5.1 definition of the partial derivative the partial derivative of f(x,y) with respect to x and y...

5.1 Definition of the partial derivative

yxy

xyx

fy

f

y

yxfyyxf

y

f

fx

f

x

yxfyxxf

x

f

)(),(),(

lim

)(),(),(

lim

0

0

the partial derivative of f(x,y) with respect to x and y are

Chapter 5 Partial differentiation

for general n-variable

i

ninii

xi

n

x

xxxxfxxxxxf

x

xxxxfi

),...,...,(),..,...,(lim

),....,,,( 2121

0

321

second partial derivatives of two-variable function f(x,y)

yxxy

yyxx

fxy

f

x

f

yf

yx

f

y

f

x

fy

f

y

f

yf

x

f

x

f

x

22

2

2

2

2

)( )(

)( )(

zRe

111

2

211

121

1

)(.....)(.........)(

is torespect with )...,...,( of derivative total the

)( given afor ,,......,3,2,1 , variables

dx

dx

x

f

dx

dx

x

f

dx

dx

x

f

x

f

dx

df

xxxxxf

xxxnix

n

n

i

i

ni

iii


5.2 The total differential and total derivative

..............

),...,( function variable-nfor

is aldifferenti total the ,0 and 0 as

]),(),(

[]),(),(

[

),(),(),(),(

),(),(

and

22

11

21

nn

n

dxx

fdx

x

fdx

x

fdf

xxxf

dyy

fdx

x

fdf

dfyx

yy

yxfyyxfx

x

yyxfyyxxf

yxfyyxfyyxfyyxxf

yxfyyxxff

fffyyyxxx

Ex: Find the total derivative of with respect to , given

that

2/121

2/12

2/12

)1(

3sin32

)1(

1332

)1(

1 ;3 ,32

x

xxx

xxyx

dx

df

xdx

dyx

y

fyx

x

f


xyxyxf 3),( 2 x

xy 1sin

5.3 Exact and inexact differentials

If a function can be obtained by directly integrating its total differential, the differential of function f is called exact differential, whereas those that do not are inexact differential.

aldifferentiinexact existdoesnot ),( function

3 (2)

aldifferentiexact ),()1( (1)

yxf

ydxxdydf

xxyyxfdxyxdydf

Inexact differential can be made exact by multiplying a suitable function called an integrating factor

exist g(x) ),( suitable no (2),(1)for

-(2)---)(),(

-(1)---)(3),(33

3

yh

xgxyyxfxdydyy

fx

y

f

yhxyyxfydxdxx

fy

x

f

dyy

fdx

x

fydxxdydf


Ex: Show that the differential xdy+3ydx is inexact.

Properties of exact differentials:

x

yxB

y

yxA

x

B

yx

f

y

A

xy

f

yxBy

fyxA

x

fdfdyyxBdxyxA

),(),(

),( and ),(),(),(

22

for n variables


ipsrelationsh )1(2

1, pairs allfor

if exact, is ),.......,,( 211

nnjix

g

x

g

dxxxxgdf

i

j

j

i

in

n

ii

Ex: Show that (y+z)dx+xdy+xdz is an exact differential

czyxzyxf

y

g

z

g

z

g

x

g

x

g

y

g

xgxgzyzyxg

)(),,( inspection by

0 ,1 ,1

, ,),,(

321321

321

5.4 Useful theorems of partial differentiation

dyy

zdx

x

zdzyxzz

dzz

ydx

x

ydyzxyy

dzz

xdy

y

xdxzyxx

xy

xz

yz

)()(),(

)()(),(

)()(),(

relation cyclic 1)()()( relation yreciprocit )()(

0constant a is if 0constant a is if

])()()[()()(

1

zyxzz

yxzzz

y

x

x

z

z

y

x

y

y

x

dxxdzz

dzz

x

z

y

y

xdx

x

y

y

xdx


5.5 The chain rule

du

dx

x

f

du

dx

x

f

du

dx

x

f

du

dx

x

f

du

df

uxxxxxf

du

dy

y

f

du

dx

x

f

du

dfdy

y

fdx

x

fdf

uyyuxxyxff

n

n

in

i i

iin

........

)( and ),....,,( variables manyfor

)( ,)( and ),(for

2

2

1

11

21

mju

x

x

f

u

f

uuuxxxxxff

j

in

i ij

miin

,...,2,1

),...,,( and ),....,,(

1

2121


5.6 Change of variables

Ex: Polar coordinates ρ and ψ, Cartesian coordinates x and y, x=ρcosφ,

y=ρsinφ, transform into one in ρ and φ 2

2

2

2

y

f

x

f

2

2

22

2

2

2

2

22

2

2

2

2

2222

21

2/122222

11),()( and )(

cossin ,

sincos

cos ,

sinsin

)/(1

/ ),tan

sin ,cos)(

,

ggg

y

f

x

fyxf

y

f

yyx

f

xx

f

yxxx

ρ

φ

y

φ

yx

y

xy

xy

x

φ (y/x

yyx

x

xyx

),(),( gyxf


5.7 Taylor’s theorem for many-variables functions

00 ,0

00

00

22

222

2

2

00

)],()[(!

