*Dianysmatikos logismos

 Eswteriko ginomeno
_ _
a.b=ax.bx+ay.by+az.bz
_ _  _   _
a.b=|a|.|b|.cosf

 E3wteriko ginomeno
_ _   _  _   _
c=a x b=-b x a
 _   _   _   _
|a x b|=|a|.|b|.sinf
        _   _   _
      | x   y   z  |
_   _ |            |
a x b=| ax  ay  az |
      |            |
      | bx  by  bz |

*Parametrikh parastash epifaneias

1.Sfairikh epifaneia:

  x=R.sin.cos
  y=R.sin.sin
  z=R.cos

2.Kylindrikh epifaneia:

  x=R.cos
  y=R.sin

*Epikampylia oloklhrwmata

          dq
 (x,y,z)=--   ==>
          dl

 ==> dq=(x,y,z).dl

a)
x=x(t), y=y(t), z=z(t)
 tA  t  tB

b)   dx     dy     dz
  x'=--, y'=--, z'=--
     dt     dt     dt

c)

  (x,y,z)dl=
 AB

 tB                    _________________
= (x(t),y(t),z(t)).|/ x' + y' + z' .dt
 tA
    Prepei tA<tB

*Parametrikes par. grammwn.

1.Eu8ygrammo tmhma.

 A(a1,a2,a3)->
 B(b1,b2,b3)

 x=a1+t.(b1-a1)
 y=a2+t.(b2-a2)
 z=a3+t.(b3-a3)

2.To3o kyklou.

 x=R.cost, y=R.sint

3. To3o elleiyhs.

 x=a.cost, y=b.sint

*Efarmoges epikampyliou
 oloklhrwmatos.

Maza ana monada
mhkous:        dm
      f(x,y,z)=--
               dl

1.Synolikh maza:

 M= f(x,y,z).dl
   AB

2. Kentro mazas:
    1
xc=---  x.f(x,y,z).dl
    M  AB
.....................

3. Roph adraneias:

Ws pros x:

Ix= (y+z).f(x,y,z).dl
   AB
.......................

Ws pros thn arxh
twn a3onwn:

I= (x+y+z).f(x,y,z).dl
   AB

3. Mhkos to3ou:

 L= dl
   AB

*Epikamp. olokl. dianysmatikhs
synarthshs.
    L _  _
 W=  F.dl
   K
 _
 F(x,y,z)=( P(x,y,z),Q(x,y,z),R(x,y,z) ).
  _
 dl=(dx,dy,dz).

 x=x(t), y=y(t), z=z(t).
  K  tK
  L  tL

     dx    .
  dx=--.dt=x(t).dt
     dt
  ................
 L _  _
  F.dl=
K

  tL                    .
= ( P(x(t),y(t),z(t).x(t).dt+...)
 tK

 Den einai aparaithto
 tK < tL.

*Idiothtes.

1.
 B _  _   A _  _
  F.dl=-  F.dl
A        B

*rotF
1.
    _     _   _
 rotF=curlF=xF=

  R   Q  _    P   R  _   Q   P  _
=(-- - --).x + (-- - --).y +(-- - --).z
  y   z       z   x      x   y

2.      _
 Ean rotF=0 tote:
     B _  _
 a)   F.dl = U(xb,yb,zb)-U(xa,ya,za)
    A
   Ane3arthto diadromhs
   oloklhrwshs.
      _  _
 b)  F.dl=0
    C

 c) Yparxei U(x,y,z):
 _
 F=gradU=U.

 d)               _  _
dU=P.dx+Q.dy+R.dz=F.dl

*Diplo oloklhrwma.
   dq
 =-- ==> dq=(x,y).dx.dy
   dS

 q=(x,y).dx.dy=
    

    
= ( (x,y).dy) dx
   

*Alladh metablhtwn sto
 diplo oloklhrwma.

1.Epilegoume
  x=x(u,v), y=y(u,v)

2.
g(u,v)=f( x(u,v),y(u,v) )

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

3.
                        |(x,y)|
f(x,y)dx.dy= g(u,v).|------|.du.dv
             D         |(u,v)|


Epilogh metablhtwn.
a) Kuklikos diskos.
  x=rcos, y=rsin.

b) Elleiptikos diskos.
x=a.r.cos,y=b,r,sin

c)Synoro se polikh morfh.
 x=rcos, y=rsin.

d)Otan sto synoro
 emfanizetai mia
parastash twn x,y
2 fores thn 8etoume
ish me u.

e)Otan mia parastash
emfanizetai sthn f kai
sto synoro thn 8etoume
ish me u.

