analyse s iii

8/12/2019 Analyse S III

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D


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V = [a1, b1] [a2, b2] [a3, b3] V R3


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C

C R2

z=x+iy x y z z= ei tan() = y

x =|z|

z= 0 z= iy = 0 (x= 0) =

2 2k

d(z1, z2) =|z1 z2| .

zo r >0

D(zo, r) = {z C / |z zo| < r}.

zo C U C

r >0

D(zo, r) U


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O C

C

zo r >0

D(zo, r) ={z C |z zo| r}

U C U U U

U C U f :U C

f(z= x+iy) = u(x, y) +iv(x, y).

u v F ={(x, y) R2/x +iy U}

f :U C zo U

l C f l z zo U limzzo, zUf(z) = l > 0 = () >0

|z zo| < z U |f(z) l| < .

f zo limzzof(z) = f(zo) U z U


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z= x +iy zo = xo+ iyo l= a+ib

f(z) =u(x, y) +iv(x, y)

f

zo

u

v

(xo, yo)

limzzo, f(z) =l

lim(x,y)(xo,yo)

u(x, y) = a lim(x,y)(xo,yo)

v(x, y) = b.

f zo

lim|z|+

f(z) =l

> 0 A= A() >0

|z| > A |f(z) l| < .

z C +

Ozn

n!

C ez z

z C ez C


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ez1+z2 =ez1ez2. e0 = 1 ez = 1

ez

2i

ez+2i =eze2i =ez(cos2+isin2) =ez.

sin cos

eix = cosx +isinx; x R

cosx = (eix) =+p=0

(1)px2p

2p!.

sinx = (eix) =+p=0

(1)p x2p+1

(2p + 1)!.

sin cos

cosz=+p=0

(1)pz2p

2p!. sinz=

+p=0

(1)p z2p+1

(2p + 1)!.

sin cos sin cos

C


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cosz= eiz+eiz

2 =chiz.

sinz= eizeiz

2i = 1

ishiz.

cos2z+sin2z= 1 sin cos

2

zo = 0 o [0, 2[ o ]0, +[ zo = oe

io

ez =zo

{z=LogR(o) +i(o+ 2k); k Z}.

[2] zo

LogC(zo) = Logo+io.

cosz= sinz= ; ; C.

cosz= 2


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U C C f : U C zo U

f zo C zo l C

limh0hC

f(zo+h) f(zo)

h =l.

l= f(zo)

f zo U f zo U

U C C f : U C z = x+iy f f(z) =f(x + iy) =P(x, y) + iQ(x, y) f

f U


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C1 ba

|(t)| dt = .

f : U C C

C1 U

f(z)dz=

ba

f((t))(t)dt.

[a, b]

=pk=1k

f(z)dz=

pk=1

k

f(z)dz.

k 1 k p

f(z)dz=

of(z)dz=

f(z)dz.

M >0

|

f(z)dz| M

ba

|(t)| dt.


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r > 0 a [0, 2[ f r > 0

limr+ zf(z) = 0 ( limr0 zf(z) = 0).

limr+

f(z)dz= 0 ( limr0

f(z)dz= 0).

D C D z1; z2 D

D z1; z2 D D

D D

D C f : D C

D D

a C

f(z)

z adz = 2if(a) a .

= 0

D C f D D f(z)dz


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D C f D

1

f(z)dz= 2

f(z)dz

1; 2


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f :D CC D zo D f

zo

r > 0

(an)nN

|z zo |< r = f(z) =+n=0

an(z zo)n.

f : D C C D D f D

D zo D f D\{zo} f zo (cn)nZ

f(z) =+

n=

cn(z zo)n.

zo

D C

zoD

f D\{zo} f zo (cn)nZ

f(z) =+

n=

cn(z zo)n.

zo


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c1 f f zo

c1= 1

2i

f(z)dz.

f :D C zo f f zo zo f

f zo

ck= 0 k


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a r >0 [0, 2[ f |z|>

lim|z|=r+

f(z) = 0.

limr+

eizf(z)dz= 0 ]0;+[ .

