*Xrhsima oria
     sinx
 lim------=1,
 x0   x
      x__
 lim |/x =1
 x

*Xrhimes anisotikes sxeseis:
 x+y2xy, |sinx||x|

*Idiothtes mer. paragwgwn:

1.Mia merika paragwghs
imh synarthsh den
einai kai synexhs
se shmeio (x0,y0).

2.Exw

f(x,y)= fx(x,y)dx+c1(y)

f(x,y)= fy(x,y)dx+c2(y)

3.Exw fxy=fyx.

*Mer. paragwgoi syn8etwn synarthsewn:
1.Exw f(x),x(t)=>
 df   df   dx
 -- = -- * --
 dt   dx   dt

2.Exw f(x,y),x(t),y(t)=>
 df     dx       dy
 --=fx* -- + fy* --
 dt     dt       dt

3.Exw f(x,y,t),x(t),y(t)=>
df          dx       dy
--=ft + fx* -- + fy* --
dt          dt       dt

4.Exw f(x,y),x(u,v),y(y,u)=>
 fu=fx*xu + fy*yu
 fv=fx*xv + fy*yv

5.Exw f(W),W(x,y)=>
     df          df
 fx= --* Wx, fy= --* Wy
     dW          dW

*Omogeneis synarthseis:

 Prepei V(x,y),
  t>0:(t konta sto 1)
               
 f(*x,*y) =  * f(x,y)

Idiothtes:
  V(x,y) isxyei:
 x* fx+y* fy = *f

*Paragwgos kata kateu8ynsh:
     _
 .   a    f .
 a= ---,  --=a*(gradf)
    |a|   a

*Oliko diaforiko synarthshs:
  df=fx*dx + fy*dy

1.H synarthsh f(x,y)
einai diaforisimh sto
(a,b) otan:
 a)Yparxoun oi fx,fy

 b)
     f(x,y)-f(a,b)-[(x-a)* fx(a,b)+ (y-b)* fy(a,b)]
lim ------------------------------------------------ = 0
xa           /----------------
yb         |/ (x-a) + (y-b)

2.An f(x,y) synexhs
kai paragwgisimh ==>
f(x,y) diaforisimh.

3.An f(x,y) diaforisimh
==> f(x,y) synexhs kai
yparxoun oi fx,fy.

*Teleio diaforiko:
H synarthsh
 P(x,y)*dx + Q(x,y)*dy
einai to oliko diaforiko
mias synarthshs f(x,y)
otan isxyei:
 P,Q synexeis kai
 Qx = Py

Gia poio polles metablhtes
 P(x,y,z)*dx+Q(x,y,z)*dy R(x,y,z)*dz
 8a exw:
 Py = Qx, Pz=Rx, Qz=Py

*Oliko diaforiko syn8eths synarthshs:
 Exw f(x,y), x(u,v),
     y(u,v).
 Tote:
  df=fx*dx + fy*dy
  df=fu*du + fv*dv

*Diaforiko anwterhs ta3hs:

 df=d(df)=
  =d(fx*dx+fy*dy)=
=fxx*dx+2*fxy*dx*dy+fyy*dy+fx*dx+fy*dy

 Opou dx=(dx)=dx*dx
  kai d(dx)=dx

df=(fx*dx+fy*dy)+fx*dx+fy*dy

Opou to "" shmainei
paragwgish gia tis
sunarthseis kai uywsh
sto tetragwno gia ta
diaforika.

*Anaptygma Taylor                        v
           x-a  ,   (x-a) ,,       (x-a) (v)
f(x)=f(a)+-----f(a)+------f (a)+...+-----f (a)+...
            1!        2!              v!

               1
f(x,y)=f(a,b)+----[(x-a)*fx(a,b)+(y-b)*fy(a,b)]+
               1!
    1
 + ---[(x-a)*fxx(a,b)+2(x-a)(y-b)*fxy(a,b)+(y-b)*fyy(a,b)]+
    2!
 +........................................+
   1                             (v)
 +---[(x-a)*fx(a,b)+(y,b)*fy(a,b)]+....
   v!

