4.3 THÉORÈME DE GREEN

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4.3 THÉORÈME DE

GREEN

(2)

Au dernier cours, nous avons vu

(3)

Au dernier cours, nous avons vu

Théorème fondamentale des intégrales curvilignes.

(4)

Au dernier cours, nous avons vu

Indépendance de chemin.

(5)

Au dernier cours, nous avons vu

Indépendance de chemin.

Déterminer si un champ de vecteur est conservatif.

(6)

Aujourd’hui, nous allons voir

(7)

Aujourd’hui, nous allons voir

Théorème de Green

(8)

Le théorème fondamental du calcul et le théorème fondamental des intégrales curvilignes disent que pour évaluer une intégrale il suffit de

connaitre une primitive sur les bords de la région d’intégration.

(9)

Z _b

a

f ⁰(x)dx = f (b) f (a)

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(10)

Z _b

a

f ⁰(x)dx = f (b) f (a)

Z

C rf · d~r = f (~r(b)) f (~r(a))

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(11)

Z _b

a

f ⁰(x)dx = f (b) f (a)

Z

C rf · d~r = f (~r(b)) f (~r(a))

(12)

Que veut-on dire par le bord d’une région d’intégration d’une intégrale double?

(13)

(14)

R<latexit sha1_base64="NeRUGML7STubCEb4M5AQOMIz+GA=">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</latexit>

(15)

C

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(16)

C

On note souvent le bord d’une région

(17)

C

@<latexit sha1_base64="mhdHlRqlNb7qY6hbALvPDlombv8=">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</latexit> R = C

(18)

C

(pas une dérivée partielle!)

(19)

C

(20)

C

Un peu comme pour un intervalle, on doit spécifier un sens de parcours.

(21)

C

(22)

C

(23)

C

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C

(25)

Théorème

(Théorème de Green)

Si le bord de la région est une courbe plane simple et fermée orientée positivement et que et ont des dérivées partielles continues sur

R<latexit sha1_base64="qQiNzekjzX8orQSelmt5QwQHD5E=">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</latexit>

P<latexit sha1_base64="cSd6xaRwgzFHOFcHNtFkevh+vX0=">AAADHXicjVLLSsNAFD2N7/qqunQTLIKrkqqgy6IblxWsCrVIkk7r0DQJyaQQil/gVtd+jTtxK/6B/oV3rqOoxceEJGfOvefO3IcXBzJVjvNUsMbGJyanpmeKs3PzC4ulpeXjNMoSXzT8KIiSU89NRSBD0VBSBeI0ToTb9wJx4vX2tf1kIJJURuGRymPR6rvdUHak7yqiDuvnpbJTcXjZo6BqQBlm1aOlwhjO0EYEHxn6EAihCAdwkdLTRBUOYuJaGBKXEJJsF7hEkbQZeQnycInt0bdLu6ZhQ9rrmCmrfToloDchpY110kTklxDWp9lszziyZn+KPeSY+m45/T0Tq0+swgWxf+nePf+v0/eXXJffM1boYJcz1d4xM7oGvjkr49rp/OxPuSuKEBOncZvsCWGfle/dsFmTcoV0B1y2P7OnZvXeN74ZXkwuAgOOmn/kOOROS7LHXPGckKIv95Imp/p9TkbB8WalulXZPNwu1/bMDE1jFWvYoDnZQQ0HqKPBt7nCNW6sW+vOurce3lytgtGs4MuyHl8BLQaiig==</latexit> Q<latexit sha1_base64="t4+B0e7iJCkADuS+thuLIyS6/n8=">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</latexit>

(26)

Théorème

‰

@R

P dx + Q dy =

¨

R

@Q

@x

@P

@y dA

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(27)

Théorème

‰

@R

P dx + Q dy =

¨

R

@Q

@x

@P

@y dA

‰

@R

F~ · d~r =

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(28)

Théorème

Preuve:

‰

@R

P dx + Q dy =

¨

R

@Q

@x

@P

@y dA

‰

@R

F~ · d~r =

(29)

Théorème

Preuve:

Commençons pour des régions de type I et II.

‰

@R

P dx + Q dy =

¨

R

@Q

@x

@P

@y dA

‰

@R

F~ · d~r =

(30)

Théorème

Preuve:

‰

@R

P dx + Q dy =

¨

R

@Q

@x

@P

@y dA

a<latexit sha1_base64="HXNi2ziZI5xVOeyio5Y21rCNhno=">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</latexit>

b<latexit sha1_base64="D+WRdc6TbX/xKWz3mmXpP6rTDt8=">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</latexit>

g(x)

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h(x)

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‰

@R

F~ · d~r =

(31)

Théorème

Preuve:

‰

@R

P dx + Q dy =

¨

R

@Q

@x

@P

@y dA

‰

@R

P dx =

¨

R

@P

@y dA

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h(x)

‰

@R

F~ · d~r =

(32)

Théorème

Preuve:

‰

@R

P dx + Q dy =

¨

R

@Q

@x

@P

@y dA

‰

@R

P dx =

¨

R

@P

@y dA

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b<latexit sha1_base64="D+WRdc6TbX/xKWz3mmXpP6rTDt8=">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</latexit>

g(x)

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h(x)

‰

@R

F~ · d~r =

(33)

Théorème

Preuve:

‰

@R

P dx + Q dy =

¨

R

@Q

@x

@P

@y dA

‰

@R

P dx =

¨

R

@P

@y dA

h(x)

‰

@R

F~ · d~r =