of matters =hich are perhaps some=hat astract.

As a rule5 a theorem =hich can e proved for functions of t=o variales5 can e e1tendedto functions of more than t=o variales =ithout any essential chan+es in the ar+ument.

>ence5 in the seuel5 =e shall usually confine ourselves to functions of t=o variales5 andshall only discuss functions of three or more variales =hen some special aspect isinvolved.

2.$. TH C45CPT 4P )65CTI45 I5 TH CA, 4) ,VRA1 VARIA;1,

2.$.$ )unctions and t'eir Ranges of Definition&uations of the form

assi+n a functional #alue uto a pair of values @x,y.6n the first t=o of these e1amples5 a

value of uis assi+ned to e#erypair of values @x,y,=hile in the third the correspondencehas a meanin+ only for those pairs of values @x5y for =hich the ineualityx"y N % istrue.

6n these cases5 =e say that uis a function of theinde"endent #ariablesx andy.-e usethis e1pression in +eneral =henever some la= assi+ns a value of uasde"endent#ariable,correspondin+ to each pair of values @x5y)elon+in+ to a certain specified set.Similarly5 =e say that uis a function of the n varialesx%5x!5 PPP5xn,if there e1ists for
http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01a.htm#9.%20If%239.%20Ifhttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#becausehttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#becausehttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/anshints.htm#1.4http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#CHAPTER%20Ihttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/02.htmhttp://mpec.sc.mahidol.ac.th/RADOK/physmath/MAT6/Calcul10.htm#10.1%20The%20Concept%20of%20Function%20in%20the%20Case%20of%20Several%20Variableshttp://mpec.sc.mahidol.ac.th/RADOK/physmath/MAT6/Calcul10.htm#10.1%20The%20Concept%20of%20Function%20in%20the%20Case%20of%20Several%20Variableshttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01a.htm#9.%20If%239.%20Ifhttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#becausehttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/anshints.htm#1.4http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/01.htm#CHAPTER%20Ihttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/02.htmhttp://mpec.sc.mahidol.ac.th/RADOK/physmath/MAT6/Calcul10.htm#10.1%20The%20Concept%20of%20Function%20in%20the%20Case%20of%20Several%20Variables


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every set of values @x%5x!5PPP5xn elon+in+ to a certain specified set a correspondin+ valueof u.

*hus5 for e1ample5 the #olumeuxyzof a rectan+ular parallelepiped is a function of thelen+ths of the three sidesx5y5z?the magnetic declinationis a function of the latitude5

the lon+itude5 and the time? the sumx%x! PPP xn isa function of the ntermsx%5x!5 PPP5xn.

6n the case of functions of t=o variales5 =e represent the pair of values @x5y y a pointin a t=o,dimensional5 rectan+ular co,ordinate system =ith the co,ordinatesxandy,and=e occasionally call this point the argument "oint of the function. 6n the case of thefunctions uHx"yand uHxy5 the ar+ument pointcan ran+e over the entirexy,plane and=e say that these functions are defined in the entirexy,plane. 6n the case of the function uHH lo+ @% ,x- y,the point must remain =ithin the circlex" y N % and the function isdefined only for points inside this circle.

As in the case of functions of a sin+le variale5 the ar+uments in the case of functions ofseveral variales may e either discontinuous or continuous. *hus5 the avera+epopulation per state of the Bnited States depends on the numer of states and on thenumer of inhaitants5 oth of =hich are inte+ers. On the other hand5 len+ths5 =ei+hts5etc.5 are e1amples of continuous variales. 6n the seuel5 =e shall deal almost e1clusively=ith pairs of continuously #ariable arguments? the point @x,y =ill e allo=ed to varyin a definite region@or domain of the 1y,plane5 correspondin+ to the inter#al in thecase of functions of one variale. *his re+ion may consist of the entirexy,plane or of aportion of the plane ounded y a sin+le closed cur#e(=hich does not intersect itself @asim"ly%connected region?.0i+.%? or it may e ounded y several closed curves. 6n thelast case5 it is said to e a multi"ly%connected region5 the numer of the oundarycurves yieldin+ the so,called connecti#ity? for e1ample5 0i+. ! sho=s a tri"ly%connectedregion.


