cours 27
4.3 THÉORÈME DE
GREEN
Au dernier cours, nous avons vu
Théorème fondamentale des intégrales curvilignes.
Indépendance de chemin.
Déterminer si un champ de vecteur est conservatif.
Aujourd’hui, nous allons voir
Théorème de Green
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.
Z
ba
f
0(x)dx = f (b) f (a)
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Z
C
r f · d~ r = f ( ~ r (b)) f (~ r (a))
<latexit sha1_base64="rb0vmjWsiIsCxZGHXv2BlOaMQYw=">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</latexit>
Le théorème de Green dit à peu près la même chose, mais pour les
intégrales doubles.
Que veut-on dire par le bord d’une région d’intégration d’une intégrale double?
C
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Un peu comme pour un intervalle, on doit spécifier un sens de parcours.
Le sens antihoraire est considéré positif.
R
<latexit sha1_base64="NeRUGML7STubCEb4M5AQOMIz+GA=">AAAC9nicjVJNS8NAEH2NX/W76tFLsAieSqqCHotePLZiW0GLJHHVpWkSkk0xFH+BV716E6/+Hf+B3vwJzo5b8QPRDdm8fTNvMjM7XhzIVDnOU8EaGR0bnyhOTk3PzM7NlxYWW2mUJb5o+lEQJYeem4pAhqKppArEYZwIt+cFou11d7W93RdJKqPwQOWx6PTc81CeSd9VRDX2T0plp+Lwsn+CqgFlmFWPSq84xiki+MjQg0AIRTiAi5SeI1ThICaugwFxCSHJdoErTJE2Iy9BHi6xXdrP6XRk2JDOOmbKap/+EtCbkNLGKmki8ksI67/ZbM84smZ/iz3gmDq3nL6eidUjVuGC2L90Q8//6nQtCmfY5hok1RQzo6vzTZSMu6Iztz9VpShCTJzGp2RPCPusHPbZZk3Kteveumx/Zk/N6rNvfDO8mCwF+hw1/8h+wHcoyR5zL3NCina+JRqJ6vcB+Ala65XqRmW9sVmu7ZjhKGIZK1ijCdhCDXuoo8nZXOMGt9aldWfdWw/vrlbBaJbwZVmPb4TAmxo=</latexit>@
<latexit sha1_base64="mhdHlRqlNb7qY6hbALvPDlombv8=">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</latexit>R = C
On note souvent le bord d’une région
(pas une dérivée partielle!)
C
<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit>
R
<latexit sha1_base64="NeRUGML7STubCEb4M5AQOMIz+GA=">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</latexit>Théorème
Preuve:
(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>R
<latexit sha1_base64="qQiNzekjzX8orQSelmt5QwQHD5E=">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</latexit>Commençons pour des régions de type I et II.
‰
@R
P dx + Q dy =
¨
R
@ Q
@ x
@ P
@ y dA
<latexit sha1_base64="7OvRhyQPAgZ7acKYSNlEKCUr1kE=">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</latexit>
‰
@R
P dx =
¨
R
@ P
@ y dA
<latexit sha1_base64="1xVue+Aik05jW9w9Ylzy8X0F6Zw=">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</latexit>
¨
R
@ P
@ y dA
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=
ˆ
b aˆ
h(x) g(x)@ P
@ y (x, y ) dydx
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=
ˆ
b a(P (x, h(x)) P (x, g (x)) dx
<latexit sha1_base64="kFf9Mf6p8xVY2tu0O/q1ItfocwE=">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</latexit>
a
<latexit sha1_base64="HXNi2ziZI5xVOeyio5Y21rCNhno=">AAAC9nicjVJNS8NAEH2NX7V+VT16CRbBU0lV0GPRi8cW7AeoSJJudTFNQrIpluIv8KpXb+LVv+M/0Js/wdlxK2oR3ZDN2zfzJjOz48WBTJXjPOesicmp6Zn8bGFufmFxqbi80kyjLPFFw4+CKGl7bioCGYqGkioQ7TgRbs8LRMu7PND2Vl8kqYzCIzWIxWnPPQ9lV/quIqrunhVLTtnhZY+DigElmFWLim84QQcRfGToQSCEIhzARUrPMSpwEBN3iiFxCSHJdoFrFEibkZcgD5fYS9rP6XRs2JDOOmbKap/+EtCbkNLGBmki8ksI67/ZbM84smZ/iz3kmDq3AX09E6tHrMIFsX/pRp7/1elaFLrY4xok1RQzo6vzTZSMu6Izt79UpShCTJzGHbInhH1Wjvpssybl2nVvXba/sKdm9dk3vhleTZYCfY46+Mx+yHcoyR5zLweEFO18SzQSlZ8DMA6aW+XKdtmp75Sq+2Y48ljDOjZpAnZRxSFqaHA2N7jFnXVl3VsP1uOHq5UzmlV8W9bTO6qqmyc=</latexit>b
