partial differentiatiion

8/12/2019 Partial Differentiatiion

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UNIT PARTIAL DIFFERENTI TIION

5 1 IntroductionObjectives5.2 Functions of Several Variables5.3 Limits and Continuity5.4 Partial Derivatives5 5 Th e Total Differential5.6 The Chain Rule

5 6 1 Chain Rule for Functions of two Variables5 6 2 Implicit Differentiation

5.7 Maxima and Minima5.8 Jacobians5.9 Surfaces5.10 Summary5.11 Solutions/Answers

5 1 INTRODUCTIONIn Unit 1 you learnt about the concept of functions of one independant variable and someoperation s concerning functions of one variable. In many physical situations you feel theabsolute necessity to express quan tities of interest in terms of two three or more variables.In such circumstances you feel the need of extending the concept of functions of onevariable to functions of more than on e variable.In this unit you will learn how to extend concep ts you already learnt conce rning functionsof o ne variable to functions of more than one variable. You will also learn some analogousresults wh en functions of one variable are extended to functions of more than one variablein genera l and tw o or th ree variable s in particular.

Afte r reading this unit you should be able tofind partial der ivatives of functions of tw o or three variables.find the total differential of fu nctions of two or three variables.find directional derivative in any given direction.find wher e a function of two variables is having relative ma ximum o r relativeminimum.to evaluate Jacobians of type

a x , y ) ln a x , y , z )a u , v ) a 14 v,

5 2 FUNCTIONS OF SEVER L V RI BLESIn this section you will know how to extend the concept of function of one variable to af u n ~ t i o n f two three or more variables.


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5 21 Functionsof woVariablesR denotes the set of real numbers.Suppose D is a c ollection of pairs of real num bers x , y ) D is a subset of@ ). Then a realvalued fundio n of two variables of f is a rule that assigns to each point x, y ) in D a uniquereal number denoted by f x, y ).The setD is called the domain off.The set x, y ) x , y )E D ),which is the set of values the func tionf takes on, is calledthe range off:W e generally use the letterz to denote the values that a function of two variables takes.Then we have

= f x , y )We call the symbol z the dependent variable off andf as the fuoction of two independentvariables:x and y.xample

The volume of a ri ht circular cone of base radius r and height h is given byV = - l r ? h

Here we call Va s the dependent variable and r and h as the ind epende nt variables.We now de fine a function of three variables as follow sSupposeD be a collection of triple of real numbers x, y, z ). D is a subset of R ~ . )Th en a real valued function of three variablesf is a rule that assigns to each pointx, y, z ) in D unique real number denoted by f x , y, z ).

The set D iscalled the dom ain off .The set

f x , y , z ) : x , y , z ) E D ) ,which is the se t of values the functionf takes on, is called the range offWe often use the letter w to denote the values that a function of three variablestakesThen we have

w = f x , y , z )Here we call the symbol w the dependen t variable of functionf and x, y, z theindependent variables off.

xample2Th e volume of a rectangularbox of sides x, y, z isgiven by

v = x y zHare V is the depend ent variable and x , y, z are the independe nt variables.In general, we can define a function of n variables X I , z, .... x,, in a similarmanner.In this unitwe consider functions of two, three or more variables. These func tionsare obviously single valued.


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5.3 L M TS ND CONTINUITYIn this section, you will learn about the concepts of limit and continuity of functions ofseveral variables. In this unit we confine most of our discussion of functions of severalvariables to those of two variables. We can easily extend these ideas in a similar manner tofunctions of more than two variables.You have learnt the concepts of open and closed intervals in connection with functions ofone variable. Now, we extend these ideas in connection with functions of two variables.Let .Qo ) be a point in D, which is a subset of2Consider the equation

Since x, y ) and xo, yo ) are in R2,we get~ x , Y ) - x a Y o ) l = ~ x - x a Y - Y o ) l = ~ ~ ~ - ~ o ~ ~Y-YO)~ 2)

From 1) and 2), we get

Squaring both sides of 3), we get2~ - x o ) ~ + ~ - y o ) ~r . 4)

We note that 4) represent a circle with centre xo, yo ) and radius r.We also see that the set of points whose coordinates x , y satisfy the inequality

is the set of points in R2 interior to and on the circle given by 4) as shown in Figure 5.1.From the above discussion we formulate the following defmition

i) The open disc D centred at a o )with radius r is the subset of R2 given by~ x , Y ~ : I ~ ~ ~ Y ~ - ~ ~ o ~ y o ~ l ~ ~ ~ .

ii) The closed disc D centred at xc, yo )with radius r is the subset of2 given by{ 4Y): l x ,Y)- xo,Yo) l r .

iii) The boundary of the open or closed disc defined in i) or ii) is the circle{ x,Y):I x,Y)- XO,YO)I r}.

iv) A neighbourhood of a point .Q o) inR2 is an open disc centred atxo, yo .