52/735

*he oundary curve and5 in fact5 every curve considered hereafter =ill e assumed to esectionally smoot',i.e.5=e assume once and for all that every such curve consists of a

finite numer of arcs5 each one of =hich has a continuously turnin+ tan+ent at each of itspoints up to and includin+ the endpoints. >ence5 such curves can at most have a finitenumer of cornersor cus"s.

*he most important types of re+ions5 =hich recur over and over a+ain in the study offunctions of several variales5 are @% the rectangular region @0i+.E5 defined yineualities of the form

in =hich each of the independent variales is restricted to a definite interval5 and the

argument "oint varies in a rectan+le? @! the circular region @0i+.F),defined y anineuality of the form

in =hich the argument "ointvaries in a circle =ith radius rand centre @, .

A pointP=hich elon+s to a re+ion/is said to e an internal "oint,if =e can find acircle =ith centre atPlyin+ entirely =ithin/.6fP is an interior point of/,=e also saythat/is a neig'bour'ood ofP.*hus5 any nei+hourhood ofP=ill contain a sufficientlysmall circle =ithPas centre.

-e may riefly point out that correspondin+ statements hold in the case of functions ofmore than t=o independent variales5 e.+5. of three varialesx,y,z. 6n this case5 theargument "ointvaries in a three,dimensional instead of in a plane re+ion. 6n particular5this re+ion may e a rectangular region,defined y ineualities of the form


53/735

or a s"'erical region,defined y an ineuality of the form

6n conclusion5 =e note a finer distinction5 =hich5 =hile scarcely essential for the presentpurposes5 is nevertheless of importance in more advanced studies. -e sometimes mustconsider re+ions =hich. do not contain their oundary points5 i.e.5 the points of the curvesoundin+ them. Such re+ions are called o"en regions @Appendi1. *hus5 for e1ample5 there+ionx "y ?% is ounded y the circlex "y %5 =hich does not elon+ to there+ion? the re+ion is therefore o"en. On the other hand5 if the oundary points do elon+to the re+ion5 as =ill e the case in most of the e1amples to e discussed5 =e say that there+ion is closed.

-hen =e are dealin+ =ith more than three independent variales5 sayx,y,z,4,ourintuition fails to provide a +eometrical interpretation of the set of independent variales.Still5 =e shall occasionally ma3e use of +eometrical terminolo+y5 spea3in+ of a system ofnnumers as a point in n%dimensional s"ace. ;aturally5 =e mean y rectan+ular andspherical re+ions in such a space systems of points the co,ordinates of =hich satisfyineualities of the form

or

respectively.

-e can no= e1press our definition of the concept of function precisely in the =ords: 6f Ris a re+ion in =hich the independent varialesx5y5 PPP may vary5 and if a definite value uis assi+ned to each point @x5y5 PPP of this re+ion accordin+ to some la=5 then uHf@x5y5 PPPis said to e a function of the continuous independent varialesx5y5 PPP.

6t is to e noted that5 4ust as in the case of functions of a sin+le variale5 a functionalcorrespondence associates a uni0ue value of u=ith the system of independent variales

x,y,PPP. *hus5 if the functional value is assi+ned y an analytical e1pression =hich ismulti%#alued5 such as artan @y/x5 this e1pression does not determine the functioncompletely. On the contrary5 =e have still to specify =hich of the several possile valuesof the e1pression is to e used5 that is5 =e must still state that =e are to ta3e the value of

artan#y/x =hich lies et=een ,! and !5 or the value et=een # and 5 or =e mustma3e some other similar specification. 6n such a case5 =e say that the e1pression definesseveral different single%#alued branc'es of the function @7ol.%5%.!E. 6f =e =ish toconsider all these ranches simultaneously =ithout +ivin+ anyone of them preference5 =e
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may re+ard them as formin+ a multi%#alued function. >o=ever5 =e shall ma3e use ofthis idea only in 8hap. 7666.

2.$.2 T'e ,im"lest Ty"es of )unctions&Vust as in the caseof functions of one variale5the simplest functions are the rational integral functions or "olynomials.*he most

+eneral polynomial of the first de+ree @linear function has the form

=here a5 and care constants. *he general "olynomial of t'e second degreehas theform

*he general "olynomial of any degreeis a sum of terms of the form a$nx$yn, =here the

constants a$nare aritrary.