<latexit sha1_base64="D+WRdc6TbX/xKWz3mmXpP6rTDt8=">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</latexit>g (x)
<latexit sha1_base64="1ZLBwVgD5sCUD4yHL4cZSk9KwcU=">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</latexit>
h(x)
<latexit sha1_base64="XhsvBssyl/h/UmV8C1FFbcMH/xQ=">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</latexit>
‰
@R
F ~ · d~ r =
<latexit sha1_base64="houNCxEpRB/5LcFiOalOPTt0OOk=">AAADUnicjVLLbhMxFD3J8CilQFqWbCwiJFbRpFQqG6QKpIplQaSt1Kkij+MWK8545PEERVF/hK9hS9fd8AfwC6y4vrgIqHh4NDPH595z7fsoa2uakOefOt3s2vUbN1durd5eu3P3Xm99Y79xrVd6pJx1/rCUjbam0qNggtWHtddyVlp9UE5fRPvBXPvGuOpNWNT6eCZPK3NilAxEjXtbhTNVUMEr69T0nWn0eFnU0gcjrXh9JkQx10rsFmrigpjwxj8b9/r5IOclroJhAn2ktefWOxkKTOCg0GIGjQqBsIVEQ88RhshRE3eMJXGekGG7xhlWSduSlyYPSeyUvqe0O0psRfsYs2G1olMsvZ6UAo9I48jPE46nCba3HDmyf4q95Jjxbgv6lynWjNiAt8T+S3fp+f+6eH/Ddfl7xgEneMqZRu+amVgDlc5quXYxP/FT7oEi1MRFPCG7J6xYedkNwZqGKxQ7INn+mT0jG/cq+bb4knLRmHPUxY8cl9xpQ/aaK74gFOjLvaTJGf4+J1fB/uZg+GSw+Wqrv/M8zdAKHuAhHtOcbGMHL7GHEd3mPT7gI867F92vWSfLvrt2O0lzH7+sbO0bNO601A==</latexit>
a
<latexit sha1_base64="HXNi2ziZI5xVOeyio5Y21rCNhno=">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</latexit>b
<latexit sha1_base64="D+WRdc6TbX/xKWz3mmXpP6rTDt8=">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</latexit>g(x)
<latexit sha1_base64="1ZLBwVgD5sCUD4yHL4cZSk9KwcU=">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</latexit>
h(x)
<latexit sha1_base64="XhsvBssyl/h/UmV8C1FFbcMH/xQ=">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</latexit>Preuve: Commençons pour des régions de type I et II.
‰
@R
P dx =
¨
R
@ P
@ y dA
<latexit sha1_base64="1xVue+Aik05jW9w9Ylzy8X0F6Zw=">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</latexit>
¨
R
@ P
@ y dA
<latexit sha1_base64="dDv/Eus224cI1JZtpe/PEzYvOBQ=">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</latexit>
=
ˆ
b a(P (x, h(x)) P (x, g (x)) dx
<latexit sha1_base64="kFf9Mf6p8xVY2tu0O/q1ItfocwE=">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</latexit>
C
1<latexit sha1_base64="0FVLC26qoEqGCcARn5PiYjZH6Mk=">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</latexit>
C
<latexit sha1_base64="URWvDrb+4sVmMiv8fu/gH5vw9d8=">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</latexit> 2C
3<latexit sha1_base64="00Ga640wgh0b/VAbH+Y1ZU3/ZuA=">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</latexit>
= ˆ
C1
P dx
ˆ
C3
P dx
<latexit sha1_base64="DYF0kv5x1k7mAXONDDpj9lYNVTg=">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</latexit>
=
ˆ
b aP (x, g (x)) dx
ˆ
b aP (x, h(x)) dx
<latexit sha1_base64="yz9QleH57KIK04RxS6StbugqZ/g=">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</latexit>
=
ˆ
b aP (x, h(x)) dx
ˆ
b aP (x, g (x)) dx
!