+

P(x)

Q(x)dx.

P, Q d Q d P2 Q

f(z) = P(z)Q(z)

0 R >0 Imz0 [R, R] f Imz 0

+

dx1+x4

dx

+

f(x)eixdx ]0, +[.

f lim|z|+f(z) = 0 g(z) =eizf(z)

+0 cosx1+x4dx +0 sinxx dx


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I=

20

R(cosx, sinx)dx.

R(., .)

0 1 z C(0, 1)

z= ei ;d=dz

iz ; cos=

1

2(z+

1

z) ;sin=

1

2i(z

1

z).

I z

C(0,1)R(

1

2(z+

1

z),

1

2i(z

1

z))

dz

iz.

20

d2+cos

I1 =

+0

R1(x)

x dx ; I2 =

+0

R2(x)Ln(x)dx.

+

0

xLnx

(1 + x)3

dx.

0 <


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f : [a, b] R

Sn [a, b]

a= xo < x1 < x2 < ..... < xn= b

xk =a +kban

1 k n

wk [xk1, xk]

(f, Sn) =n

k=1

f(wk)(xk xk1).

|Sn

|=sup1kn

(xk

xk1

) ban

f


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[a, b] ba

f(t)dt= lim|Sn|0

nk=1

f(wk)(xk xk1).

f [a, b]

f(wk) xk xk1 II 1 f Ox {x= a} {x= b} II 1

f D R R

R

f D D R2

R R= [a, b] [c, d] D II 2

wij = [a+ (i 1)h1; a +ih1] [c + (j 1)h2; c +jh2]

n1 n2 h1= ban1

; h2= cdn2

[a, b] h1 [c, d] h2 h= (h1, h2) vij = (i, j) wij Rij =wij D n1 n2

RijD

h1h2f(i, j)


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D z = f(x, y) D D Oz xoy

f D D

f(x, y)dxdy= limh=(h1,h2)(0,0)

RijD

h1h2f(i, j)

D= [a, b] [c, d] R D D Rij D

n1i=1

n2j=1

h1h2f(i, j)

[a,b][c,d]

f(x, y)dxdy= limh=(h1,h2)(0,0)

n1i=1

n2j=1

h1h2f(i, j)

f(x, y) = 1

Aire(D) =

Ddxdy.

f D R R R

f D D


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f 0 D z = f(x, y) D Oz xOy D

V =

D

f(x, y)dxdy

f=f1+f2 D

f(x, y)dxdy=

D

f1(x, y)dxdy+

D

f2(x, y)dxdy.

D= D1 D2

D1

D2= D

f(x, y)dxdy=

D1

f(x, y)dxdy+

D2

f(x, y)dxdy.

f0 D

Df(x, y)dxdy 0 Aire(D) =

D

dxdy

D =[a, b] [c, d]

f

f D D

D

f(x, y)dxdy=

ba

(

dc

f(x, y)dy)dx=

dc

(

ba

f(x, y)dx)dy.

f(x, y) = k(x)g(y)

D f(x, y)dxdy= (

ba k(x)dx)(

dc g(y)dy).


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D

D D

D= {(x, y) , x [a, b] k1(x) y k2(x)}

D= {(x, y), y [c, d] h1(y) x h2(y)}

k1

k2

h1 , h2

[a, b] [c, d]

D Dx Dy D

D

f(x, y)dxdy

D

f(x, y)dxdy=

dc

F(y)dy F(y) =

k2(y)k1(y)

f(x, y)dx.

D

f(x, y)dxdy=

ba

G(x)dx G(x) =

h2(x)h1(x)

f(x, y)dy.

f D

D

f(x, y)dxdy=

ba

(

h2(x)h1(x)

f(x, y)dy)dx


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D

f(x, y)dxdy=dc(k2(y)k1(y)

f(x, y)dx)dy

D

D f(x, y)dxdy=

ba ( h2(x)h1(x) f(x, y)dy)dx=

dc (

k2(y)k1(y) f(x, y)dx)dy.

f f(x, y) =f(x, y) D x= 0 Oy

D

f(x, y)dxdy= 0.

x y

D (x, y)

m=

D

(x, y)dxdy.