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We na te thatm open disc does not contain any point on its boundary just a s an openinterval does not contain its end points. A closed disc contains all points on its boundary justas the ose interval contaius all its boundary points.Unit

If the values of the functionz = x , y ) can be made as close as w e like t a fixednumber L by taking the point x , y ) close t he poin t xo, yo ), but not equal tx a y ), then we say that L is the limit of as the point x , y ) approaches thepoin t (xo ,~0) .We write this in symbols s

We read this as he limit off as x , y ) approaches (xo , yo ) is L.l r lh~s i iud is tanceof x ,y )~m xo ,yo)mmely) l r xo12 0 - yo)is small in some sense, then we mean that 4 y ) s close to xo, yo ).Since

and

we have

which will be small if 6 is sufficiently small. Therefore when we calculate limits,we may think either in terms of distance in the plane o r in terms of differences inwordhates as shown n Figure5 2below.

TbeopensquarcIx-xoI< b,(y- yoJ


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Tbe limit off x , y ) as x y ) approaches xo, y ) is the numberL iffor any e > 0, the re exists a 8 >O such that for all points either( i ) 0 < 4 ( x - xo12 0 - yo)2 < ti i m p ~ i e g t h a t l f x , y ) - L I < E , O ~( i i ) O < I x - xo I < ba ndO < I y-yoI


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(iv) lim [K f x, y ) ] K L 1 K being any number, andxy)-txo,h)

functionf (x, y ) is said tobe continuous at a point a,o ) if(i) f is defined at ( a ,yo )(ii) lim f x, y ) exists, and(+Y)-.( ,Yo)(iii) lim fC r, y) =f( a, yo)( x Y ) ~ . Y ~

xampleDetermine the points at whichf (x, y) is continuous if

olutionWe approach (0,O) along the line y = m x, m being an arbitrary constant. Then

x m x mlim f (x, y) = l rrp 1mcry - 3 ) v)40 x ~ y )o.o) 2 1 m2) =GHere we get different limits for different values ofmTherefore, lim f (x, y ) does not exist .

( r y ) - ( O , O )So is not continuous at O,0 )when( b )* (0 ,0 ) , h 2 *Oand

ablim f(x,y) = a2 b2 f ( a , b )(+Y)Therefore is continuous at all points U b ) * O, 0). So we conclude thatf iscontinuous at all points except the origin.Here we make a note f one or more of three conditions in the definition ofcontinuity off (x,y ) fails to hold, then is discontinuous at the point underconsideration.

ElTest the continuity at the origin for the Function


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Determine the po ntsof continuity of the functionif x, 090)

5.4 P RTI L DERIV TIVESIn t is section you will learn how to extend the concept of derivative of a function of onevariable to functions of more than one variable For simplicity we consider here thefunctions of two variables only. Based on similar arguments we can get the correspondingresults involving functions of three or more variables.

Take two pointsPo xo ,y and P XI, l ofD with value atPo and wl at PIrespectively as shown in Figure5 3 Then the increment in w in going fiom rn at Po to wl

Aw = Wl-wo = f x1,y1)-f xo,yo). 1)corresponding toAr = XI - o and Ay = yl mWe keep b ixed and makeP approach

Y A

m- 53some specific. mooth curve in the x y plane We now s u h e hatP approaches

d w A w= lim lim f x l * Y l ) - f x o , Y o ) 2)d s P I P O A s PI- ~ A X ) ~ + A ~ ) ~