Rational fractional functionsare uotients of polynomials? this class includes5 fore1ample5 thelinear fractional function

-e pass y extraction of roots from rational to certain algebraic functions5 fore1ample5

6n the construction of more complicated functions of several variales5 =e almost al=aysrevert to the =ell,3no=n functions of a sin+le variale @cf. compound functions

2.$.+ Geometrical Re"resentation of )unctions&6n 8hapter Z of 7ol. 65 =e havediscussed the t=o principal methods for representin+ a function of t=o independentvariales5 namely @% y means of the surface uHf@x,y inxyu,space5 descried y the

point =ith co,ordinates @x5y,u)as @x5y ran+es over the re+ion of definition of thefunction u,and @! y means of the curves @contour lines in thexy,plane alon+ =hich uhas a definite fi1ed value 7.-e shall not repeat this discussion here. 6f the student is notalready perfectly familiar =ith these methods of +eometrical representation5 he =ould e=ell advised to turn to %#.E.%of the first volume.

2.2 C45TI56IT


55/735

2.2.$ Definition&*he reader5 =ho is acuainted =ith the theory of functions of a sin+levariale and has seen =hat an important part is played in it y the concept of continuity5=ill naturally e1pect that a correspondin+ concept =ill fi+ure prominently in the theory offunctions of more than one variale. "oreover5 he =ill 3no= in advance that thestatement that a function u Hf@x5y is continuous at the point @x,y=ill mean5 rou+hly

spea3in+5 that for all points @, near @x5y the value of the functionf@, =ill differ utlittle fromf@x,y.-e e1press this idea more precisely as follo=s:

*he functionf@x5y5 defined in the re+ionR5 is continuous at the point @5 ofR5

provided that it is possile to find for every positive numer a positive numer

= () @in +eneral5 dependin+ on and tendin+ to # =ith such that for all points of the

re+ion the distance of =hich from @, is less than , i.e.5 for =hich

Or5 in other =ords5 the relation

is to hold for all pairs of values @%53 such that % 7 andthe point @ %, 7 elon+s to the re+ion/.

6f a function is continuous at every point of a re+ionR,=e say

that it is continuousinR.

6n the definition of continuity5 =e can replace the distance condition % 3 y theeuivalent condition:

*here shall correspond to every M # t=o positive numers %and !such that

=henever I%I%andI7I !.

*he t=o conditions are euivalent. 6n fact5 if the ori+inal condition is fulfilled5 so is thesecond5 if =e ta3e

and5 conversely5 if the second condition is fulfilled5 so is the first one5 if =e ta3e for the

smaller of the t=o numers % and !.


56/735

*he follo=in+ facts are almost ovious:

*he sum5 difference and product of continuous functions are also continuous. *heuotient of continuous functions is continuous e1cept =here the denominator vanishes.8ontinuous functions of continuous functions are themselves continuous. 6n particular5

all polynomials are continuous and so are all rational fractional functions e1cept =herethe denominator vanishes

Another ovious fact =orth statin+ is the follo=in+: 6f a functionf@x5y is continuous in a re+ionRanddiffers from ero at all interior pointsPof the re+ion5 it is possile to mar3 off aoutPa

neig'bour'ood5 saya circle5 elon+in+ entirely toR5 in =hichf@x5y does not vanish any=here. 6n fact5

if the value of the function atPis a5 =e can mar3 off aoutPa circle so small that the value of the functionin the circle differs from ay less than a/! and therefore is certainly not ero.

A function of several variales may have discontinuitiesof a much more complicatedtype than a function of a sin+le variale. 0or e1ample5 discontinuities may occur alon+entire arcs of curves5 as in the case of the function uHy/x5 =hich is discontinuous alon+

the entire linexH #. "oreover5 a functionf@x5y may e continuous inxfor each fi1edvalue ofyand continuous inyfor each fi1ed value ofxand yet e a. discontinuousfunction ofxandy.*his is e1emplified y the functionf@x5y!xy/@xy. 6f =e ta3e anyfi1ed non,ero value ofy,this function is oviously continuous as a function ofx,sincethe denominator cannot vanish. 6fyH #5 =e havef@x5 # H #5 =hich is a continuousfunction ofx.Similarly5f@x5y is continuous inyfor each fi1ed value ofx.Lut at everypoint on the lineyx,e1cept the pointxHyH #5 =e havef@x5y H %? and there are pointsof this line aritrarily close to the ori+in. >ence the function is discontinuous at the pointxy#.

