M-102 Unit- 2M-102 Unit- 2SessionSession

Prepared byPrepared by

Mr. Kishor PokarMr. Kishor Pokar

Euler’s theorem for function of two variables

Corollaries of Euler’s theorem Theorem on Total Differentials Corollary of Theorem on total

differentials. Euler’s theorem for function of three

variables.

Topics Covered

Euler’s Theorem for homogeneous function of two variables

2

Q. StateandproveEuler'stheoremfor the

homogeneousfunctionof twovariables

Or

beareal valued functiondefined

on .Supposethat f

( , )

homoisa

.

gene

,

ous

deg

x yx

x y

Let

E

func

z f x y

reen f andf existstionof If

thenprovethat

onE

z¶

Í

=

¶+¶

¡

nzzy

=¶

Proof:Sincez ( , ) homf x y isa ogeneousfunction=

n yf(x,y) x .

xz g

æö÷ç= = ÷ç ÷çè ø

n-1 n2

y y ynx x .

x x x xz

g g¶ æö æö æ ö÷ ÷ ÷ç ¢ç ç= + × -÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç çè ø è ø è ø¶

n n-1y yx nx x .

x x xz

g yg¶ æö æö÷ ÷ç ¢ç= -÷ ÷ç ç÷ ÷ç çè ø è ø¶

n y 1x .

y x xz

g¶ æöæö ÷÷ ç¢ç= × ÷÷ çç ÷ ÷ç çè ø è ø¶

n-1

n n-1 n-1

n

yx .

y xy y y

x y nx x x .x x x x

ynx

xn f(x,y)

nz.

zy yg

z zg yg yg

y

g

¶ æö÷¢ç= ÷ç ÷çè ø¶¶ ¶ æö æö æö÷ ÷ ÷ç ¢ç ¢ç+ = - +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç çè ø è ø è ø¶ ¶

æö÷ç= ÷ç ÷çè ø

=

=

Corollary – 1Statement

2

2

Let beareal valued function

real valued functiondefinedon .Supposethat

fi sa functionof degreenandthat

all thesecondorder partialderivativeoff are

existson

z=f(

andc

x,y)

homogene

ontinuous.Then provethat

ous

x

definedon

E Í ¡

2 2 2

22 2

2xy y n(n 1)z

xx.

z z z

y y

¶ ¶ ¶+ + = -

¶ ¶¶ ¶

2 2

2 2

2

22

z=f(x,y) is ahomogeneous functionof x,yof degreen,

byEuler'stheorem

x y nzx

Diff. . . .x

x y n ,x x x x

x x xy nx ,x x x x

(n 1)xx

z zy

w r t

z z z zy

z z z zy

z

¶ ¶+ =

¶ ¶

¶ ¶ ¶ ¶+ + =

¶ ¶ ¶ ¶ ¶

¶ ¶ ¶ ¶+ + =

¶ ¶ ¶ ¶ ¶

¶= -

¶

2 2

2 2

2 2

2 2

2 2 2

2

22

22

22

2 22 2

Similarly, y xy (n 1)yx y x

x y x

x xy =(n 1)xx x x

xy y (n 1)yy x x

x xy y (n 1) x yx x x x

n(n 1)z.

z z zy

z zWKT

y

z z zy

z z zy

z z z z zy y

¶ ¶ ¶+ = -

¶ ¶ ¶ ¶

¶ ¶=

¶ ¶ ¶ ¶

¶ ¶ ¶+ -

¶ ¶ ¶ ¶

¶ ¶ ¶+ = -

¶ ¶ ¶ ¶

ì ü¶ ¶ ¶ ¶ ¶ï ïï ï+ + = - +í ýï ï¶ ¶ ¶ ¶ ¶ ¶ï ïî þ

= -

Corollary – 2Statement

2

Letu=u(x,y)beanonhomogeneousreal valued function

real valued functiondefinedon .but z= (u)be

homogeneous functionof degreen.

(u)x y n , (u) 0 for any(x,y)

x (u)

definedon

E functionand

Thenprovethat

u uprovided

y

Í

¶ ¶ ¢+ = ¹¢¶ ¶

¡

E.Î

Proof:- Since z= (u) isahomogeneous functionof x,yof degreen,

byEuler'stheoremwe ,

x y nz=n (u).x

u ux (u) y (u) n (u)

x y

(u)u ux y n .

x y (u)

have

z zy

¶ ¶+ =

¶ ¶

æ öæ ö¶ ¶ ÷ç÷ç ¢ ¢Þ + ÷=÷ çç ÷÷ç ç ÷è ø¶ ¶è ø

æ öæ ö¶ ¶ ÷ç÷çÞ + ÷=÷ çç ÷÷ç ç ÷è ø ¢¶ ¶è ø

Corollary - 3

2

2 2 2

22

2 2

2

Letu=u(x,y)beanonhomogeneousreal valued function

definedon . z= (u)behomogeneous

functionof degreen.

u u ux xy y (u) (u) 1 ,

xx

(u)(u) n ,

(u)

defined

E Butfunction

Thenprovethat

y y

where prov

Í

¶ ¶ ¶ é ù¢+ + = -ê úë û¶ ¶¶ ¶

=¢

¡

(u) 0 for any(x,y) E.ided ¢ ¹ Î

Proof:

2 2

2 2

2 2

2 2

2 2 2

2

2

2

22

22

2 22 2

u ux y (u),

x

x y (u) ,x x x x

x y (u) 1 ,x x x

x xy (u) 1 x ,x x x

xy y (u) 1 y ,y y y

x xy y (u) 1 xx x y x

y

u u u uy

u u uy

u u uy

u u ux

u u u uy

¶ ¶+ =

¶ ¶

¶ ¶ ¶ ¶¢+ + =¶ ¶ ¶ ¶ ¶

¶ ¶ ¶é ù¢+ = -ë û¶ ¶ ¶ ¶

¶ ¶ ¶é ù¢+ = -ë û¶ ¶ ¶ ¶

¶ ¶ ¶é ù¢+ = -ë û¶ ¶ ¶ ¶

¶ ¶ ¶ ¶é ù¢+ + = - +ë û¶ ¶ ¶ ¶ ¶y

y

(u) 1 (u).

u

ì ü¶ï ïï ïí ýï ï¶ï ïî þ

é ù¢= -ë û

Theorem on Total Differentials

Stateandprovetheoremontotaldifferentials.