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exists, itsvalue is called the directional derivative of w = f x,y ) at xo,yo ) in thedirection ofL. Herewe use the adjedive 'directional' becruse the answer in 2) depends onthe functionatPI , he pomt Po and the direction in whichPI approaches Po. We will takeup two special cases of the directional derivative. nSection5.7, we discuss e general caseof the directional derivative n any dhection.We f h t makePI approachPo along the liney = yo parallel to the x-axis. Seumdly wemakeP I approachPo along the line x =xo parallel to the y-axis. In the first case wherePIapproachesPo along the liney = yo we get

We d l he resulting limit the partial derivative of w = f x, y ) with respect to x atPo xo,yo ) From (3)we note that this is just the ordinary derivative with respect to x of theEunctionf (x ) =f A o obtained fromf (x, y )by holding the variable y as constant. Thisalso measures the instantaneous rate of change at Po ofthe function w = f x y )per unit

a wchange in We use the notation r to denote the partial derivative of w with respecta xtox By deleting the subscript0 every where in 3) ,we obtain the partial derivative of w atx y )and the same is given by

We also note that in order to calculate such a partial derivative iom the equation for w, wesimply apply the rules of ordinary differentiation, treating y as constant.Considering the second case and proceeding o similar lines, we get the partial derivatives

a wof w =f 4 )with respect toy denoted by f or sa

We make the fdlowing observations(1) The definition allows us to calculate partial derivatives much in the same way as wecalculate the ordinary derivatives by allowing only one of the variables to vary at a

time. Sowecan use all those formulae learnt on differentiation of functions of onevariable forpartial derivatives.

a w a w) The partial derivatives nd -give us the rate of change of w with respect toa x a yeach of the variables x andy with the other one held constant. We will discuss inSection5.7 about how w changes when bothx and y change simultaneously.

a w a w3 ) Even though we calculate -and -by holding one of the variables as constant,a x a ywe note that each is a function of both the variables x and y.

Nowwe can easily obtain the partial derivatives of a sum, difference, product and quotientof two functions much in the same way as that of ordinary derivatives in view ofobsewadon1 Weh n now easily extend the definition of partial derivativesto functions ofthree ormore variables.igherOrderPnrtiolDerivatives: he partial derivatives of a given function of two

variablesrre also functions of two variables and we can some timesdifferentiate themfurther to get higher order partial derivatives.Let w = x , y )and its partial derivatives be defined over a domainD. Then for 4 y ) lwe define the second order partial derivatives as


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a2w a wa i - 5 . - ( a x ) .

If w = 5 y and its first and s ew nd o rder partial derivatives are defined over a domainDthen for 5 y ED w e define third order partial derivatives as

a 2 w a 2 wW e call the partial deriva tives s mixe partial derivatives of second ordera x a y a y a xand a3w a w a3 3 w etc. as the mixed partial derivatives ofa x a y a x a x a ' a y 2 a x g a y a x a ythird order.On sim ilar lines we can defme partial derivatives of orders higher than three.xample

Obtain all the second or der partial derivatives for w = yZ xySSolution

Find ux,u,, u,, u iri nd u, if u = log(ux by), where a and are arbitraryconstants.


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EObtain all thenine secondorder partial derivativesandverify k t ll the threepairs of mixed partial derivatives areequal if u = n S ?jcz.

We note that if w = x, y ) a d ts partial detivatives are continuous, the order ofdifferentiation s immaterial that is

xample6 :Considerz =x cosy y cos

a z= cosy y s h x , - x s i n y - a x ,a x a

2 zzWe note here that a x a y a y a x5.5 THE TOTAL DIFFERENTIALWe firstprove the increment theorem for the functionof two variables.

Let the functionw = f x,y be continuous and possesspartial derivativesthroughouttheregioaR: x-xo


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We can use mean value theorem of a function of one variable as all conditionsrequired are now satisfied.We note that the increment Aw is the change inf fromA xo , ) toB(xo+ Ax,yo+ Ay)inR.We write

Aw = Awl Aw;where

Awl = f(xo+Ax,Yo)-f(xGYo) 5 )and

Here Awl is the change in w fromA to C and Awz is the change in w from C to B.In Awl ,we hold y =yo fixed and have an increment of a function of x that iscontinuous and differentiable. Applying mean value theorem, we get

Awl = ~ ( X O + A X , ~ O ) - ~ ( . ~ ~ , Y O )x ( x i , ~ o ) ~ 7)for some xl between ro and xo Ax i.e. xo