Other e1amples of discontinuous functions are +iven in7olume %.

2.2.2 T'e Conce"t of limit in t'e Case of ,e#eral Variables&*he concept of the limitof a function of t=o variales is closely related to the concept of continuity.


57/735

6n order to emphasie this5 =e read this statement: *he doule limit off@x5y asxtends to

andy to is l.

6n the lan+ua+e of limits5 =e can say that a functionf@x5y is continuous at a point @, if5 and only if5

-e can see the matter in a ne= li+ht if =e consider se0uences of "oints. -e shall say

that a seuence of points @x%5y%5 @x!5y! PPP 5 @xn5yn tends to a limit point @5 5 if the

distance tends to # as n increases. -e can then sho= at once

@7olume % that5 iff@x5y l as @x5y @,5then for every

seuence of points @xn5yn in/ =hich tends to @5 .*he converse is also true5 if

e1ists and is eual to lfor every seuence @xn5yn of points in/tendin+ to

@,, then the doule limit off@x5y asxand y e1ists and is eual to %. -eomit the proof of this statement.

6n our definition of a limit5 =e have allo=ed the point @x5y to vary in the re+ion/.>o=ever5 if =e so desire5 =e can impose restrictions on the point @x5y. 0or e1ample5 =e

may reuire it to lie in a su,re+ion/!of/,or on a curve (5 or in a set of points1 in/.6n this case5 =e say thatf@x5y tends to las @x5y tends to @5 in/@or on (5 or in1.

;aturally5 it is implied that/!@or (,or1 must contain points aritrarily close to @5 in order that the definition may e applicale.

Our definition of continuity then implies the t=o follo=in+ reuirements:

@% As @x,ytends to @, in/,the functionf@x,y) must possess a limitl?

@! this limit l must coincide =ith the value of the function at the point @, .

6t is ovious that =e could define in the same manner the continuity of a function notonly in a re+ion/5 ut also5 for e1ample5 alon+ a curve (.

2.2.+T'e 4rder to ('ic' a )unction #anis'es @may e omitted at a first readin+.&6f

the functionfXx,y is continuous at the point @5 5 the differencef@x5y ,f@, tends to

ero asxtends to andyto. Ly introducin+ the ne= variales 7 Hx - and 7 y, 5

=e can e1press this statement as follo=s: *he function @%5 7 Hf@" %, " 7),f@5 )of the variales %and 7tends to #.
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-e shall freuently encounter functions such as @%5 7 =hich tend to ero G as do %and7. As in the case of one independent variale5 it is useful for many purposes to descrie

the ehaviour of @%5 7 as % # and 7# more precisely y distin+uishin+ et=een

different orders of #anis'ingor orders of magnitude of @%5 7. 0or this purpose5 =ease our comparisons on the distance

of the point =ith co,ordinatesx %andy 7from the point =ith co,ordinates

and 5 and ma3e the statement:

A function @h5 3 vanishes as # to the same order as or5 more

precisely5 to at least the same order as5 provided that there is a constant (5 independentof %and 75 such that the ineuality

holds for all sufficiently small values of5 or5 more precisely5 =hen there is a M # such

that the ineuality holds for all values of %and 7such that .

"oreover5 =e say that @h5 3 vanishes to a hi+her orderGG thanif the uotient @h5

3/ tends to # as#. *his is sometimes e1pressed y the symbolic notation @h5 3 H

o@GGG.

G 6n some =or35 the phrases @%, 7beco$es infinitely s$all as % and 7 door @%, 7 is infinitesimalare alsofound. *hese statements have a perfectly definite meanin+5 if =e re+ard them simply as another =ay of

sayin+ @%, 7 tends to # =ith %and 7. ;evertheless5 =e prefer to alto+ether avoid the misleadin+ e1pressioninfinitely small.

GG6n order to avoid confusion5 =e point out e1pressly that a hi+her order of vanishin+ for #implies

smaller values in the nei+hourhood of#? for e1ample5 vanishes to a hi+her order than,since is

smaller than,=henis nearly ero.

GGG;aturally5 the letter ois chosen5 ecause it is the first letter of the =ord order.6f =e =ish to e1press the

statement that @%,7 vanishes to at least the same order as,hut not necessarily to a hi+her order5 some

authors the letter instead of o5 =ritin+ @h53 H @.-e shall not use this symol.