1),(

:is variables twofor series sTaylor' full *

),(at evaluated be to are sderivative the all

,

]2[!2

1),(),(

:variables twofor

yxn

n

yxfy

yx

xn

yxf

yx

yyyxxx

yy

fyx

yx

fx

x

fy

y

fx

x

fyxfyxf

])3(16)3)(2(48)2(27[2!

1

)3(7)2(93),(

2/ ,2/ ,/

/ ,/

226

666

22222322

2

yyxxe

yexeeyxf

exyyeyxfyexxeyfeyxf

xyeeyfeyxfxyxyxyxyxy

xyxyxy

Ex: The Taylor’s expansion of f(x,y)=yexp(xy) about x=2, y=3.

5.8 Stationary points of many-variables functions

yyxxxyyyxx

yyxxxyyyxx

yyxxxyyyxx

xx

xyyy

xx

xyxx

yyxyxx

fffff

fffff

fffff

f

ffy

f

yfxfyxfyxf

y-y ΔΔxxx

fyyfxfxyxfyxf

y

f

x

f

2

2

2

222

00

00

2200

or sign opposite have and if :point saddle (3)

and negative are and both if :maximum (2)

and positive are and both if :minimum (1)

)]()([2

1),(),(

grearrangin , ,

]2[!2

1),(),(

0 and 0


two-variable function about point ),( 00 yx

Ex: has a maximum at the point , a minimum at and a stationary point at the origin whose nature cannot be determined by the above procedures.


)exp(),( 223 yxxyxf )0,2/3()0,2/3(

Sol:

minimum a :)0,2/3( maximum a: )0,2/3(

point stationary edundetermin an: )0,0(

0 ,)2/3exp(2/33 ,)2/3exp(2/36)0,2/3( pointsat (2)

0)00(point at (1)

)exp()32(2

)exp()24(

)exp()6144( are sderivative second the

)0,2/3( ,)0,2/3( ,)0,0(at are points stationary the

0yor 00)exp(2

2/3or 00)exp()23(

2222

2223

2235

223

2242

xyyyxx

xyyyxx

xy

yy

xx

fff

fff,

yxxyxf

yxyxf

yxxxxf

xyxyxy

f

xxyxxxx

f

for a n-variable function at all stationary points


),.......,( 21 nxxxf

signs mixed have seigenvalue all :point saddle (3)

zero than less are seigenvalue all 02

1 :maximum (2)

zero thangreater are seigenvalue all 02

1 :minimum (1)

2

1

2

1

2

1

2

1

and

rseigenvecto orthogonal and seigenvalue real hasit symmetric, and real is

2/ elements withmatrix a define

2

1)()( :expansion sTaylor'

2

2

2

2

2

0

rr

r

rr

r

rr

rr m

rmmmrmmr m

Trmr

mmm

r

Trr

rr

rrssTrrrr

rr

T

jiij

jii j ji

a f

af

aaaeeaaeaMeaf

eaxeeeMe

en nM

xMxfxx

fM

xxxx

fxfxff

i

i

xx

f allfor 0


Ex: Derivative the conditions for maxima, minima and saddle points for a function of two variables, using the above analysis.

yyxxxyxyyyxxyyxxyyxx


yyxxxyxyyyxxyyxx

yyxxyyxx


xyyyxxyyxx

xyyyxxyyyx

xyxx

yyyx

xyxx

ffffffffff

ffffffffff

ffffffff

ffff

fff fffffff

fffff

fffff

ff

ff

ffM

222

222

2222

222

22

2

04)()(0 if

0 4)()(0 if

sign mixed have seigenvalue two :point saddle (3)

4)-( )(

00 and 0 :maxima (2)

4)( and 0 , 0

positive and real are seigenvalue two :minima (1)

]4)()[(2

1

0))((0

Find the maximum value of the differentiable function

subject to the constraint

Lagrange undetermined multipliers method


5.9 Stationary values under constraints

cyxg ),(

),( yxf

),(

points stationary theat and of values the and find 0

0

that such choose can we dependent, are and

multiplier edundetermin Lagrange :

0)()()(

0 and 0 maximize

cyxg

yxy

g

y

fx

g

x

f

dydx

dyy

g

y

fdx

x

g

x

fgfd

dyy

gdx

x

gdgdy

y

fdx

x

fdff

Ex: The temperature of a point (x,y) on a circle is given by T(x,y)=1+xy. Find the temperature of the two hottest points on the circle.


