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FONDAMENTAL DES INTÉGRALES

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Texte intégral

(1)

cours 26

4.2 THÉORÈME

FONDAMENTAL DES INTÉGRALES

CURVILIGNES

(2)

Au dernier cours, nous avons vu

(3)

Au dernier cours, nous avons vu

Intégrale curviligne de champs scalaire

(4)

Au dernier cours, nous avons vu

Intégrale curviligne de champs scalaire Champs vectoriels conservatifs

(5)

Au dernier cours, nous avons vu

Intégrale curviligne de champs scalaire Champs vectoriels conservatifs

Intégrale curviligne de champs de vecteurs

(6)

Aujourd’hui, nous allons voir

(7)

Aujourd’hui, nous allons voir

Théorème fondamental des intégrales curvilignes.

(8)

Aujourd’hui, nous allons voir

Théorème fondamental des intégrales curvilignes.

Indépendance de chemin.

(9)

Aujourd’hui, nous allons voir

Théorème fondamental des intégrales curvilignes.

Indépendance de chemin.

Déterminer si un champ de vecteur est conservatif.

(10)

Le théorème fondamental du calcul dit

(11)

Le théorème fondamental du calcul dit

Z b

a

f 0(x)dx = f (b) f (a)

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(12)

Le théorème fondamental du calcul dit

Z b

a

f 0(x)dx = f (b) f (a)

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On va voir aujourd’hui une première généralisation du fait que pour évaluer une intégrale, il suffit d’évaluer une primitive sur les bords de

la région d’intégration.

(13)

Z

C rf · d~r

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(14)

Z

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=

Z b

a rf · ~r 0(t)dt

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(15)

Z

C rf · d~r

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=

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a

✓ @f

@x , @f

@y , @f

@z

·

✓ dx

dt , dy

dt , dz dt

dt

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=

Z b

a rf · ~r 0(t)dt

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(16)

Z

C rf · d~r

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=

Z b

a

✓ @f

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·

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dt , dy

dt , dz dt

dt

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=

Z b

a rf · ~r 0(t)dt

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(17)

Z

C rf · d~r

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=

Z b

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=

Z b

a

✓ @f

@x

dx

dt + @f

@y

dy

dt + @f

@z

dz dt

dt

<latexit sha1_base64="9i/RYMZms+5iNBCTSX8T2MgeoDY=">AAADkXicjVLbTtwwEJ2QXii0ZWEf+xJ11Yqq0jZLkYAHpFX7UokXKnUBidCV4ziLhTeJHAeRjfIj/Fn/oH3rJ3Q8eHsBUXAU5/jMORPP2HGhZGnC8Ju34D94+Ojx4pOl5afPnq90VtcOyrzSXIx4rnJ9FLNSKJmJkZFGiaNCCzaNlTiMzz7a+OG50KXMsy+mLsTJlE0ymUrODFLjzuVuJDMzZl/jIFIiNetRqhlvooJpI5kK0vYPvmivgslF2ySmffsfaT2X1ndKZ3PpjKSRlpNT8yZIzLjTC/shjeAmGDjQAzf2885PiCCBHDhUMAUBGRjEChiU+BzDAEIokDuBBjmNSFJcQAtL6K1QJVDBkD3DeYKrY8dmuLY5S3Jz/IvCV6MzgFfoyVGnEdu/BRSvKLNlb8vdUE67txq/scs1RdbAKbJ3+ebK+/psLQZS2KYaJNZUEGOr4y5LRV2xOw/+qspghgI5ixOMa8ScnPM+B+QpqXbbW0bx76S0rF1zp63gh9ulgHPKWv/efUNnKDFeUC9rRAZnOiW8EoPrF+AmONjoD973Nz5v9oYf3OVYhBfwEtbxBmzBED7BPoyAe+C99t55od/1d/yh77QLnvN04Z/h7/0C0wvVQg==</latexit>