‰
@R
P dx
<latexit sha1_base64="JLKP7HuC/EbvzDGq/P/1zy4OhP4=">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</latexit>
= ˆ
C1
P dx + ˆ
C2
P dx + ˆ
C3
P dx
<latexit sha1_base64="cOyzKhGEiEDeAy9w30LlGkVsRjM=">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</latexit>
a
<latexit sha1_base64="HXNi2ziZI5xVOeyio5Y21rCNhno=">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</latexit>b
<latexit sha1_base64="D+WRdc6TbX/xKWz3mmXpP6rTDt8=">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</latexit>g(x)
<latexit sha1_base64="1ZLBwVgD5sCUD4yHL4cZSk9KwcU=">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</latexit>
h(x)
<latexit sha1_base64="XhsvBssyl/h/UmV8C1FFbcMH/xQ=">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</latexit>Preuve: Commençons pour des régions de type I et II.
‰
@R
P dx =
¨
R
@ P
@ y dA
<latexit sha1_base64="1xVue+Aik05jW9w9Ylzy8X0F6Zw=">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</latexit>
¨
R
@ P
@ y dA
<latexit sha1_base64="dDv/Eus224cI1JZtpe/PEzYvOBQ=">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</latexit>
=
ˆ
b a(P (x, h(x)) P (x, g (x)) dx
<latexit sha1_base64="kFf9Mf6p8xVY2tu0O/q1ItfocwE=">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</latexit>
C
1<latexit sha1_base64="0FVLC26qoEqGCcARn5PiYjZH6Mk=">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</latexit>
C
<latexit sha1_base64="URWvDrb+4sVmMiv8fu/gH5vw9d8=">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</latexit> 2C
3<latexit sha1_base64="00Ga640wgh0b/VAbH+Y1ZU3/ZuA=">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</latexit>
=
ˆ
b aP (x, h(x)) dx
ˆ
b aP (x, g (x)) dx
!
<latexit sha1_base64="7bq5rIg21ev9XT/GARWUNMi/FI8=">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</latexit>
‰
@R
P dx
<latexit sha1_base64="JLKP7HuC/EbvzDGq/P/1zy4OhP4=">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</latexit>
=
¨
R
@ P
@ y dA
<latexit sha1_base64="Y0bBIF5WhuSXRF0PJMjc90R8wwA=">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</latexit>
‰
@R
P dx + Q dy =
¨
R
@ Q
@ x
@ P
@ y dA
<latexit sha1_base64="7OvRhyQPAgZ7acKYSNlEKCUr1kE=">AAADkXicjVLbbhMxEJ3tcknLLaWPvFhESEhAtClIwANSgBckXpKKtJW6VeR1nNaKs17Z3tJVtB/aP4CvgPHIpaUVF0fxHp85Z+wZu6i0cj7LzpK19MbNW7c76xt37t67/6C7+XDXmdoKORFGG7tfcCe1KuXEK6/lfmUlXxZa7hWLjyG+dyKtU6b84ptKHi75UanmSnCP1LTrcqNKL7wV2ojFV+XkdJVX3HrFNdtpGRux/Pns9NkY5+ZdrlA83cnnlosL2bi9wKftiyvR0aVo02Ka99NuL+tnNNh1MIigB3GMzGaSQg4zMCCghiVIKMEj1sDB4e8ABpBBhdwhrJCziBTFJbSwgd4aVRIVHNkFzke4OohsieuQ05Fb4C4a/xadDJ6gx6DOIg67MYrXlDmwf8q9opzhbA1+i5hriayHY2T/5TtX/r8vnF9RX/5esYc5vKFKg7oiJvRAxL1q6l2oj12q3WOGCrmAZxi3iAU5z2+DkcdRh8INcIp/I2Vgw1pEbQ3fYy0STihr86vGFd20wnhFHW8QeZzpLvHlDK6+k+tgd7s/eNnfHr/qDT/EN9SBR/AYnuI7eQ1D+AQjmOBpzuBH0knW0630bTpMo3YtiZ4t+G2kn38CiD/Lbg==</latexit>
Preuve: Commençons pour des régions de type I et II.