*Synarthsiakes Orizouses

 (u,v)   | ux   uy |
 ------ = |         |
 (x,y)   | vx   vy |

1.Isxuei:

 (u,v)       1
 ------ = ---------
 (x,y)    (x,y)
           ------
           (u,v)

2.
  (u,v)   (u,v)   (t,w)
  ------ = ------ * ------
  (x,y)   (t,w)   (x,y)

3.u(x,y), v(x,y) einai
 synarthsiaka e3arthmenes
 otan yparxei F:

 F(u,v)=0. Alliws einai
           ane3arthtes

4.Einai u,v synarthsiaka
e3arthmenes <===>

 (u,v)
 ------ = 0
 (x,y)

*Peplegmenes sxeseis:

 Gia na epilyetai h
F(x,y)=0 monoshmanta
sto (a,b) dhladh na
orizetai mia y=f(x)
prepei:

 F(a,b)=0
 Fy(a,b)0


 Gia epilyetai h
F(x,y,z) dhladh na
orizetai z=z(x,y)
prepei.

 F(a,b,c)=0
 Fz(a,b,c)0

*Systhma peplegmenwn sxesewn

Estw:
 F(x,y,u,v)=0
 G(x,y,u,v)=0

 Ean
  F(a,b,c,d)=0
  G(a,b,c,d)=0

 kai
  (F,G)
  ------  0
  (u,v)

 Tote to systhma
epilyetai monoshmanta
dhladh yparxoun oi
 u=u(x,y), v=v(x,y).

*Akrotata synarthsewn

 Estw
  f(x1,x2,...xv)
 kai A(a1,a2,...av) h
 lysh tou shsthmatos

 fx1=0, fx2=0...fxv=0.

Sth 8esh A ypologizoume
ta:

 D1=fx1x1

    | fx1x1   fx1x2 |
 D2=|               |
    | fx2x1   fx2x2 |
 .....................
    | fx1x1 ... fx1xv |
 Dv=| ............... |
    | fxvx1 ... fxvxv |

 An Dk>0 gia ka8e k
  to topiko elaxisto sto A
        k
 An (-1) *Dk>0 tote
  topiko megisto sto A

*Desmeumena akrotata:

 Exoume thn
  f(x1,x2,...xv)
 kai
  q1(x1,x2,...xv)=0
  q2(x1,x2,...xv)=0
  .................
  qk(x1,x2,...xv)=0
 Opou k<v.

a) 8ewroume thn

 g=f+1*q1+2*q2+...+v*qv

b) Lynoume to
 gx1=0, gx2=0,...,gxv=0

 q1=0,q2=0,...,qk=0

 Estw lysh tou systhmatos
x1=a1,x2=a2,...,xv=av,
1=b1,2=b2,...,k=bk

c) Ypologizw ton pinaka
    _                    _
   | q1   q1       q1  |
   | ---   ---  ...  ---  |
   | x1   x2       xv  |
 _ | ...................  |
 A=|                      |
   | qk   qk       qk  |
   | ---   ---  ...  ---  |
   | x1   x2       xv  |
   |_                    _|

Sto shmeio x1=a1,x2=a2,..xv=av

 Ean exei estw kai mia
ypoorizousa kxk mh
mhdenikh exoume pi8ano
akrotato

d) Estw h orizousa

    |                         q1     qk |
    | gx1x1  gx1x2  .. gx1xv  ---  .. --- |
    |                         x1     x1 |
    | ....   .....     ....   ...     ... |
    |                                     |
    |                         q1     qk |
    | gxvx1  fxvx2  .. gxvxv  ---  .. --- |
    |                         xv     xv |
    |                                     |
D1= | q1               q1               |
    | ---  ..........   ---    0  .... 0  |
    | x1               xv               |
    | ...    ......     ....   .........  |
    |                                     |
    | qk               qk               |
    | ---  ............ ---    0 ..... 0  |
    | x1               xv               |

 Sto x1=a1,x2=a2,...xv=av, 1=b1,...k=bk

Diagrafoume 1h seira
kai 1h sthlh kai pame
sth D2 kai pame etsi
mexri thn Dv-k.

e)
 i)An k artios kai
D1>0 ... Dv-k>0 tote
topiko elaxisto

ii)An k peritos kai
D1<0,....Dv-k<0 tote
topiko elaxisto

iii)An v artios kai
    p
(-1) *Dp<0, opou
 p=1,2,....,v-k tote
topiko elaxisto

iv)An v peritos kai
    p
(-1) *Dp>0 tote
topiko megisto.