2/12/1 ,2/1

2/32/1 ,2/1

(1) intoput and 2/1(2) and (1) from

-(3)---020)()1(

-(2)---020)()1(

(1)-----1),( constraint

min

max

22

22

22

Tyxxy

Tyxxy

xy

yxyxy

xyy

xyyxx

xyx

yxcyxg

the stationary points of f(x,y,z) subject to the constraints g(x,y,z)=c1, h(x,y,z)=c2.

0)(

0)(

0)(

z

h

z

g

z

fhgf

z

y

h

y

g

y

fhgf

y

x

h

x

g

x

fhgf

x

Ex: Find the stationary points of subject to the

following constraints:


333),,( zyxzyxf

0),,( and 1),,( (ii)

1),,( (i)222

222

zyxzyxhzyxzyxg

zyxzyxg

2/13/2 1 ,023 ,023

0 and 0 ,0 (a)

:zero is and , of somefor :conditionanother :note *

3/1at occur points stationary

2/31 intoput 3/2

023)(

023)(

023)( (i)

2222

222

2

2

2

zyzyzzyy

zyx

zyx

zyx

zyxzyx

zzgfz

yygfy

xxgfx


)0,1,0()001( ,)0,2/1,2/1( :0

)0,0,1( ,)1,0,0( ,)2/1,0,2/1( :0for points stationary

0 and 0 case thefor ,Similarily

)0,1,0( ,)1,0,0( ),2/1,2/1,0( :points stationary

11 ,0230 ,0 (c)

11 ,0230 ,0 (b) 22

22

, ,,z

y

zy

yyλyyyzx

zzλzzzyx

6/2 ,6/1 ,6/1161 from

20 from , if (a)

0)(2)(30)(2)(3)2()1(

(3)---023)(

(2)---023)(

(1)---023)( (ii)

2222

22

2

2

2

zyxxzyx

xzzyxyx

yxyxyxyx

zzhgfz

yyhgfy

xxhgfx


ntinconsistedifferent all are and , (d)

sconstraint by prohibited (c)

(a) condition in included

02)(3

02)(3

02)(3 if (b)

6/1 ,6/1 ,6/2 if

61 ,6/2 ,6/1 if ,similarily

zyx

zyx

zxzx

zyzy

yxyx

zyxzy

/zyxzx


5.11 Thermodynamic relations

Maxwell’s thermodynamic relations:

P: pressure V: volume T: temperature S: entropy U: internal energy

XY

U

YX

UdY

Y

UdX

X

UdU

PdVTdSdU

XY

22

and )()(

:Y and Xof variables twofor

micsthermodyna of lawfirst the

Ex: Show that VS SPVT )/()/(

VS

SV

SV

S

P

V

T

VS

U

SV

U

PV

UT

S

U

dVV

UdS

S

UdUPdVTdS

)()(

)( and )(

)()(

22


Ex: Show that VT TPVS )/()/(

VT

TVVT

VT

VVTT

VT

VT

VT

T

P

V

STV

ST

T

ST

VT

P

VT

ST

V

S

T

U

VV

U

TTV

U

VT

U

T

ST

T

UP

V

ST

V

U

dTT

UdV

V

U

PdVdTT

SdV

V

STPdVTdSdU

dTT

SdV

V

SdS

TVdUdS

)()(

])([)()()(

)()( since

)() ( and )()(

)()(

])()[(

)()(

and variablest independen two with and aldifferenti totalconsider

22

22


5.12 Differentiation of integrals

dtx

txf

x

txFdttxf

xdt

x

txF

t

x

txf

t

txF

xx

txF

ttx

txF

xt

txF

txfx

txFdttxftxF

),(),(),(]

),([

),(]

),([]

),([

),(),(

),(),(

),(),(

22

indefinite integral

dtx

txf

dx

dI

dtx

txfdt

x

txf

x

uxF

x

vxF

dx

xdI

uxFvxFdttxfxI

v

u

uv

vt

ut

),(

),(),(),(),()(

),(),(),()(

definite integral

(1) The integral’s limits are constant:


(2) The integral’s limits are function of x

dt x

txf

dx

duxx,uf

dx

dvxvxf

dx

dI

dttxfxdx

duxuxf

dx

dvxvxf

x

I

dx

du

u

I

dx

dv

v

I

dx

dI

xuxfu

Ixvxf

v

I

xuxFxvxFdttxfI

xv

xu

xv

xu

xvt

xut

)(

)(

)(

)(

)(

)(

),())((-))(,(

),())(,())(,(

))(,( ,))(,(

))(,())(.(),(

Ex: Find the derivative with respect to x of the integral

dtt

xtxI

x

x2 sin

)(

)sin2sin3(1

|sinsinsin2

cos)1(

sin)2(

sin

2323

2

2

3

2

2

xxxx

xt

x

x

x

x

dtt

xtt

x

xx

x

x

dx

dI

xx

x

x

5.1 definition of the partial derivative the partial derivative of f(x,y) with respect to x and y...

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