(xG, yG)

xG= 1

m

D

x(x, y)dxdy ; yG= 1

m

D

y(x, y)dxdy.

D

I =

D

d(M, )2(x, y)dxdy.

d(M, ) M

R R g C1 D (u, v) g(u, v) = (x, y) = (a(u, v); b(u, v))


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g (u, v)

J(u, v) =

au

(u, v) av

(u, v)bu

(u, v) bv

(u, v)

.

g J DJ(u, v)

DJ(u, v) = J(u, v) =a

u(u, v)

b

v(u, v)

a

v(u, v)

b

u(u, v).

R ; RR g: R C1

|DJ(u, v) | = 0 DJ g

g(u, v) = (x, y) = (a(u, v); b(u, v)).

f :R R R

f(x, y)dxdy =

f(a(u, v); b(u, v)) |DJ(u, v) |dudv

=

f(g(u, v)) |DJ(u, v) |dudv.


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II

f : [a, b] [c, d]R

f [a, b] [c, d] h= (h1, h2) (0, 0)

Ih =n1i=1

n2j=1

h1h2f(i, j)

h1= b a

n1; h2 =

c d

n2 wij = [a + (i 1)h1; a + ih1] [c + (j 1)h2; c +jh2] (i, j) wij f(x,y,z) V R3

V

f(x,y,z)dxdydz

V = [a1, b1] [a2, b2] [a3, b3] II f D


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V

R3

V R3 P = [a1, b1] [a2, b2] [a3, b3] V

Vh= {wijk V ; 1 i n1 ; 1 j n2 ; 1 k n3}.

h = (h1, h2, h3) hm1 m 3

f V

V

f(x,y,z)dxdydz = limh(0,0,0)

wijkV

h1h2h3f(i, j, k)

= limh(0,0,0)

wijkVh

h1h2h3f(i, j, k).

h (0, 0, 0) Vh V V V

V

dxdydz

f V R R R R

f V V


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f=f1+f2 V

f(x,y,z)dxdydz=

V

f1(x, y, z )dxdydz+

V

f2(x,y,z)dxdydz.

V =V1 V2

V1

V2=

V

f(x,y,z)dxdydz=

V1

f(x,y,z)dxdydz+

V2

f(x,y,z)dxdydz.

f0 V V

f(x,y,z)dxdydz 0.

V olume(V) = Vdxdydz f(x,y,z) V

f(x, y, z )dxdydz V

V

V

V

V ={(x,y,z) R3 a3 z b3 ; (x, y) Dz}.

Dz V xOy z Dz z a3 V b3 V

Vf(x, y, z )dxdydz=

b3a3 (

Dz f(x,y,z)dxdy)dz.


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V

V ={(x,y,z) R3 (x, y) z(x, y) ; (x, y) Dz}.

V

f(x,y,z)dxdydz=

Dz

(

(x,y)(x,y)

f(x,y,z)dz)dxdy.

Dz

V

(x, y, z ) V

x [a, b]y [(x), (x)]z [(x, y), (x, y)]

V

V

f(x,y,z)dxdydz= ba (x)

(x) (x,y)

(x,y)

f(x, y, z )dz dydx. V z= 0

xOy f(x,y, z) = f(x,y,z) (x, y) = (x, y)

V

f(x,y,z)dxdydz= 0.

x,y,z V


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(x,y,z) V V

m=

V

(x,y,z)dxdydz.