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and

where ~1 and EZ both approach zero as Ax and Ay approach zero.Making use of 11) and (11 ) in 9),we get

AW = f , x q y o ) A x + f , x q y o ) A y + ~ 1 A ~ + ~ 2A ywhere^^ and ez+Oas Axand Ay-0.Now we can extend (2) easily to functions of three variables as follows :Let w=f (x, y, z )be continuous and possess partial derivativesf,, fv , i at and insome neighbourh60d o f a e point m yo, zo )whose partial derivatives arecontinuous at.that point. Then we have

where EI ~ 2 ,3 0 as AX,Ay, A 0.Now we define the differentialof w = f x, y )by the formula

dw = fxAx+fYAy.We can write, in view of (3), that

In the increment Ax and A y are small, then the differential dw is a goodapproximation for the change Aw in the value of w = f (x, y) as we move from thepoint(x,y)to (x+A x, y+Ay),since

Aw - d w E ~ A X EZAY - 0 as Ax, Ay O .As in the case of functions of one variable, writing Ax and A y as x and dyrespectively, we get

It is of interest to note here that the function w = f x, y ) is differentiable inRwhenever (13) holds.Similarly, for a function of three variables w = f (x, y, z ), we defme thedifferential w by

Note :aw w wIn (14), the separate terms ,- y and - z are sometimes calledax z

partial differentials of w with respect to x, y and z respectively.We can then call the sum of these partial differentials of w as the total differentialdw So dw iscalled as the total difPerentialof w, when w is a function of morethan one variable.

xample :Find the tot l differential of w=x sin y- sin x.


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Solutionw = , y - y e o s x , w =xmsy s inxa x a y

To give you an idea about the use of differentials, we consider a rectangle withsides x and y. The area of this rectangle is

If we increase the dimensions of the rectangle to x Ax and y Ay, the change inarea is

The differential estimate for this change in area isr :

The error of our estimate, the difference between the actual change and theestimated change, is the difference d = x Ay

xampleUse the differential to estimate

SolutionWe know

= 5 Vian = 10.What we need to find is an estimate for the increase of

x, y) = xH yY3fromx=25, y = OOOtox=27, y=1021.

gives1 1 45d f = - ~ - ~ ~ ~ d x + - xdy.2 3

Putting x =25, y = 1000, dx = 2, dy = 21, we have

Hence


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DiilerQ0l.l EFind the total differential of w 2 2xy y2.

Use differentials to fmd the approximate value of m

EEstimate by a differential the change in the volume of a right circular cylinder ifthe height is increased from 12to 12 1 m and the radius is decreased from 6 to 5.8cm

5 6 THE CHAIN RULEIn this sedion we obtain the chain rules for finding partial derivatives of functions of two orthree variables. In a similar way we can get the chain rules of partial differentiation forfunctions of more than three variables.

5.6.1 Chain Rule for Functions of Tw o VariablesIf w x, y) has continuous partial derivatives x and f nd if x x t), y y t) aredifferentiable functions oft, then the compositew x t), y t)) is a differentiablefunction of 1In this case we get,

Wetaketoasanyvalueoft.Thenxo=x to)andyo=y to).Ifwenowtake Ax, Ayaschanges that muin x and y when t is changed from to to to A1 we then get from 13) ofSection 5 5 that

where el p O as Ax, Ay4


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Dividing both sides of (2)by At ,we get

We also note that, as f 0,

So we get

The equations we want to prove are simply the statements that (4) holds at every admissiblevalue oft, which infact it does.We extend (4) to functions of three variables as follows ;If w = ( y , z ) hascontinuouspartial derivativesandx=x t),y=y t),z=z t )aredifferentiable functions oft, then the compositef ( x ( ), y (t ), z t ) ) is a differentiablefunction of t. In this case we get

If w =f (x, y ) and if x =x r s ),y =y r, s ), re any functions, then the compositef x r, ), y r, s ) ),when defined, is a function of r and s Ifx, y andf have continuouspartial derivatives, then the partial derivatives of w with respect to r and s exist and in thiscase we get .

aw a x aw*- - (6)ar ax ar arand

aw a ~ a x + a w a y 9as ax s asWe can extend (6) and (7) to functions of three variables as followsIfw=f (x,y,z)andifx - x( r , s ) , y = y(r ,s ) , z = z r,s)areanyfunctions,thenthecomposite function w = f x (r, s), y (r, s), z (r, s)), when defined, is a function of r and s. Ifx y, z and f have continuous partial derivatives, then the partial derivatives of w with respectto r and s exist and we get

~ a w a x + a w a y + a ~ a ~ar ax ar ar az ar 8)aw a x a w * a w a z- -- - - -a~ ax a~ a~ a~ a~ 9)

To show what variables are assumed independent in computing a partial derivative, weemploy the following notations.