OR

Letz= f(x,y)bedefinedonE.Then provethat

xxz z

dz d dyy

¶ ¶= +

¶ ¶

Proof:-

( )Fix (x,y) E. Let ( x, ), .

(x+ x,y+ )

denoteachangein(x,y).

Let z f(x+ x,y+ ) bethecorrespondingchangeinz.

Then

f(x+ x,y+ ) f(x,y)

= f(x+ x,y+ ) f(x+ x,y) f(x+ x,y) f(x,y)

y E

Let y

z y

z y

y

Î = Ì

D D Î

+D = D D

D = D D -

é ù é ùD D - D + D -ê ú ê úë û ë û

N N

N

y 1

x 2

y 1 x 2

x 2 y 1

, '

f(x+ x,y+ ) f(x+ x,y)= f (x+ x,y+ )

f(x+ x,y) f(x,y) f (x+ x,y) x

f (x+ x,y+ ) f (x+ x,y) x

f (x+ x,y) x f (x+ x,y+ )

Next byLagrange stheorem

y y y

z y y

z y y

D D - D D D D

D - = D D

\ D = D D D + D D

\ D = D D + D D D

2

1

2 1

0 0

x 2 x

y 1 y

x y

x y

, f (x+ x,y) = f (x,y)

f (x+ x,y+ )= f (x,y)

( f (x,y)) x ( f (x,y))

( x, ) ( , )

f (x,y) x f (x,y)

. .,

x .x

Now

y

z y

As y

dz d dy

i e

z zdz d dy

y

D +

D D +

\ D = + D + + D

D D ®

= +

¶ ¶= +

¶ ¶

Corollary of Theorem on Total Differentials

Let u and v be two functions of x and y. Then prove that

duvdvuuvd )(

Proof:

(uv) (uv)(uv) x

x

v u v u= u v x u v

x x y y

v u v uu x v x u v

x x y y

v v u uu x v x

x y x yu v v u

Fromthetheoremontotal differentialswehave

d d dyy

d dy

d d dy dy

d dy d dy

d d

¶ ¶= +

¶ ¶

é ùé ù¶ ¶ ¶ ¶ê ú+ + +ê ú ê úê ú¶ ¶ ¶ ¶ë û ë û¶ ¶ ¶ ¶

= + + +¶ ¶ ¶ ¶

é ù é ù¶ ¶ ¶ ¶ê ú ê ú= + + +ê ú ê ú¶ ¶ ¶ ¶ë û ë û

= +

Euler’s theorem for homogeneous function of three variables

3

x y z

Q. StateandproveEuler'stheoremfor thehomogeneous

functionof threevariables

Or

f( x,y,z) beareal valuedhomogeneous

function degree n definedon .

Iff , f and f existsonE,then provethat

x y z nH.x

LetH

of E

H H Hy z

=

Í

¶ ¶ ¶+ + =

¶ ¶ ¶

¡

Proof:-

n

SinceH f(x,y,z) homogeneous function degree n,

y zf(x,y,z) x , .

x xy z

v= .x x

isa of

H

whereu and

=

æ ö÷ç\ = = ÷ç ÷çè ø

=

( )

( )

1

1

- n n2 2

- n -1 n-1

y znx u,v x x

x x u x v

x nx u,v x y x zx u v

n

n

H

H

¶ ¶ ¶= - -

¶ ¶ ¶¶ ¶ ¶

= - -¶ ¶ ¶

( ) ( )

( )

( )

1

1

1

- n

- n

- n2 2

,

nx u,v xx x

vnx u,v x

u x v xy z

nx u,v xx u x v

n

n

n

Hence

H

u

¶ ¶= +

¶ ¶æ ö¶ ¶ ¶ ¶ ÷ç= + + ÷ç ÷çè ø¶ ¶ ¶ ¶æ ö¶ ¶ ÷ç= + - - ÷ç ÷çè ø¶ ¶

10

1

n

n

n

n-1

vx

y u y v y

xu x v

xy u x

y x yy u

H u

H

H

é ù¶ ¶ ¶ ¶ ¶ê ú= +ê ú¶ ¶ ¶ ¶ ¶ë ûé ù¶ ¶

= + ×ê úê ú¶ ¶ë û

¶ ¶=

¶ ¶¶ ¶

=¶ ¶

10

1

n

n

n

n-1

vx

z u z v z

xu v x

xz v x

z x zz v

H u

H

H

é ù¶ ¶ ¶ ¶ ¶= +ê ú

ê ú¶ ¶ ¶ ¶ ¶ë ûé ù¶ ¶

= × +ê úê ú¶ ¶ë û

¶ ¶=

¶ ¶¶ ¶

=¶ ¶

( )nx y z nx u,v =nH.xH H H

y z¶ ¶ ¶

+ + =¶ ¶ ¶