V

xG =

1m

V

x(x,y,z)dxdydzyG =

1m

V

y(x, y, z )dxdydzzG =

1m

V

z(x,y,z)dxdydz

V R3 : V

C1

(u,v,w) = (x,y,z) = (a(u,v,w); b(u,v,w); c(u,v,w))

(u,v,w) x =a(u,v,w)y =b(u,v,w)

z =c(u,v,w)

J(u,v,w) =

au (u,v,w) av (u,v,w) aw (u,v,w)b

u(u,v,w) b

v(u,v,w) b

w(u,v,w)

cu

(u,v,w) cv

(u,v,w) cw

(u,v,w)

.

DJ(u,v,w) J(u,v,w)

g(u,v,w) = f (u,v,w) = f(a(u,v,w), b(u,v,w), c(u,v,w)).


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V

f(x,y,z)dxdydz=

g(u,v,w) |DJ(u,v,w) |dudvdw.

| DJ(u,v,w) |

M (,,)

(O,i ,

j ,

k)

i Ox

j Oy

k Oz

OM

OM

xOy

= OM

=

(Ox,OM)

=

(OM,

OM)

(x, y, z ) M (O,i ,

j ,

k)

x = cos cos

y = cos sin

z = sin

DJ=2cos f V

(,,) (x,y,z) V.


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V

f(x, y, z )dxdydz=

f(coscos, cossin, sin)2 |cos| ddd.

M Oz

(,,z) (x,y,z) V.

x = cos

y = sin

z = z

DJ= f V

(,,z) (x,y,z) V.

V

f(x, y, z )dxdydz=

f(cos, sin, z)dddz.


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V V A

d(M, A) = distance(M, A) =distance((x,y,z), A) =d((x, y, z ), A)

(x,y,z) M V V A

JA=

V

d((x,y,z), A)2(x,y,z)dxdydz

(x, y, z ) V M A

M


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R3 (O,

i ,

j ,

k)

g : D R3 R3 u= (x, y, z ) D

g(u) = P(u)

i +Q(u)

j +R(u)

k

g(u) = (P(u), Q(u), R(u))

2

f :D R3 R f D f

gradf(u) =g(u) = (

f

x(u),

f

y(u),

f

z(u)).

f h


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grad(f+ h)(u) =

gradf(u) +

gradh(u).

S

f(x,y,z) = 0

mo= (xo, yo, zo) S

gradf(mo) S mo T S mo

m T gradf(mo)

mom

gradf(mo).mom= 0

f : D R3 R f(x,y,z) = 0 mo

(x xo)f

x(mo) + (y yo)

f

y(mo) + (z zo)

f

z(mo) = 0.

g : D R3 R3

f : D R3 R u D g((u) =

gradf(u) g f g(u) =(P(u), Q(u), R(u))

df(u) = P(u)dx +Q(u)dy+R(u)dz.

df(u) = P(u)dx +Q(u)dy+R(u)dz

g

fx

(x,y,z) =P(x,y,z)fy

(x,y,z) =Q(x,y,z)fz

(x,y,z) =R(x, y, z )

R f(x,y,z) =


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g : D R3 R3

(P,Q,R) g g

rot(g) =

Ry

Q

z,

Pz

Rx

,Qx

Py

.

rot(g) =

x

yz

PQ

R

.

a = (a1, a2, a3)

b =(b1, b2, b3)

a b =

a2 a3b2 b3i

a1 a3b1 b3j +

a1 a2b1 b2k .

f f :D R3 R

V1,

V2

rot(

V1+

V2) =

rot(

V1) +

rot(

V2)

rot(

V1) =

rot(

V1)

rot(f

V1) =f

rot(

V1) +

gradf

V1.

M(t) Oz (r,,z) M(t) r z (t) = M(t)

OM(t) = x(t) =rcos(t)

y(t) =rsin(t)z(t)


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V(M(t)) =

dOM(t)

dt =

rsin(t)rsin(t)

0

=

y(t) x(t)

0

.