8onsider no= a fe= e1amples. Since


59/735

the components %and 7of the distancein the directions of thex-andy,a1es vanish to at

least the same order as the distance itself. *he same is true for a linear 'omogeneousfunctiona%b7=ith constants aand bor for the functionsin@%/. 0or fi1ed values of

lar+er than %5 the po=ernof the distance vanishes to a hi+her order than?

symolically5 H o@ for M %. Similarly5 a homo+eneous uadratic polynomial a%

b%7 c7 in the variales %and 7vanishes to a hi+her order than as #:

"ore +enerally5 the follo=in+ definition is used. 6f the com"arison function @%5 7)is

defined for all non,ero values of @%,7 in a sufficiently small circle aout the ori+in andis not eual to ero5 then


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!. >o= many constants does the +eneral form of a polynomialP@x,y of de+ree ncontainU

E. 2rove that the e1pression

vanishes atxy# to at least the same order asE@x y E/!.

F. 0ind the condition that the polynomial

is of e1actly the same order as in the nei+hourhood ofx#5y# @i.e.5 othP/ and

/Pare ounded.

'. Are the follo=in+ functions continuous atxHy#U

(. 0ind a @ @!.!.% for those functions of 1. ' =hich are continuous.

>ints and ans=ers

2.+ TH DRIVATIV, 4) A )65CTI45
http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/02.htm#2.2.1%20Definition%232.2.1%20Definitionhttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/anshints.htm#2.1http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/02.htm#2.2.1%20Definition%232.2.1%20Definitionhttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/anshints.htm#2.1


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2.+.$ Definition. Geometrical Re"resentation&6f =e assi+n in a function of severalvariales definite numerical values to all ut one of the variales and allo= only thatvariale5 sayx5 to vary5 the function ecomes a function of one variale. 8onsider afunction u f@x,y of the t=o varialesxandyand assi+n toythe definite fi1ed valueyy#c.*he function uf@x,y# of the sin+le varialex5 =hich is thus formed5 may erepresented +eometrically y lettin+ the planeyy#cut the surface uf@x,y @0i+s. (/$.*he curve of the intersection thus formed is represented in the plane y the euation uf@x,y#. 6f =e differentiate this function in the usual =ay at the pointxHx#@=e assumethat the derivative e1ists5 =e otain the"artial deri#ati#e off7xy8 (it' res"ect toxatthe point @x#,y#. Accordin+ to the usual definition of the derivative5 this is the limit

)eometrically spea3in+5 this partial derivative denotes the tan+ent of the an+le et=een aparallel to thex,a1is and the tan+ent line to the curve uf@x,y#. >ence it is the slo"e oft'e surface u=f7xy8 in t'e direction of t'ex%axis. 6f @x#,y# is a point on the oundary

of the re+ion of definition5 =e ma3e the restriction that5 in the passa+e to the limit5 thepoint @x %5 y# must al=ays remain in the re+ion.

Several different notationsare used for these partial derivatives such as


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6f =e =ish to emphasie that the partial derivative is the limit of a difference uotient5 =edenote it y

-e use here a special round letter instead of the ordinary dused in the differentiationof functions of one #ariable5 in order to sho= that =e are dealin+ =ith a function ofseveral variales and are differentiatin+ =ith respect to one of them. 6t is sometimesconvenient to use Canc'y:s symbolDand =rite

ut =e shall seldom use this symol.

6n e1actly the same =ay5 =e define the partial derivative off@x,y =ith respect toyat thepoint @x#,y# y the relation

*his represents the slope of the curve of intersection of the surface uHf@x,y =ith theplanexx#5 perpendicular to thex,a1is.


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and5 in +eneral5 the n,th derivatives y

6n practice5 the performance of partial differentiations involves nothin+ =hich the studenthas not encountered already5 ecause5 accordin+ to the definition5 all the independentvariales are to e 3ept constant e1cept the one =ith respect to =hich =e aredifferentiatin+. -e therefore must merely re+ard the other variales as constants andcarry out the differentiation accordin+ to the rules y =hich =e differentiate functions ofa sin+le independent variale. *he student may nevertheless find it helpful to study thee1amples of partial differentiation +iven in 7olume %.

Vust as in the case of one independent variale5 the possession of derivatives is a specialproperty of a function5 not even en4oyed y all continuous functions. *he termdifferentiable is considered in detail in !.F.%All the same5 this property is possessed yall functions of practical importance5 e1cludin+ perhaps isolated exce"tional "oints.

