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INTRODUCTION

Analysisis the process or result of modeling and analyzing a phenomenon. In mathematics, itdepends upon the concepts of limits and convergence.

Mathematical analysis has its beginnings in the rigorous formulation of calculus.

UNIT I. DIFFERENTIAL CALCULUS OF FUNCTIONS OF MORE THAN ONE VARIABLE

Objectives pon the end of the unit, a student must be able to!. determine limits and continuity of a function at a point and over a set"#. find partial derivatives $e%plicit, implicit, chain rule, higher&order'

Revie!(uadric )urfaces, *imits, +ontinuity, ifferentiability

O"tline-unctions of More than ne /ariable*imits and +ontinuity0artial erivativesifferentiability and Total ifferential+hain 1uleHigher&order 0artial erivatives

Re#e$ence!#.! to !#.2, T+

%.% F"ncti&ns M&$e t'an One Va$iable

4hat 5e 6no5 $or should7ve 6no5n since MATH !' -unctions of a single variable

( )xfy= yis uni9ue for eachx

If RR:f , then ( )xfy= represents a curve on the plane containing the points( )( )xf,x .

R the 1eal :umber *ine

( ){ }

Ry,xy,xR =#

0lane

!

Analysis

+ontinuity

ifferentiability

Integration

over -:+TI:) on the set of 1eal:umbers or +omple% :umbers

MATH 38

Infinite )eries

Multivariable-unctions

*imits and +ontinuity

ifferentiation

Multiple Integration

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( ){ }Rz,y,xz,y,xR =3 )pace

In general, ( ){ }Rx,...,x,xx,...,x,xR nnn = #!#! or the n (i)ensi&nal

n")be$ s*ace5here ( )nx,...,x,x #! is a point on nR .

Distance in nR ;iven points ( )nx,...,x,xA #! and ( )ny,...,y,yB #! ,

( ) ( ) ( ) ( )#####

!! nn yx...yxyxABd +++= .

)ometimes, ( )ABd is also 5ritten as BA .

In 3R , sets of ordered triples $points' may represent surfaces.

De#initi&n + F"ncti&ns M&$e t'an Va$iable,

A #"ncti&n nva$iablesis a set of ordered pairs ( )w,P , 5here nRP , such that not5o ordered pairs have the same first component. In other 5ords, w is uni9ue for each P .

( )Pfw= w is a function of P :D f set of all admissible P :R f set of all resulting w

If f is a function of nvariable, then the -$a*' f is the set of all points( ) !#!

+ nn Rw,x,...,x,x for 5hich( ) fn Dx,...,x,x #! and ( )nx,...,x,xfw #!= .

A function of # variables, say ( )y,xfz= , represents a surface in 3R . A graph of a function ofmore than # variables cannot be holistically represented.

De#initi&n + Level C"$vesS"$#aces,

The level c"$ve ( )y,xfz= at cz= is the set of all points ( )y,x such that( ) cy,xf = .

The level c"$ve ( )z,y,xfw= at cw= is the set of all points ( )z,y,x such that( ) cz,y,xf = .

#

E/a)*les

!. )phere ( ) #2### =++ zyxz,y,x

#. 0lane ( ){ }

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A c&nt&"$ )a*of a function f is a set of level curves $or level surfaces' by considering

different values of c .

00000000000000000000000000

%.1 Li)its an( C&ntin"ity

4hat 5e 6no5 $or should7ve 6no5n since MATH 3=' De#initi&n Li)it a F"ncti&n

*et f be defined on some open interval containing a e%cept possibly at a , itself.

Then( ) Lxflim

ax=

if for each >> , ho5ever small, there e%ists >> such that

if , then ( ) such

that

if , then ( ) > e%cept

possibly at ( )>> y,x , itself. Then ( ) ( )( ) Ly,xflim

y,xy,x=

>>if for each >> , ho5ever small,

there e%ists >> such that if ( ) ( )

#

>> yyxx , then ( )

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T& (ete$)ine t'e li)it "sin- t'e (e#initi&n2

solve for $dependent on ' such that if , then ( ) > y,x , e%cept possibly at >> y,x itself, and

( ) ( )

( ) Ly,xflim

y,xy,x

=

>>

. Then

> y,x as an accumulation point, then

( ) ( )( )

SinP

y,xflimy,xy,x >> e%ists and is also e9ual to L .