=

k

V(M(t)) =

OM(t)

g : D R3 R3 C1 (P,Q,R) g

rot(g)(u) = 0 u D D

h : D R3 R3 g : D R3 R3

u D

rotg(u) =

h (u) h g h (P1, Q1, R1) g = (P,Q,R) h

P1 =R

y

Q

z Q1 =

P

z

R

x R1=

Q

x

P

y.

g : D R3 R3 (P,Q,R) g

div(g)(u) = Px

+Q

y +

R

z (u).


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h , g : D R3 R3

f :D R

R div(

h + g) =div(

h ) +div(g).

div(fg) =fdiv(g) +grad(f).g

div(h g) =g .

rot(

h )

h .

rot(g)

h: D R3 R3 C1

(P,Q,R) h

C2

div(h)(u) = 0 u D.

D

Rn

: L(Rn;R).

f : Rn R C1 f df

df :x dfx=n

j=1

f

xj(x)dxj .


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dx1;dx2; ... ; dxn {ej}; 1 j n dxi(

nj=1 xjej) = xi x

(x) =n

i=1

ai(x)dxi; x= (x1, x2,...,xn) .

Ck ai C

k k N k = 0

C0 Rn f

Ck = df f

C1 U Rn

(x) =n

i=1

ai(x)dxi; x= (x1, x2,...,xn) U.

U

aidxj

(x) =ajdxi

(x) x= (x1, x2,...,xn) U.

2 3

= 2y2(x +y)dx + 2xy(x + 3y)dy

g: U Rn R g(x1, x2,...,xn) g

= 2xzdx 2yzdy (x2 y2)dz


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R2 R3

: [a, b]

t (t)

(t) = (x(t), y(t)) (t) = (x(t), y(t), z(t)) C1 [a, b[ \{(b)}

C1

(a) =(b)

f : D R D

R2

R3 f

I=

f dl=

f(x, y)dl=

ba

f((t)) (t) dt.

I= f dl=

f(x,y,z)dl=

ba f((t))

(t) dt.

R2 R3 I=

f dl f dl=(t) dt

L L = dl =

ba

(t) dt

L= ba

(x

(t))2

+ (y

(t))2

dt

L= ba

(x

(t))2

+ (y

(t))2

+ (z

(t))2

dt.


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2

f(x, y)dx=

ba

f(x(t), y(t))x(t)dt .

f(x, y)dy=

b

a

f(x(t), y(t))y(t)dt.

M=

(x, y)dl M=

(x, y, z )dl G G

xG= 1

M

x(x, y)dl yG= 1

M

y(x, y)dl.

3

U Rn

(x) =n

i=1

ai(x1,...,xn)dxi; x= (x1, x2,...,xn) U.

C1 U

t [a, b]OM(t) =

ni=1

xi(t)ei.

=

(M(t))dM

dt dt= b

a

n

i=1

ai(x1(t),...,xn(t))x

i

(t)dt.


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C1 m Cj C

1

=m

j=1

Cj

.

n= 2 = P dx +Qdy

P dx +Qdy=

ba

(P(x(t), y(t))x(t) +Q(x(t), y(t))y(t)) dt.

+=

C

(AB)

(AB)

=

(AD)

+

(DB)

.


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C1

V U f

V =

f C1

(AB) A

B U (AB)

V(M).

dM=f(B) f(A).

V U

C1 U

V(M).

dM= 0

V(M).

dM= 0

R2

K R2 (C) C1 (C) (C) K (C)

U R2

(x, y) = P(x, y)dx +Q(x, y)dy (x, y) U.


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KU C C1

(C)(P(x, y)dx +Q(x, y)dy) =

KQ

x

P

y (x, y)dxdy.

K

K={(x, y) R2 a x b (x) y (x) }.

K={(x, y) R2

c y d

h(y) x g(y) }.

K R2 (C)

C1

A K

A=1

2

(C)

(xdy ydx).

A=

K

dxdy=1

2

(C)

(xdy ydx).


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D R2 (C) 1 2 r= r() A D

A= 12

2

1

r()2d.

x y x= r()cos y= r()sin.