2.+.2. Continuity and t'e xistence of Partial Deri#ati#es (it' res"ect to xandy&6nthe case of functions of a sin+le variale5 =e 3no= that the e1istence of the derivative ofa function at a point implies the continuity of the function at that point @7olume %. 6ncontrast5 the possession of partial derivatives does notimply the continuityof a functionof t=o variales? for e1ample5 the function u@x,y H !1y/@x" y,=ith u@#5# H # haspartial derivatives every=here and yet =e have already seen in !.!.!that it isdiscontinuous at t'e origin. )eometrically spea3in+5 the e1istence of partial derivatives
http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat6/calcul10.htm#10.3.1%20Definition.%20Geometrical%20Representation:http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/02.htm#2.4.1.%20The%20Concept%20of%20Differentiability%232.4.1.%20The%20Concept%20of%20Differentiabilityhttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/02.htm#2.3%20THE%20DERIVTATIVES%20OF%20A%20FUNCTION%232.3%20THE%20DERIVTATIVES%20OF%20A%20FUNCTIONhttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/02.htm#2.2.2%20The%20Concept%20of%20limit%20in%20the%20Case%20of%20Several%20Variables:%232.2.2%20The%20Concept%20of%20limit%20in%20the%20Case%20of%20Several%20Variables:http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat6/calcul10.htm#10.3.1%20Definition.%20Geometrical%20Representation:http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/02.htm#2.4.1.%20The%20Concept%20of%20Differentiability%232.4.1.%20The%20Concept%20of%20Differentiabilityhttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/02.htm#2.3%20THE%20DERIVTATIVES%20OF%20A%20FUNCTION%232.3%20THE%20DERIVTATIVES%20OF%20A%20FUNCTIONhttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/02.htm#2.2.2%20The%20Concept%20of%20limit%20in%20the%20Case%20of%20Several%20Variables:%232.2.2%20The%20Concept%20of%20limit%20in%20the%20Case%20of%20Several%20Variables:


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respect toyand then =ith respect tox.*his oservation depends on the follo=in+important t'eorem:

6f the mi1ed partial derivativesfxyandfyxof a functionf#x, y) are continuous in a re+ion/,then the euation

holds throu+hout the interior of that re+ion5 that is5 the order of differentiation =ithrespect toxand toyis immaterial.

*he proof5 li3e that in the precedin+ section5 is ased on the mean #alue t'eorem of thedifferential calculus. -e consider the four points @x5y,@x %5y,@x,y 7 and @x %,y

7),=here %# and 7 #.6f @x5y is an interior point of the re+ion/and %and 7aresmall enou+h5 all these four points elon+ to/.-e no= form the e1pression

6ntroducin+ the function

of the varialexand re+ardin+ the varialeymerely as a "arameter5 =e can =rite thise1pression in the form

*ransformin+ the ri+ht,hand side y means of the ordinary mean #alue t'eorem of thedifferential calculus5 =e otain

=here # N N %. >o=ever5 the definition of @x yields

since =e have assumed that the mixed second order "artial deri#ati#efyxdoes e1ist5 =ecan a+ain apply the mean #alue t'eoremand find that


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=here and !denote t=o numers et=een # and %.

6n e1actly the same =ay5 =e can start =ith the function

and represent&y the euation

-e thus arrive at the euation

and5 euatin+ the t=o e1pressions for&, otain

6f =e no= let %and 7tend simultaneously to # and recall that the derivativesf1y@x5y andfy1@x5y are continuous at the point @x5y,=e otain immediately

=hich =as to e proved.

0or more refined investi+ations5 it is often useful to 3no= that the t'eorem on t'e re#ersibility of

t'e order of differentiation can e proved =ith =ea3er assumptions. 6n fact5 it is sufficient to

assume that5 in addition to the first partial derivativesfxandfy5 only one mixed "artial deri#ati#e,sayf1y5 e1ists,and that this derivative is continuous at the point in uestion.6n order to prove this5 =e return

to the aove euation

divide y %7and then let 7alone tend to #. *hen the ri+ht,hand side has a limit5 =hence also the left,hand

side has a limit and

0urther5 it has een proved aove =ith the sole assumption thatf1ye1ists that


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6t is of fundamental interest to sho= bymeans of an e1ample that5 in the asence of the assumption of the

continuity of the second derivativefxyorfyx5 the theorem need not e true and that5 on the contrary5 the t=o

derivatives can differ. An e1ample is +iven y the functionf@x,y H 1y@x,y/@x " y5f@#5 # #,for

=hich all the partial derivatives of second order e1ist5 ut are not continuous. -e find that

=hence

*hese t=o e1pressions differ5 =hich5 y the aove theorem5 can only e due to the discontinuity offxyat theori+in.