C
<latexit sha1_base64="0FVLC26qoEqGCcARn5PiYjZH6Mk=">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</latexit> 1C
<latexit sha1_base64="URWvDrb+4sVmMiv8fu/gH5vw9d8=">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</latexit> 2C
3<latexit sha1_base64="00Ga640wgh0b/VAbH+Y1ZU3/ZuA=">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</latexit>
‰
@R
P dx + Q dy =
¨
R
@ Q
@ x
@ P
@ y dA
<latexit sha1_base64="7OvRhyQPAgZ7acKYSNlEKCUr1kE=">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</latexit>
c
<latexit sha1_base64="hKmzLwSK/+hOyRzsiTHGnf+0j7E=">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</latexit>d
<latexit sha1_base64="AfogDpvZjXquVAdBYExLr36wwNc=">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</latexit>f
<latexit sha1_base64="Y8B+HbBfcPyVYYdvp4FYz09nhT4=">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</latexit>(y )
k (y )
<latexit sha1_base64="2b8aEJrf8/n2Yy6ZVLGbwZoM4Ug=">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</latexit>
‰
@R
Q dy =
¨
R
@ Q
@ x dA
<latexit sha1_base64="a4zzbkFbNWHihsIxXpIt4UF++Is=">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</latexit>
¨
R
@ Q
@ x dA
<latexit sha1_base64="kt2yDT0zhesooSuQq7MMocGrjtA=">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</latexit>
=
ˆ
d cˆ
k(y) f(y)@ Q
@ x (x, y ) dxdy
<latexit sha1_base64="xTppfWJBKxOwhUeM9aCrKWIg52Q=">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</latexit>
=
ˆ
d cQ(k(x), y ) Q(f (y, x)) dy
<latexit sha1_base64="h7btZW4JC7/Kc5D0ztPU0Njig/A=">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</latexit>
Preuve: Commençons pour des régions de type I et II.
C
<latexit sha1_base64="0FVLC26qoEqGCcARn5PiYjZH6Mk=">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</latexit> 1C
<latexit sha1_base64="URWvDrb+4sVmMiv8fu/gH5vw9d8=">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</latexit> 2C
3<latexit sha1_base64="00Ga640wgh0b/VAbH+Y1ZU3/ZuA=">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</latexit>
c
<latexit sha1_base64="hKmzLwSK/+hOyRzsiTHGnf+0j7E=">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</latexit>d
<latexit sha1_base64="AfogDpvZjXquVAdBYExLr36wwNc=">AAADHXicjVLLSsNAFD2N7/qqunQTLIKrkqqgy6IblxbsA2qRJJ3qYJqEZFIoxS9wq2u/xp24Ff9A/8I716moxceEJGfOvefO3IcXBzJVjvOcsyYmp6ZnZufy8wuLS8uFldV6GmWJL2p+FERJ03NTEchQ1JRUgWjGiXB7XiAa3uWhtjf6IkllFJ6oQSzaPfc8lF3pu4qoauesUHRKDi97HJQNKMKs42glN4FTdBDBR4YeBEIowgFcpPS0UIaDmLg2hsQlhCTbBa6QJ21GXoI8XGIv6XtOu5ZhQ9rrmCmrfToloDchpY1N0kTklxDWp9lszziyZn+KPeSY+m4D+nsmVo9YhQti/9KNPP+v0/eXXJffM1boYp8z1d4xM7oGvjkr49rp/OxPuSuKEBOncYfsCWGflaNu2KxJuUK6Ay7bX9hTs3rvG98MryYXgT5HHXzkOOROS7LHXPEBIUVf7iVNTvn7nIyD+napvFParu4WKwdmhmaxjg1s0ZzsoYIjHKPGt7nGDW6tO+veerAe312tnNGs4cuynt4AY3qing==</latexit>f