8wmeans -with x and y independent.ax

x and z independent.

with y,x and t independent.


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xample9I fw ~ + y - r + s h t a n d x + y t,Bad r) and ii)ax)yJ z)4z

olutioni) Witb x, y, z independent, we have

t x+y, w 2+y-z+sin x+y).Therefore

ii) Witb x t, ndependent, we havey t -X,W g+ t -x ) +s i nt .

Thus

olution

Therefore we have

Similarlywe have


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If z e a function of x andy and x = - , = ,prove that

~f v = rove that2,

XHint:TakeE = - q = : then v = f 5 , q )z '

E l lIff x y ) 0 , rove that


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5.6 2 Implicit ifferentiationYou would recall that the method of implicit differentiation was used in Unit 2 to find thederivative of a function defined implicitly by an equation. In this section you will see howthe derivative of an implicitly defmed function can be obtained through the use of partialderivatives.Let

whereY = Q , ( x )

By the chain rule (withx = x, y = x)) ,we get

e ave assumed that y is a differentiable function of x.Example

If Q is a differentiable function such that y = Q ( x ) satisfies the equation+ y sinxy = 0 ,

fmd rSoluthon

Let

Then

Hence

Find ,given .haty = 4 r ) satisfies each equation.a5i) x s e c y + y s e c x - 6 = 0

(ii) 21ny- 2 4.


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PuttlDaPe renti.t&5 7 M XIM ND MINIMIn this section you will learn about the maxima and minima of functions of two variables.We give some definitions to startwith.

A function of two variablesf x , y ) is said to have an absolute or globalmaximum at Crayo in a region iff x , y ) s f xo,yo)forall x,y)inR.An absolute or global minimum of f in occursat xo, yo ) if

f x ,y) f xo,yo) forall 4 x 9 ~ n R

Definition2 :

A relative or localmaximumoff occursat xo yo ) if a circleD aboutm,yo)eXistswithf x,y) sf (xo,yo)forall(x,y)intheinteri~~ofD.

A relative or localminimum off occursat ~ a o)if a circleD about~~,yo)existswithfx,y) zf(xo,yo)forafl(x,y)intheinteriorofD.We refer to both maxima and minima as extrpma,and relative maximumand relative minimum as relative e m m a

A functionf of two variables is said to have a crltic lpoint at xo, y )inthe domain off if eitheri) both partial derivatives of fare zero at xo, yo )

orii) at least one of tpe partial derivatives fails to exist at xo yo ).

Suppose that a relative maximum value of the function z =f x , y ) occursat aninterior point a, b ) of the domain of the function and that t and both exist

x ayat a, b ) as shown in the figure55.Then we observe the followingf cts :1) x =a is an interior point of the domain of the curve I =f x, b ) in which

theplaney= bcutsthesurfacez= f x, y).2) The function z =f x, b )bas a relative maximum at x = a.3) The value of the derivative of z=f x , b ) at x = a is therefore zero.

Since this derivative is precisely q 4 b ) ,we condude thatOn the lines of similar argument with the function z =f a, y ), we get

Thuswehavefx a, b) =O = f , a , b).So we have a necessary condition forf to have an extreme value at an interior pointa, 6 ) :


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efinition

Th e point where there is no relative maximum or no rela tive minimumbut x and f are both zero is called a saddlepo nt

We apply the second derivative test to determine whether a functionz - x , y ) has a relative maximum or minimum va lue at a point P a, b ) wherex and f both vanish.We assume thatf and its first and second order partial derivatives are continuou s insome region R about P a, ) as shown in the figure 5.6. We takeS a h , b k ) to be a point close enough toP so that PS ies in R. We take theparametric equation of P o be

W e study the values of x y ) along P S by con sidering the functionF t ) = f a + h t , b + k t )