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3

Ck ; k N

F

U R2

R3


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Ck F

M(u, v) U F(M(u, v))

x= x(u, v); y= y(u, v); z= z(u, v).

x; y; z Ck

U V R2 F : U R2 G :V R2 F G : U V Ck

F =G .

F F F Ck F

R3

R3 S R3 F(U) Ck U

F G S = F(U)

V

S= G(V)

z= x2 y2

S= {(x, y, z ) R3 z= x2 y2}

F(R2)

F : R2 R3 F(x, y) = (x,y,x2 y2)


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C

G:: R2 R3 G(u, v) = (u+v, u v, 4uv)

G F (x, y) = (x+y; x y) G(R2)

z= f(x, y) f(x, y, z ) = 0

z= f(x, y) f R

x= x, y= y, z= f(x, y).

S f(x, y, z ) = 0 f

Mo = (xo, yo, zo) S f Mo U (xo, yo) : U R

3

G(x, y) = (x,y,(x, y)).

S


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S F M(u, v) S F M(u, v) 2 S

F : (u, v) M(u, v) = (x(u, v); y(u, v); z(u, v)).

M

u =

x

u

i +

y

u

j +

z

u

k

M

v =

x

v

i +

y

v

j +

z

v

k

Mo(uo, vo)

S Mo Mo Mu

Mv

n1 =

M

u

M

v .

M

u ,

M

v .

M

n =n1

n1 .


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z = f(x, y) (x, y) K K C1

S M(x, y, f (x, y)) (x, y)

z=

1 x2 y2

(x, y) K={(x, y) x2 +y2 1

4}.

x= sincos ; y= sinsin ; z= cos .

(, )D = [0, 2[[0, ] S

AB,

AC

AB

AC

C1 u v u v

M(u, v) uMu

vMv

d=

Mu

Mv

uv.


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={M(u, v) ; (u, v) D}.

A(

) =

D

d=

D

M

u

M

v uv.

x= sincos ; y= sinsin ; z= cos .

D= D = [0, 2] [0, ]

z= f(x, y) (x, y) D.

g R3 R U S C1 g S D

S={M(u, v) ; (u, v) D}.

D

g(M(u, v)))

M

u

M

v

uv.

=

S

g(M)d.

g

Sg(M)d S


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S

m =

S

(M)d

xG = 1m

S

x(M)d

yG = 1

m

S

y(M)d

zG = 1

m

S

z(M)d

S

I=

S d(M, )

2

(M)d.

V

U S C1

V

S D

= {M(u, v) (u, v) D}

P(V) =

D

M

u

M

v uv.

n

P

(

V) = P

d.


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S R3 n

S

C1

S

V C1 V1, V2; V3

V S

V

S

rot

V .n d=

V .

dM=

(V1dx +V2dy+V3dz).

Du u P(u) (C)

V(u) M

(u, t)

OM=

OP(u) +t

V(u).

x(u, t) = a(u) +t(u)

y(u, t) = b(u) +t(u)

z(u, t) = c(u) +t(u)

x(u, t) = u+t

y(u, t) = u2 + 2t

z(u, t) = 1 + 4tu


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u

x= a(u), y= b(u), z= c(u).

u= i +

j +

k

x(u, t) = a(u) +t

y(u, t) = b(u) +t

z(u, t) = c(u) +t

H M S (HM)

P t

HM=tHP

OM= (1 t)

OH+t

OP .

x= a(u) y= b(u) z=c(u)

OH =

i +

j +

k

x(t, u) = (1t)+ta(u); y(t, u) = (1t)+tb(u); z(t, u) = (1t)+tc(u).


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D R3 M A D D

M

D D = (Oz) (xOz)

x= a(u) y = 0 z= c(u).

x(u, v) = a(u)cosv; y(u, v) = a(u)sinv; z(u, v) =c(u).

x(u, t) = (2 + cos(u))cos(v)

y(u, t) = (2 + cos(u))sin(v)(

z(u, t) = u (u, v) [, ] [, ].

analyse s iii

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