<latexit sha1_base64="Y8B+HbBfcPyVYYdvp4FYz09nhT4=">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</latexit>(y )
k (y )
<latexit sha1_base64="2b8aEJrf8/n2Yy6ZVLGbwZoM4Ug=">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</latexit>
‰
@R
Q dy =
¨
R
@ Q
@ x dA
<latexit sha1_base64="a4zzbkFbNWHihsIxXpIt4UF++Is=">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</latexit>
¨
R
@ Q
@ x dA
<latexit sha1_base64="kt2yDT0zhesooSuQq7MMocGrjtA=">AAADRnicjVLLTttAFD1xgFJoIbTLbiyiSiyqyAEkuoR2wxKqBpAwisaTCR3h2NZ4jIii/EK/hi2s+QX+oN10h9j2zu3wKiplLNtnzr3nztxHUqS6tFF0WQvqE5NTL6Zfzsy+ej0331h4s1PmlZGqI/M0N3uJKFWqM9Wx2qZqrzBKDJJU7SZHn51991iZUufZVzss1MFAHGa6r6WwRHUbS7HWme1+iftGyFFcCGO1SMPt8R0+GccfehvdRjNqRbzCx6DtQRN+beULtTpi9JBDosIAChks4RQCJT37aCNCQdwBRsQZQprtCmPMkLYiL0Uegtgj+h7Sbt+zGe1dzJLVkk5J6TWkDPGeNDn5GcLutJDtFUd27L9ijzimu9uQ/omPNSDW4hux/9PdeD5f5+6vuS5PZ2zRx0fO1HkXzLgaSH9WxbVz+YX3crcUoSDO4R7ZDWHJyptuhKwpuUKuA4LtP9jTsW4vvW+Fnz4XhWOOOrzNccSd1mQvuOJDQpa+3EuanPbfc/IY7Cy32iut5e3V5vonP0PTeIdFLNGcrGEdm9hCh27zHac4w3lwEfwKroLrP65BzWve4sGq4zcId7Jj</latexit>
=
ˆ
d cQ(k(x), y ) Q(f (y, x)) dy
<latexit sha1_base64="h7btZW4JC7/Kc5D0ztPU0Njig/A=">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</latexit>
‰
@R
Q dy
<latexit sha1_base64="m863GGRFvIvzQkZGz6OOtcgmrN8=">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</latexit>
= ˆ
C1
Q dy + ˆ
C2
Q dy + ˆ
C3
Q dy
<latexit sha1_base64="sqbW+YDDA3IoVWhpmIFyy5ZUq/U=">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</latexit>
= ˆ
C2
Q dy
ˆ
C3
Q dy
<latexit sha1_base64="REY55jLTAq71xLVCAkKJpRHScPU=">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</latexit>
=
ˆ
d cQ(k(y, y )) dy
ˆ
d cQ(f (y ), y ) dy
<latexit sha1_base64="fjuT3tqdi8OVOTP6G9PR7odits4=">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</latexit>
Preuve: ‰
@R
P dx + Q dy =
¨
R
@ Q
@ x
@ P
@ y dA
<latexit sha1_base64="7OvRhyQPAgZ7acKYSNlEKCUr1kE=">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</latexit>
Si la région n’est pas type I et II; on la découpe!
R
<latexit sha1_base64="ukDIwB8NuiDI9fVmO1tCzYMaJGw=">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</latexit> 1R
<latexit sha1_base64="bOsl8mPEt1kCi0m46jJ2VS5XjOg=">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</latexit> 2R
<latexit sha1_base64="bMCZTAJETVm9Z7EaLCXJfTeYU3Y=">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</latexit> 3¨
R
@ Q
@ x
@ P
@ y dA =
¨
R1
@ Q
@ x
@ P
@ y dA +
¨
R2
@ Q
@ x
@ P
@ y dA +
¨
R3
@ Q
@ x
@ P
@ y dA
<latexit sha1_base64="s5Sl4Nnwonv7Nx9tIyciTT5Q9as=">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</latexit>
C
<latexit sha1_base64="0FVLC26qoEqGCcARn5PiYjZH6Mk=">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</latexit> 1C
<latexit sha1_base64="URWvDrb+4sVmMiv8fu/gH5vw9d8=">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</latexit> 2C