W e know hat F is a differentiable function of t since the first partial derivativesare con tinuous and x = a th, y = b tk are differentiab le functions of t. Using


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\the chain rule, we g et

Further, we know thatF ' is continuous on the closed interval 0 1 inceand are co ntinuous throughout R. nce again, making use of chain rule, we

aget

F t) h aF t ) aF t )x + k y

Now, F ' ( t ) satisf y the cond itions of Taylor s theorem on the closed interval[ 0 , l 1 namely, F and F are continuous in ( 0 , l ) and F s differentiable n openinterval ( 0 , l ) . S o w e h av e

f o rs o m e c i n O < c < 1F 1) = f (a h, b k)

F O ) ( a , b )F ( 0 ) y ; ( a , b ) + y , ( a , b )

Substituting these values in 4), we getf (a h, b k) f (a, b) hfx (a, b) k fAa, b)

Supposingfx = 0, f, = 0 at ( a, b ), we wish to determine whetherf (x y ) has amaximum or m inimum at ( a, b ). W e write (5 as

Since a m inimum o r maximum valire of f at (a,b ) depends on the sign off ( a + h , b + k ) - f a,b),weareledtoconsiderthesignof

or, conside ring c very sm all,Q(0) = h2f= (a, b) 2 h kfv a, b) 2fwa, b) 7)

Mu ltiplying both side s of 7) by fn a, b) , we getxQ (0) (hf, + kfv)2 + (firfw - fv 2, 2 8)

The sign of Q(0) can be determined from 7). Consequently we g et the following


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ri for the behaviour of f x, y ) at ( a b ).1) 1 f f a < 0 o r f , < 0 r n d f a f , - f v 2 > 0 at a ,b) , thenQ O) ~ a n d f , f , - f v 2 > O a 4 b) , then Q O ) > O f o r a l lsmall non zerovalues of h and k and f has a relative maximum v alue at ( a b ).(3) If f f,- f < t (a, b ), then it can be shown that there arecombinations of arbitrary small non zero values of h and k for which

Q (0 )>0 and ( 0 ) < 0 .Thus, a rbitrarily close to the pointPo ( 4 b, f a ,b) )onthesur facez f ( y)therearepointsabovePaand also points below Po . The function f therefore has a saddle point at4 ) -

4) Finally, iff= fw - v Q we can draw no con clusion about the sign ofQ ( 0 ) and we r e q u h some other test t o settle this case.ow we give a brief summary of maximum and minimum tests as follows

If z = f 4 ) s continuous, then extreme values of f may occur ati) boundary points of the domain of fii) interior points where r = f y = 0iii) points wherefr orfy fails to exist.

Further, iff has continuous first and second order partial derivatives on some opendisccontaining a, b)andif f r a , b ) = f y 4 b ) = 9 heni) f < orf 0 at a, b ) implies that f has a localmaximum at (a, b ).ii) f >0 or f > 0 and f,f, -f&> 0 a t a , b ) implies that f has localminimum at (a, b ).iii) f, f - g


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Find the maximum and minimum values of xy 7 i .4

Point x x v v - x ~ x x- 163 4

( 0 , - 1 ) 3 4( 1 9 0 ) 3 8

( - 1 , ) 3 - 8

Investigate the m aximum and minimum values of2 y 2 5 x 2 - 8 x y - 5 y 2

5 8 J COBI NSIn this section, you will learn about Jaw bians. Jawbian s play an important role inevaluation of m ultiple integrals when the variables of an integral are changed by a suitabletransformation.If x = g u, v ) and y = h u v be differentiable then th e Jacobian of x and y withrespect to u and v, denoted by is given by( 4 v)

a x a x( x , Y ) = ~ a vu,V ) a u a v

If x g ( u , v, w ) , y h 4 , w), = j ( u , v, w ) aredifferentiable thenwedefmethe Jacobian of the trans form ation Erom a region U in u v w -space to a region W in

a x x a xu a v~ ( x , Y , z ) 2(u, v , W ) a u a v a wt t a z~ ~ ~

max / min / saddle pointsaddle pointminimum value -minimum value - 1maximum value 1maximum value 1


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In a simil rmanner we can defme Jacobians of order higher than three.ote

-is sometimes denoted bya u, V,W)We now prove som e important properties of Jacobians. We prove these propertiesfor Jacobians of second order only. You can get sim ilar such pro perties forJacobian s of order higher than 2

Thearem

thenJJ 1.