#. If the function f has different limits as ( )y,x approaches >> y,x through t5o

distinct set of points having >> y,x as an accumulation point, then

( ) ( )( ) Ly,xflim

y,xy,x=

>>does not e%ist.

De#initi&n + C&ntin"ity,

*et f be a function of nvariables and nRP . Then f is c&ntin"&"s at P if thefollo5ing conditions are satisfied

i. ( )Pf is defined

ii. ( )Xflim

PX e%ists

iii. ( ) ( )PfXflim

PX=

:ote that ( )nx,...,x,xX #! and ( )na,...,a,aP #! .

T'e&$e)s

!. 0olynomial functions are continuous every5here.

#. 1ational functions are continuous over their respective domains.

3. If gis a function of t5o variables and ( ) ( )( ) by,xglim

y,xy,x=

>>, and f is a functions

of a single variable continuous at b , then ( ) ( )( )( ) ( )bfy,xgflim

y,xy,x=

>>.

2

E/a)*les. @valuate the follo5ing.

!. ( ) ( )( yxxlim

,y,x3#

#

!#

+ 3. ( ) ( )

zyxlim,,z,y,x

+

3# #

23#

#.( ) ( ) xyx

yxlim

,y,x +

#

##

#!!#

E/e$cises. )ho5 that the follo5ing do not e%ist

!.( ) ( ) ##>> yx

yxlim

,y,x

+

#.( ) ( ) ##

3

>> yx

yxylim

,y,x +

+

3.( ) ( ) #> 3# yx

yxlim

,y,x +

E/a)*les.

!.( ) ( ) ##

##

>> yx

yxlim

,y,x +

#.( ) ( ) yx

yxlim

,y,x #

3#

>>

+

3. )ho5 that( ) ( ) ##

##

>> yx

yxlim

,y,x +e%ists.

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Ty*es Disc&ntin"ity

!. Essential if( )Xflim

PX does not e%ist

#. Re)&vable if( )Xflim

PX e%ists but ( )Pf is undefined or( ) ( )PfXflim

PX

T'e&$e)

If f and gare functions on nvariables that are continuous at nRP , then the

follo5ing are also continuous at P gf+ , gf , gf andg

f, provided ( ) >Pg .

=

E/a)*les. etermine 5hether the follo5ing are continuous ordiscontinuous at the given point. Also, if discontinuous, identifythe type of discontinuity.

!. ( )##

>,

#. ( ) ( ) ( )

( ) ( )

=

+

+

=

>>>

>>##

,y,xif

,y,xifyx

yx

y,xf at ( )>>,

E/e$cises. At 5hat points are the follo5ing discontinuousBThen, identify the type of discontinuities.

!. ( ) ( )xylnxy,xf =

#. ( )

=

x

ytanArcy,xg

3. ( )zyx

xyzz,y,xh

++=

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%.5 4a$tial De$ivatives

De#initi&ns

*et f be a function of t5o variables xand y.

The (e$ivative f !it' $es*ect t& x2 denoted by xf , is given by

( ) ( ) ( )

h

y,xfy,hxflimy,xfh

x+

=>

.

The (e$ivative f !it' $es*ect t& y2 denoted by yf , is given by

( ) ( ) ( )

h

y,xfhy,xflimy,xfh

y+

=>

.

In general, if f is a function of nvariables, nx,...,x,x #! ,

( ) ( ) ( )h

x,...,x,xfx,...,hx,...,x,xflimx,...,x,xf nnkh

nxk#!#!

>#! +=

.

ther :otations for xf !f , fDx ,x

f

for yf #f , fDy ,y

f

for kxf kf , fD kx ,

kx

f

Another techni9ue to solve for the partial derivatives 5ith respect to a certain variable is to treat othervariables as constants. Then, use differentiation techni9ues from previous math courses.

E/a)*le.

+onsider ( ) yxxy,xf ## += . erive xf and yf using the

definition.

E/e$cises. se the definition to solve for the partial derivatives.

!. ( ) #yxy,xg += "x

g

,

y

g

#. ( )yx

yxy,xh

+

= "

x

h

,

y

h

E/a)*les. )olve for the partial derivativesx

f

,

yf

.

