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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) [, ] [, ].