C o n s i d e r u = + x , y ) a n d v=Ip x,y).Wecansolvex,yintermsof u nd va nd ge t x = + i y v)and y= Ip i u , v ).We also get

andav avdv r - dyax ay

Sowe get thatau au1 = au0, 1 and -au av av av

Now we have


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av ax av ~ ax av yx au ay au ax av ay av= I

heoremIf u and v are functions o f p and q andp and q are functions of x and y then

h o fSince and v are functions o f p and q we get

We also get

and

Now consider

by interchanging rows and columns of the second determinant.

So we get now


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If the functionsu and v of two independent variables x and y be such that4 ) 0, thena 0.a x,Y)

Proofincef u , v)=O,wehave

and

Eliminating a rom the above two equationsa ~ vwe get

by interchanging rows and columns of the determinantTherefore,

xample 3I f x= rw sO ,y = r s inO, z=z ,f ind a x , Y 2)-a r, 0, Z)

olution

a , , h oa r, 0, Z) a r e a z


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findd ind XI , X Z, 1 3 )X I , ~ 2 , 3 ) Cvl I

find x9YJ nd u, V , Wu, V , W x, Y, 2


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5.9 SURFACESIn t is section, you will leam'about some basic ideas about surfaces in general.We fhst give its definition.

A swface is tbe locus of a point whose cartesian coordinatesx y, z are funetions of two independent parameters u and v.-

So we get

to be the parametric equations of a surface. When we represent a surface in this way, we callthis as the Gaussian form of representing a surface; we also call the two parameten u and vas the c~i l i ni aroordinatesof a current point on a surface.For any point x , , on the surface, we can uniquely determine the values of u and v andwe refer to this as u ).If we eliminate parameters u and v in the parametric equations of a surface, then the relationso obtained is of the form

and this also represents a surface.When a surface is represented in this fom, we call this as the constraint equation of asurface Consider

Eliminatingu, v in I), we get2 - = z .

Next take

Again on elimination of u and v in (2). we get

Finally considerx = u & h v , y = u s i n h v , z = ~

on elimination of u andv m (3), we get2 3 = z. .

We see that the dBerent parametric equations 1), (2) and (3) give rise to thesamesurface2-9 =hmfmewe note that the parametric equations of a given surface are not

necessarily unique. Further, we obsewe that the constraint equation -y2 z ofthe surface represents the whole of i t The parametric equations of the surfacerepresent a portion of i tFur example, the rametric equations(3) of a surface represent only the part ofthesOmce*2-9= z for which z as u is real


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Therefore we note here that the parametric equation and the constraint equation toa surface are not always equivalent.If we write the equation to a surface in the form

z f (x ,y)We ca ll th is as ~o n g e sorm of the equation to a surface.In Mon ge s form of equation to surface, we can regard x and y themselves asparameters

Tangent Plane and omalLine at a Point bn a SotfaceLet

F(x ,y , z )represent a su rfice. We take 5 y, z) and Q x Ax y by, Az as two pointsclose t o each other on the su rface represented by equations (4).Let arc Q be As and chord Q bekW e now hat

From our know ledge of coordinate geometry of threedimension, we now that thedirection satics of Q re


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Axe., ,Wewrite the above as

W e a l s o n o t e t h r t c s d s O , Q P , ~ te ~ t o aangent 1inePT;Now noting that the coordinatesofmy point on arc Q rc functionsofs only weget the direction cosines of PTto be

zs ds sNow differentiating both sides of the equation 4)witb respect to s, we get

Again using our knowledge of coordin tegeometry of three dimensions,we get6) that line with direaim min es ,%, spexpendiculnto the line

a~ a~ aFhaving direction ratiosasSince we can take different curves joining Q to e get a number of tangent

~ a~ aFlines atP and the line with directionr tios will be perpendicularto allax g asuch trngent lines atP. Thus we set that all tangent line at lie in a plant rhrougbPperpendicularto he line with direction ratios

So we get the equation of the tangent plane to surfrce givenby the equation 4) atpoint to be

Where (X, Y Z )are the cwrentcoordinatesof my point on the tangent plane.

rom what we leunt in coordinategeometry of mime dimensions,we get theequation of a plane through a point ( x y zo ) having a linewitbdirection ratios a,b , c asnormal tobe

We also get the equations of a normal to he surfrce atP which is a line through Pperpendicular to the tangent plane atP as

X-x Y-y z-zIx y Ft


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xample14 -3Find the equationsof the tangent plane and the normal to he s m c e yz - a at

xlylzl)whercaisaconstant.