-ith our assumptions aout continuity5 a function of t=o variales has t'reepartialderivatives of the second order

fourpartial derivatives of the t'irdorder5

and5 in +eneral5 7n> $8partial derivatives of the n%t'order5

Oviously5 similar statements also hold for functions of more than t=o independentvariales. 6n fact5 =e can apply our proof eually =ell .to the interchan+e ofdifferentiations =ith respect toxand or =ith respect toyandz,etc.5 ecause eachinterchan+e of t=o successive differentiations involves only t(o independent variales ata time.

xercises 2.2

%. >o= many n,th derivatives has a function of three varialesU

!. 2rove that the function


70/735

satisfies the euation

E. 8alculate

F. 2rove that

'. 8onsiderin+

as a function of the nine variales a,b5 . . . 5 7,prove that

Hints and Ans(ers

2./ TH T4TA1 DI))R5TIA1 4) A )65CTI45 A5D IT, G4-TRICA1

-A5I5G
http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/anshints.htm#2.2http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/anshints.htm#2.2


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2./.$. T'e Conce"t of Differentiability&. 6n the case of functions of one variale5 thee1istence of a derivative is intimately interlin3ed =ith the possiility of appro1imatin+

the function Hf@ in the nei+hourhood of the pointxy a linear function H @P*his linear function is defined y the euation

)eometrically spea3in+ @and ein+ current co,ordinates5 this represents the tan+ent

to the curve Hf@ at the pointP=ith the co,ordinates xand f@x),analytically

spea3in+5 its characteristic feature is that it differs from the functionf@ in the

nei+hourhood ofPy a uantity o@% of hi+her order than the ascissa %H @-x)@!.E.E. >ence

or5 other=ise5

=here denotes a uantity =hich tends to ero =ith %. *he term %f!@x,the linear "art ofthe increment off@x correspondin+ to an increment of %in the independent variale5 =e

have already @!.E.9 called the differentialof the functionf#x)and have denoted it y

@or also y dyHHy!dx,since for the functionyxthe differential has the value dydx

%%.-e can no= say that this differential is a function of the t=o independent varialesxand %5 and need not restrict the variale %in any =ay. Of course5 this concept ofdifferential is5 as a rule5 only used =hen %is small5 so that the differential %f!@x forms anappro1imation to the differencef@x%-f#x)=hich is accurate enou+h for the particularpurpose.

8onversely5 instead of e+innin+ =ith the notion of the derivative5 =e could have laid the

emphasis on the reuirement that it should e possile to appro1imate the function H

f@ in the nei+hourhood of the pointPy a linear function such that the differenceet=een the function and the linear appro1imation vanishes to a hi+her order than theincrement %of the independent variale. 6n other =ords5 =e =ould reuire that there

should e1ist for the functionf@ at the point x a uantity&,dependin+ onxut noton %,such that
http://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/02.htm#2.3.3.%20Change%20of%20the%20Order%20of%20Differentiation%232.3.3.%20Change%20of%20the%20Order%20of%20Differentiationhttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat6/calc2.htm#2.3.9%20The%20Approximate%20Representation%20of%20Arbitrary%20Functions%20by%20Linear%20Functions.%20Differentiationhttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat9/02.htm#2.3.3.%20Change%20of%20the%20Order%20of%20Differentiation%232.3.3.%20Change%20of%20the%20Order%20of%20Differentiationhttp://mpec.sc.mahidol.ac.th/RADOK/physmath/mat6/calc2.htm#2.3.9%20The%20Approximate%20Representation%20of%20Arbitrary%20Functions%20by%20Linear%20Functions.%20Differentiation


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=here tends to # =ith 7.*his condition is euivalent to the reuirement thatf@x shall edifferentiableat the pointx? the uantity&must then e ta3en as the derivativef!@x atthe pointx.-e see this immediate

calcul diferential si integral 2

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