DIFFERENTIABILITY OF REAL VALUED FUNCTIONS OF TWO VARIABLES AND EULER S THEOREM. ARUN LEKHA Associate Professor G.C.G., SECTOR-11, CHANDIGARH

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1 DIFFERENTIABILITY OF REAL VALUED FUNCTIONS OF TWO VARIABLES AND EULER S THEOREM ARUN LEKHA Assciate Pressr G.C.G. SECTOR-11 CHANDIGARH

2 FUNCTION OF TWO VARIABLES Deinitin: A variable Z is said t be a unctin tw independent variables and dented b z= i t each pair values and ver sme dmain D ={: a<<bc<<d} there crrespnds a single deinite value Z. e.g. The area A a rectangle having sides lengths and is. i.e. A = is a unctin tw variables and. Dmain the unctin is D ={: > 0 > 0}.

3 PARTIAL DERIVATIVE FIRST ORDER Partial derivative Z = w.r.t. regarding as cnstant is dented b z/ r / r and prvided it eists and is inite. Similarl prvided it eists and is inite. lt 0 lt 0

4 e.g. I z =e -/ + tan -1 /. then / z e 2 2 PARTIAL DERIVATIVES SECOND ORDER The irst rder partial derivatives z/ r z/ are generall unctins and and hence we can again ind their partial derivatives w.r.t. X r. The partial derivatives thus btained are called secnd rder partial derivatives and are dented b r.

5 HOMOGENEOUS FUNCTIONS A unctin tw variables and the rm = a n +a 1 n-1 +.a n-1 n-1 +a n n in which each term is degree n is called hmgeneus unctin r i it can be epressed in the rm n g/ r n g/. e.g. = / + is hmgeneus unctin degree 1

6 EULER S THEOREM: I Z = is a hmgeneus unctin and degree n then z/ + z/ = nz E. Shw that Z =a 2 + 2h + b 2 is hmgeneus unctin degree 2 and veri Euler s therem. Sl. Z=a 2 +2h + b 2 = 2 [a+2h / + b 2 / 2 ] = X 2 g /. Z is hm. Functin degree 2. Veriicatin Euler s therem. z/ = 2a + 2h.

7 z/ = 2h + 2b then z/ + z/ = 2a+2h + 2h+2b =2a h + 2b 2 = 2z Euler s therem r a hmgeneus unctin three independent variables. I H is a hmgeneus unctin r z rder n then H/ + Y h/y + z H/Z = nh

8 Dierentiable Functin: A unctin is said t be dierentiable at i z = can be epressed in the rm z = /. + / where as 0. Remark: Cntinuit at are suicient cnditins r dierentiabilit. Ttal dierential a unctin I z = then ttal dierential Z is dented and deined b dz = z/. d + z/.d

9 Dierentiabilit The unctin Z= is said t be dierentiable at a pint i in a neighburhd it can be represented in the rm +h +k = Ah + BK + h +k were AB are independent the variables hk.

10 and 0 as hk 0 independentl. Thm: I a unctin is dierentiable at a pint then it is cntinuus at that pint. Remark: Cnverse is nt alwas true. Eample: = is nt dierentiable at 00 but cntinuus at 00. Therem: I a unctin is dierentiable at a pint then and bth eist and A h h h Lt 0 B k k k Lt 0

11 Que. Discuss the dierentiabilit 1 = + at 00 Sl. is dierentiable at 00 I 0+h0+k 00 = Ah+Bk +h +k where 0 as h k0 Nw A 00 h Lt 0 h0 h 00 h Lt h h h Lt h 0 h 1 1 i i h h 0 0

12 A des nt eist. Similarl B = 00 = k Lt 0 0 k k 00 k Lt 0 k k 1 1 k 0 k 0 B des nt eist. Hence is nt dierentiable at 00. YOUNG S THEOREM Let be deined in a dmain D R 2. Let ab be an interir D and let i and eist in the neighburhd ab ii and are dierentiable at the pint ab then = at ab

13 SCHAWARZ S THEOREM I ab be a pint the dmain DR 2 a unctin such that i ii and eist in the neighburhd the pint ab is cntinuus at ab then eist at ab and = at ab Change Variables Let Z = = uv = uv

14 Taking as cnstant Taking u as cnstant and b slving the abve equatins in z/ z/ we get their values in terms z/u z/v u v. u z u z u z.. v z v z v z..

15 Cmpsite unctins: Deinitin: Let Z = and let = t and = t then z is called cmpsite unctin t. Dierentiatin cmpsite unctins: Let Z = pssess cntinuus partial derivatives and X = t = t pssess cntinuus derivatives then dz/dt = z/. d/dt + z/. d/dt Implicit unctins: Deinitin: Let be a unctin tw variables and = be a unctin such that vanishes identicall then = is an implicit unctin deined b the unctinal equatin = 0

16 Dierentiatin implicit unctins I = 0 r c be an implicit unctin then i d/d = -/ / / = -/ 0 ii d 2 /d 2 = Implicit unctin therem Tw Variables Let be a unctin tw variables 2 and and ab be a pint its dmain deinitin such that i ab = 0 ii and eist and are cntinuus in certain nbhd. ab.

17 ii ab 0 then there eist a rectangle a-h a+h b-k b+k abut ab such that r ever in the interval [a-h a+h] = 0 determines ne and nl ne value = ling in the internal [b-k b+k] with the llwing prperties i b = a ii = 0 r ever in [a-ha+h] iii is derivable and bth and are cntinuus in [a-h a+h].

18 e.g. = and a pint 01 S that 01 = 0 and 01 = 2 0 Nw the tw pssible slutins = i = is implicit unctin in nbhd. 01 where <1 >0. ii = is implicit unctin in nbhd. 0-1 where <1 <0.