So we get the equation of the tangent plane at anypoint XI yl, ) as

i.e.,xyln+xlyzl+xlylz- 3xlylzlThe equation of normal at xl Y becomes

x1(x-x1)- y1(y-y1)- s(z-21).

Find the equ tionsof the tangent plane a d he nonnal to he s d c e3 6 3 + 9 9 + 4 z = 72 at(O.2.3).

5.10 SUMM RY1) f 4 y ) is said to tend to the limitL as (x, y) xa yo , written as lim - L, if for anyk Y ) ~ f l s Y )e > 0 there exists a 8O such that either

(2) f x, y ) is said to be continuous at a point (xa yo if( i ) isdefinedat(xo,yo)

ii ) lim f (x. y) existsC ~ ; Y - . ~ Y .

w(3) The partid derivative of w = f (x, y) with respect tox , enotedby f or- sa xa w x +A X Y )- (x, y )- - l h f (a x AX 9Ax-0

provided the limit exists.(4) The second order partid derivatives of w - f x, y)arefn,fm f* fpwheae


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5) The differential of w f (x,y ) is defined by the formula d w f dr + h y.6) If w f (x,y, z ) and i f x = x (r,s) , y = y (r,s), z z r,s , then

7) I f f (x,y ) = 0 and if y is a function of x ,then

8) I f f ( x y ) has continuous first and second order partial derivatives on s om e open d isccontaining ( a , b ) and iff , (a,b ) = 0 = f a ,b , heni) f, 0 at (a,b = f has a local maximum at ( a, b ).ii) f > 0 and f,h - > 0 at a, b )*f has a local minimum at ( a, b ).iii) f, f - < 0 a t (a,6 he point ( a , b ) s a saddle point off , .e., a, b ) s

neither a maximum nor a minimum.iv) f, f - = 0 at (a,6 the test is in conclusive; this requires furtherinvestigation.

9 . I f x = g ( u , v , w ) , y = h ( u , v , w ) , z = j u,v,w)aredifferentiable,thentbeJacobian from a region U n uvwsp ce to a region Win y z-space, denoted byo s defined asa 4 w)

(10) Th e equation of the tangent plane to tbe surface f x , , z ) = 0 a t the point5 y , z ) i s

(11) Th e equation of the nonnal line t he surfacef x, y, = 0 at the point x, y, z ) areX x Y - y Z - zIb f i

5.11 SOLUTIONS / ANSWERSl

x, y ) is discontinuous at the origin (0 ,0).

f x, y ) is continuous at all points except at the origin ( 0 ,0 . .


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2E3 a b ax = y =a + y ax +byu V = - anda f y = -(ux+by)

3u = -2Ozx +2p, u . = 6xy, u, = 0,2u, = 3y2+2xz, uu, = -5x4+2xy, uy, = 2 p = 3y az, a = -5x4+2xy

and u, = 2

V = nr 2h dV = V , d r + V , = 2zrhdr+m2dhor r = 12, h = 6, dr = 0.1 and h = 0.2,dV = 2nx12x6x0.1-nx12x12x0.2

.I4 8 ~ 28.8~ -203.

(i) Letf(x,y) xsecy+ysecx-6. . x = secy+ysecx tanx

f = xsecy tany+secxHence fi -secy+ysecx tanx

dr f, secx+xsecytanyii) Leff(x,y) = x 3 ~ n y - y 3 + x 2 - 4

dy fx=3x2ylny+2ryHence -f , ~ ~ - 3 ~ ~

The value of the function tt ins a minimum atx = y = 3 and them n mum valueis 27.


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ElFunction has maximum value 0 at 0,O and minimum value - at d ).

El5

: Equation of tangent plane at 0,2,3) so x - x I ) + ~ ~ ~ - ~ I ) + ~ z - z I )i . e . , 9 y - y l ) + r - z 1 ) =Equation of normal at 0,2,3)s

partial differentiatiion

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