3<latexit sha1_base64="00Ga640wgh0b/VAbH+Y1ZU3/ZuA=">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</latexit>
C
<latexit sha1_base64="3UMMBVM8j86DbnXFS4Eq+ZRV+Rg=">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</latexit> 4C
<latexit sha1_base64="RUwTcejRjK6XDvh4ANG7W+NzAf4=">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</latexit> 5@
<latexit sha1_base64="xU5J+sGmQuoXyACMuZPen9wT6/c=">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</latexit>R = C
1+ C
2+ C
3=
<latexit sha1_base64="PRCYHENDie0lzrtS6Td2ZR8zdHw=">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</latexit>C
1+ C
4C
4+ C
2+ C
5C
5+ C
3= ( C
1+ C
4) + ( C
4+ C
2+ C
5) + ( C
5+ C
3)
<latexit sha1_base64="TmtOf03faRnCKlx1nG0vvLne+s8=">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</latexit>
=
<latexit sha1_base64="avuoKUTfo8on2FkbJ3gt42yUIMY=">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</latexit>@ R
1+ @ R
2+ @ R
3Preuve: ‰
@R
P dx + Q dy =
¨
R
@ Q
@ x
@ P
@ y dA
<latexit sha1_base64="7OvRhyQPAgZ7acKYSNlEKCUr1kE=">AAADkXicjVLbbhMxEJ3tcknLLaWPvFhESEhAtClIwANSgBckXpKKtJW6VeR1nNaKs17Z3tJVtB/aP4CvgPHIpaUVF0fxHp85Z+wZu6i0cj7LzpK19MbNW7c76xt37t67/6C7+XDXmdoKORFGG7tfcCe1KuXEK6/lfmUlXxZa7hWLjyG+dyKtU6b84ptKHi75UanmSnCP1LTrcqNKL7wV2ojFV+XkdJVX3HrFNdtpGRux/Pns9NkY5+ZdrlA83cnnlosL2bi9wKftiyvR0aVo02Ka99NuL+tnNNh1MIigB3GMzGaSQg4zMCCghiVIKMEj1sDB4e8ABpBBhdwhrJCziBTFJbSwgd4aVRIVHNkFzke4OohsieuQ05Fb4C4a/xadDJ6gx6DOIg67MYrXlDmwf8q9opzhbA1+i5hriayHY2T/5TtX/r8vnF9RX/5esYc5vKFKg7oiJvRAxL1q6l2oj12q3WOGCrmAZxi3iAU5z2+DkcdRh8INcIp/I2Vgw1pEbQ3fYy0STihr86vGFd20wnhFHW8QeZzpLvHlDK6+k+tgd7s/eNnfHr/qDT/EN9SBR/AYnuI7eQ1D+AQjmOBpzuBH0knW0630bTpMo3YtiZ4t+G2kn38CiD/Lbg==</latexit>
Si la région n’est pas simplement connexe; on la découpe!
R
<latexit sha1_base64="ukDIwB8NuiDI9fVmO1tCzYMaJGw=">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</latexit> 1R
<latexit sha1_base64="bOsl8mPEt1kCi0m46jJ2VS5XjOg=">AAADH3icjVK5TsNAEH1xuM8AJY1FhEQVOQEJyggaynDkkAKKbLOEFY5t2etIUcQn0ELN19AhWv4A/oLZYUFAxLGW7bdv5s3uHF4cyFQ5zlPOyo+NT0xOTc/Mzs0vLBaWlhtplCW+qPtRECUtz01FIENRV1IFohUnwu15gWh6l3va3uyLJJVReKwGsTjtud1QnkvfVUQdHXYqnULRKTm87FFQNqAIs2rRUi6PE5whgo8MPQiEUIQDuEjpaaMMBzFxpxgSlxCSbBe4wgxpM/IS5OESe0nfLu3ahg1pr2OmrPbplIDehJQ21kkTkV9CWJ9msz3jyJr9KfaQY+q7DejvmVg9YhUuiP1L9+75f52+v+S6/J6xwjl2OFPtHTOja+CbszKunc7P/pS7oggxcRqfkT0h7LPyvRs2a1KukO6Ay/Zn9tSs3vvGN8OLyUWgz1EHHzkOudOS7DFXfEBI0Zd7SZNT/j4no6BRKZU3S5WDrWJ118zQFFaxhg2ak21UsY8a6nSbLq5xg1vrzrq3HqzHN1crZzQr+LKsp1cKlqMx</latexit> 2C
<latexit sha1_base64="0FVLC26qoEqGCcARn5PiYjZH6Mk=">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</latexit> 1C
2<latexit sha1_base64="URWvDrb+4sVmMiv8fu/gH5vw9d8=">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</latexit>
C
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