*Efarmoges diplwn oloklhrwmatwn.

Ta analoga me ta
apla oloklhrwmata +

6.
 Ogkos anamesa sthn S
kai thn probolh ths ()
sto Oxy:

V= |z(x,y)|dxdy
  ()

7.Embado ths S ean h
probolh sto Oxy einai
():     _____________
        /   z    z
Es= |/ 1+(--)+(--) .dx.dy
   ()      x    y

*8ewrhma Green sto epipedo.

(P(x,y)dx+Q(x,y)dy)= (Qx-Py).dx.dy
+                   ()

1. Ean Qx=Py tote:

 P.dz+Q.dy=0
+

Ean periexei mia pollaplothta
8a einai iso me
opoiadhpote allh
diadromh pou periexei
thn pollaplothta.

*Tripla oloklhrwmata.
             dq
p(x,y,z)= ----------
           dx.dy.dz


dq=p(x,y,z).dx.dy.dz.


q= p(x,y,z).dx.dy.dz=
   
  2  2  2
=  [  ( p(x,y,z).dz) dy] dx.
 1  1  1

*Allagh metablhtwn sto
 oloklhrwma.

Estw x=x(u,v,w),y=y(u,v,w),z=..

(u,v,w)=f(x(u,v,w), y(u,v,w), z(u,v,w))

 f(x,y,z).dx.dy.dz=
(V)

              |(x,y,z)|
= (u,v,w).|--------|.du.dv.dw
 (U)          |(u,v,w)|


a)Ean V einai sfaira tote:

x=r.sin.cos
y=r.sin.cos
z=r.sin
Opou 0ra
     0

(x,y,z)
--------=r.sin
(r,,)

Opou  0<<

b)Ean V kylindros.

x=r.cos
y=r.sin
z=z
Opou 0ra
     02

(x,y,z)
--------=r
(r,,z)

*Efarmoges triplou
oloklhrwmatos.

 Koita efarmoges
epikampyliou olokl.

*Epifaneiako oloklhrwma

1.Ean h epifaneia S
dinetai me e3iswsh
z=z(x,y), kai () h
probolh ths S sto
epipedo Oxy tote:

 f(x,y,z)dS=
(S)
                    _________
= f(x,y,z(x,y)) |/1+Zx+Zy.dx.dy
 xy

Analoga einai kai gia
ta Oxz kai Oyz.

2.Ean h S dinetai se
parametrikh parastash
_ _
r=r(u,v).

x=x(u,v),y=y(u,v),z=z(u,v) ==>
   _
=> r=(x,y,z).

z=z(x,y)==>
r(u,v)=(u,v,z(u,v))=(x,y,z).
    _
_  r
ru=--=(xu,yu,zu).
   u
    _
_  r
rv=--=(xv,yv,zv).
   v


 f(x,y,z)dS=
(S)
                             _    _
= f(x(u,v),y(u,v),z(u,v)).|ru x rv|.du.dv
 F

a)Ean S einai sfairikh:

x=R.sin.cos
y=R.sin.sin
z=R.cos
 _    _
|r x r|=Rsin

b)Ean S kylindrikh:

x=Rcos, y=Rsin, z=z
 _    _
|r x rz|=R

*Epifaneiako olokl.dianysmatikhs synarthshs.

1.H e3iswsh ths epifaneias
S exei do8ei parametrika:
x=x(u,v),y=y(u,v),z=z(u,v)

tote:
   _  _
 F.dS=
(S)
     _                        _    _
= F(x(u,v),y(u,v),z(u,v)).|ru x rv|.du.dv
  (D)
     _  _    _
+ an n.(ru x rv) >0 omoropa.
- diaforetika.

2.H S dinetai ws z=z(x,y).
   _  _
 F.dS=
(s)

= (-P.zx-Q.zy+R).dx.dy
  xy
     _ _
+ an n.z > 0
- diaforetika.


*To 8ewrhma Stokes.
  _  _       _  _
 F.dl= rotF.dS
C       S

*To 8ewrhma Gauss - Ostrogradsky.
   _  _        _
 F.dS= divF.dx.dy.dz
(S)     (V)