!. ( ) !>#3#

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6e&)et$ic Inte$*$etati&n 4a$tial De$ivatives

4hat 5e 6no5 $or should7ve 6no5n from MATH 3='

+onsider ( )xfy= continuous at some real number >x .

( )>xfdx

dy= slope of tangent line to the graph of ( )xfy= at ( )( )>> xf,xP .

+onsider ( )y,xfz= continuous at some point ( )b,a .

( )b,.afx

f

x=

slope of the line tangent to the curve of intersection of the surface

( )y,xfz= and the plane by= at ( )( )b,af,b,aP .

( )b,.afy

fy=

slope of the line tangent to the curve of intersection of the surface

( )y,xfz= and the plane ax= at ( )( )b,af,b,aP

8

E/e$cises. )olve for the partial derivativesx

f

,

y

f

,z

f

.

!. ( ) ( ) ( )zyc!syxsinz,y,xf +++= ## 3.

( ) zlnyy"z,y,xf xxyz += #

#. ( )zyx

yzxz,y,xf

++

+=

#

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%.7 Di##e$entiability an( t'e T&tal Di##e$ential

De#initi&n + inc$e)ent a #"ncti&n,.

If f is a function of t5o variables xand y, the inc$e)ent f at a point ( )y,x byx and y is given by ( ) ( ) ( )y,xfyy,xxfy,x,y,xf ++= .

De#initi&n + (i##e$entiability,.

If f is a function of t5o variables xand ysuch that

( ) ( ) ( ) yxyy,xfxy,xfy,x,y,xf yx +++= #!>>>>>> .

5here #! , are functions of x and y such that >#! , as( ) ( )>>,y,x ,

then f is (i##e$entiable at ( )>> y,x .

Alternatively, to conclude differentiability, use the ne%t theorem.

T'e&$e).

*et f be a function of xand ysuch that xf and yf e%ist on an open dis6 r;PB > ,

for some >>r . If xf and yf are continuous at >P , then f is differentiable at >P .

The contrapositive of the follo5ing theorem is use to conclude n&n3(i##e$entiability.

DO NOT atte)*t t& "se t'is t'e&$e) t& c&ncl"(e (i##e$entiability.

T'e&$e).

If f is a function of t5o variables xand ydifferentiable at a point, then it is continuous atthat point.

Re)a$8 $the contrapositive' If f is discontinuous at a point, then it is n&tdifferentiable at that point.

C

E/a)*le.

+onsider ( ) !## += yxy,xf . @valuate

( )>#>>!>#! .,.,,f .

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De#initi&n + t&tal (i##e$ential,

*et f be a function of t5o variables xand y. The t&tal (i##e$ential at ( )y,x by x and y is given by ( ) ( ) ( ) yy,xfxy,xfy,x,y,xfd yx += .

Re)a$8s !. ( ) ( )y,x,y,xfy,x,y,xfd $appro%imate' $actual'

#. If ( )y,xfz= , then dyfdxfdz yx += or dyy

zdx

x

zdz

+

= .

3. In general, if ( )nx,...,x,xfw #!= ,

then nndx

x

w...dx

x

wdx

x

wdw

++

+

= #

#!

!

.

E/a)*le.

)ho5 that the follo5ing are differentiable on their respective domains.

!. ( ) ##!

yxy,xf += #. ( ) y

xxlnyy,xg =

E/e$cise.

;iven ( ) ( ) ( )

( ) ( )

=

+=

>>>

>>>,fx and ( )>>,fy e%ist but

f is n&tdifferentiable at ( )>>, .

Hint se the follo5ing definitions of partial derivatives.

( ) ( ) ( )

>

>>>>>

> xx

y,xfy,xflimy,xfxx

x

=

( )

( ) ( )

>

>>>>>

> yy

y,xfy,xflimy,xfyy

y

=

E/a)*les.

!. Appro%imate ( ) ( )## >>!#CC!3 .. + using total differential.

#. The formula#!

!!!

rrR += determines the combined resistance R 5hen resistors

!r and #r are connected in parallel. se total differential to determine the effect of a

decreasing !r to the combined resistance R . $Assume that #r is fi%ed.'E/e$cises.

!. An open bo% is to have an inside dimension of !> in, 2 in and < in and thic6ness of >.2 in.Appro%imate using total differential the volume of the material to be used in constructing thebo%.

#. The period # of a pendulum of length L is given byg

L# = # 5here gis the

acceleration due to gravity $a constant'. )ho5 that

=g

dg

L

dL

#

d#

#

!.

