[PDF] [PDF] Chapitre17 : Intégrale double - Melusine

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R2 R 2 R 2

XĂR2

X X iPIm(Pi) X

A R2

m+(A) A m´(A) A

A m´(A) =m+(A)

AB m(AYB) =m(A) +m(B)

m(AYB) =m(A) +m(B)´m(AXB) m C1

D R2

iPI∆i i

ĕ ∆i

ĕ ∆iδ(∆i) =x,yP∆i}x´y}

D R2 f:DÑR

∆i

D R2 f:DÑR

Df=ř

iPIaim(∆i) ɍ(∆i)iPI

D R2 f:DÑR

I´(f) =␣ť

I+(f) =␣ť

Dfť

Df(x,y)xy

D1XD2=H ť

D

1f(x,y)xy+ť

D

2f(x,y)xy=ť

D

1YD2f(x,y)xy

D f(x,y)xy D D g(x,y)xy D xy=m(D) D

Df)ˆm(D)

D f(x,y)xyˇ D |f(x,y)|xy D f D f(x,y)xy= 0ùñ @(x,y)PD,f(x,y) = 0 t Ωuf(x,y)xy= 0 ĕ f(Ω)‰0 f D

ɍD= [a,b]ˆ[a1,b1]aăb,a1ăb1 ĜD

D f(x,y)xy=ż b a

żb1

a

1f(x,y)y!

x=ż b1 a 1 żb a f(x,y)x! y g: [a,b]ÑRh: [a1,b1]ÑR D g(x)h(y)xy=ż b1 a 1 żb a g(x)h(y)x! y= żb a g(x)x!

żb1

a

1h(y)y!

ɍ φ1,φ2[a,b]R

Df(x,y)xy=şb

a

şφ2(x)

1(x)f(x,y)y

x a(φ2(x)´φ1(x))x 1 2 b a

Df(x,y)xy=şb

a

şψ2(y)

1(y)f(x,y)x

y

D O 1

1 1

I=ť

Dxya x

2+ 4y2xy

1´x2(

I=ş1

0xş?

1´x2

0ya x

2+ 4y2y

x

Jx=ş?

1´x2

0ya x

2+ 4y2y

%u=x2+ 4y2 u= 8yy Jx=1 8 x2+4(1´x2) x 2? uu Jx=1 12 (4´3x2)?

4´3x2´x3

I=1 12 ş1

0(4´3x2)3/2x´ş1

0x4x u=4´3x2 u=´6xx 1 12 1 6 4

1u3/2u´ş1

0x4x

ĕ I=7

45

UV R2

UV UV C1

C1

C1φ:UÑR2 φ U

ɍ φ(x,y)PU

J

φ(x,y) =

Bφ1

Bx(x,y)Bφ1

By(x,y)

Bφ2

Bx(x,y)Bφ2

By(x,y)

ɍφ= (φ1,φ2)

K R2

Ŀ R2ŀ

φ UK R2 C1

φ(K) R2 f φ(K)

φ(K)f(x,y)xy=ij

K f(φ(u,v))|Jφ(u,v)|uv R2 R

φ R2R2

A=0 a c b d1 A

R2 φA

(φ1(x,y),φ2(x,y)) (x,y)PR2 0

φ1(x,y)

2(x,y)1

A =0 a c b d1 A0 x y1 A +0 x0 y 01 A

ɍ φ(0,0) = (x0,y0)

(x,y)PR20

Bφ1

Bx(x,y)Bφ1

By(x,y)

Bφ2

Bx(x,y)Bφ2

By(x,y)1

A =0 a c b d1 A

Jφ(x,y) =A‰0

KR2 fφ(K)R

φ(K)f(x,y)xy=ij

K f(φ(u,v))|A|uv

φ |A|= 1

φ(K)f(x,y)xy=ij

K f(φ(u,v))uv @(x,y)PK,(´x,y)PKf(´x,y) =f(x,y)

Kf(x,y)xy= 2ť

K

1f(x,y)xyɍK1=KX(R+ˆR)

φ(x,y) =φ(´x,y)

Oy ť

K

2f(x,y)xy=ť

K

1f(x,y)xyɍK2=KzK1

a 2+y2 b 2= 1 a ⃗i⃗j= (0,1)b⃗j a0 0b

E ȿ C E

φ(K)xy=ij

K abxy=πab

Kxy=π

φ:R2ÝÑR2

(r,θ)ÞÝÑ(rθ,rθ)

φ C1R2 (r,θ)PR2 φ

J

φ(r,θ) =

θ´rθ

θ rθ

=r K r 1 r 2 r 1 r 2

φ(K)

φ(K)

φ(K)f(x,y)xy=ij

K f(rθ,rθ)rrθ r2 r 1r f(rθ,rθ)θ! r m(φ(K)) =ż r2 r 1r r= (β´α)r22´r21 2

φ(K)f(x,y)xy=ż

żφ2(θ)

1(θ)rf(rθ,rθ)r!

D D xy=ż

żf(θ)

0 rr!

θ=1

2 f(θ)2θ

Df(x,y,z)xyz Ę

Ω Rp f: ΩÑRn C1 APΩ RpRn

Bu^ÝÝÑBM

C 1 ∆ÝÑR3 (u,v)ÞÝÑM(u,v) = (x(u,v),y(u,v),z(u,v))

ɍ∆ R2

BM

Bu(u,v)^ÝÝÑBM

Bv(u,v)'

'''ÝÝÑBM

Bu(u,v)^ÝÝÑBM

Bv(u,v)''''uv

ĕ M(u,v)S ÝÝÑBM

Bu(u,v)ÝÝÑBM

Bv(u,v)

BM

Bu(u,v)^ÝÝÑBM

Bv(u,v)''''uv γ: [a,b]ÑR3

a b S

φ(M)σ=ij

φ(x(u,v),y(u,v),z(u,v))'

BM

Bu(u,v)^ÝÝÑBM

Bv(u,v)'

''''uv

γ: [a,b]ÑR3 C1 C

ρ:CÑR+

Cm=ş

γρ(M)s

Ĝ ' GPR3 mÝÝÑOG=ş

γρ(M)ÝÝÑOMs

'%mx

G=ş

γρ(M)xMs

my

G=ş

quotesdbs_dbs23.pdfusesText_29

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