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%.9 C'ain R"le #&$ F"ncti&ns M&$e T'an One Va$iable

T'e&$e).

If $is a differentiable function of xand y, ( )y,xf$= , 5here ( )s,r%x= and

( )s,r&y= andr

y,

s

x,

r

x

ands

y

e%ist, then $is a function of rand s

and

r

y

y

$

r

x

x

$

r

$

+

=

s

y

y

$

s

x

x

$

s

$

+

=

In general, if ( )nx,...,x,xf$ #!= is differentiable and ( )my,...,y,y%ix #!= and 'i

y

x

e%ists for each n,...,,i #!= and m,...,,' #!= , then

'

n

n''' y

x

x

$...

y

x

x

$

y

x

x

$

y

$

++

+

=

#

#

!

!

If ( )nx,...,x,xf$ #!= is differentiable and ( )t%ix = and txi

e%ists for each

n,...,,i #!= , then

tx

x$

...tx

x$

tx

x$

t$ n

n

++

+

=

#

#

!

!

!!

E/e$cises.

!. *et ### zyxw ++= 5here = c!ssinrx # , = sinsinry # and

= sinc!srz . )olve forr

w

,

wand

w. @%press these in terms of

,r and .

#. *et ( )y,xfz , 5here = c!srx and = sinry . )ho5 that#

#

###!

+

=

+

z

rr

z

y

z

x

z.

3. The temperature if a metal plate at ( )y,x is yx" 3 degrees. A bug is 5al6ing

northeast at a rate of 8 units per minute $that is, #== dydx

'. -rom the bug7s

E/a)*les.

!. *et # cmDmin and the radiusis increasing at a rate of < cmDmin. -ind the rate of change of the volume 5hen theheight is 2> cm and the radius is != cm.

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The follo5ing are used for i)*licit (i##e$entiati&nof functions of several variables.

T'e&$e)s.

!. If f is a differentiable function of a single variable xsuch that ( )xfy= and f is definedimplicitly by the e9uation ( ) >=y,x% , then if

%is differentiable and ( ) >y,xy% ,

( )

( )y,xy%

y,xx%

dx

dy=

#. If z is a differentiable function of a single variable xand ysuch that ( )y,xfz= and f isdefined implicitly by the e9uation ( ) >=z,y,x% , then if % is differentiable and

( ) >z,y,x%z ,

( )

( )z,y,xz%

z,y,xx%

x

z=

and

( )

( )z,y,xz%

z,y,xy%

y

z=

%.: Hi-'e$3&$(e$ 4a$tial De$ivatives

If ( )y,xfz= such that xf and yf e%ist, the sec&n(3&$(e$ *a$tial (e$ivatives are given by

( ) ( ) ( )

x

y,xfy,xxflimy,xfx

xx

+=

!!

>

( ) ( ) ( )

x

y,xfy,xxflimy,xfx

yx

+=

##

>

( ) ( ) ( )

y

y,xfyy,xflimy,xf y

xy

+

=

!!

> ( ) ( ) ( )

y

y,xfyy,xflimy,xfy

yy

+

=

##

>

!#

E/e$cises.

?e careful of the form of the previoustheorems that you should use.

!. If ># 3#3 =+ yyxx , solve for

dx

dy.

E/a)*le.

If >## = zsinyzcosx , solve for

dy.

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Ot'e$ n&tati&ns

-or xxf , ( )fDD xx ##

x

f

( )fDD !! !!f

-or xyf , ( )fDD xy xy

f

#

( )fDD !# !#f

-or yxf , fDD yx yx

f

#

( )fDD #! #!f

-or yyf , fDD yy #

#

y

f

( )fDD ## ##f

Ill"st$ati&n.

If ( )z,y,xfw= 2 ( )( )xyy

ffDDDff xxyxxy

===

3

!!#

( )( )yzx

ffDDDff yzxyzx

===

3

#3!

T'e&$e)

)uppose that if f is a function of xand ydefined on an open dis6 ( )( )r;y,xB >> and

xf , yf , xyf and yxf are all defined 5ithin B such that xyf and yxf are continuous onB . Then, ( ) ( )>>>> y,xfy,xf yxxy =

!3

E/e$cise.

+onsider ( ) 2#3 #