=

Z b

a

d

dt f (~r(t))dt

<latexit sha1_base64="aek7iQWn9FheymNH/G6npsSGhyE=">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</latexit>

=

Z b

a rf · ~r 0(t)dt

<latexit sha1_base64="igIL5WPSNb3FBZFg8aP4/WSdXGo=">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</latexit>

(18)

Z

C rf · d~r

<latexit sha1_base64="3ccABqmZPp3bz07UWzrcBQuEJJc=">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</latexit>

=

Z b

a

✓ @f

@x , @f

@y , @f

@z

·

✓ dx

dt , dy

dt , dz dt

dt

<latexit sha1_base64="RQ/yIALza64XEDN2T905yAB7yLg=">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</latexit>

=

Z b

a

✓ @f

@x

dx

dt + @f

@y

dy

dt + @f

@z

dz dt

dt

<latexit sha1_base64="9i/RYMZms+5iNBCTSX8T2MgeoDY=">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</latexit>

=

Z b

a

d

dt f (~r(t))dt

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=<latexit sha1_base64="7Ax5RGGB+1HkAbu3cpMfG1+Jk2w=">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</latexit> f (~r(b)) f (~r(a))

=

Z b

a rf · ~r 0(t)dt

<latexit sha1_base64="igIL5WPSNb3FBZFg8aP4/WSdXGo=">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</latexit>

(19)

Z

C rf · d~r

<latexit sha1_base64="3ccABqmZPp3bz07UWzrcBQuEJJc=">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</latexit>

=<latexit sha1_base64="7Ax5RGGB+1HkAbu3cpMfG1+Jk2w=">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</latexit> f (~r(b)) f (~r(a))

On vient donc de démontrer le théorème suivant:

(20)

Théorème

Z

C rf · d~r

<latexit sha1_base64="3ccABqmZPp3bz07UWzrcBQuEJJc=">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</latexit>

=<latexit sha1_base64="7Ax5RGGB+1HkAbu3cpMfG1+Jk2w=">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</latexit> f (~r(b)) f (~r(a))

On vient donc de démontrer le théorème suivant:

(21)

Si est une courbe lisse de paramétrisation

pour , et si est une fonction différentiable dont le gradient est continu sur , alors

~r<latexit sha1_base64="WLrztjHuw1KmXRzAetfvWs9H8hU=">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</latexit> (t) a<latexit sha1_base64="4hZjV2e3bNfnZFwdZbko0TPb7Zo=">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</latexit>  t  b

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit>

f<latexit sha1_base64="SOzf6H11TrRi46mvhBUP/433KTI=">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</latexit>

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit>

Théorème

Z

C rf · d~r

<latexit sha1_base64="3ccABqmZPp3bz07UWzrcBQuEJJc=">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</latexit>

=<latexit sha1_base64="7Ax5RGGB+1HkAbu3cpMfG1+Jk2w=">AAADEXicjVJNTxRBEH0MfiB+rXiTy8SNye7BzSyawIWE4MUjJCyQACEzTS92mJ2Z9PRsstmY8B/8D171ys149Rf4D/TmT+B12RCVEOzJ9Lx+Va+mqrqyKje1S5LvM9Hsrdt37s7dm7//4OGjx60nC9t12VilB6rMS7ubpbXOTaEHzrhc71ZWp6Ms1zvZyRtv3xlrW5uy2HKTSh+M0uPCDI1KHanD1rPVYWd/rFVsO1m3+/LykHa7h6120ktkxVdBP4A2wtooW7+wjyOUUGgwgkYBR5wjRc1nD30kqMgdYErOEhmxa7zHPLUNvTQ9UrIn3I952gtswbOPWYta8S85X0tljBfUlPSzxP5vsdgbiezZ62JPJabPbcJvFmKNyDq8I3uT7sLzf3W+FochVqQGw5oqYXx1KkRppCs+8/iPqhwjVOQ8PqLdEitRXvQ5Fk0ttfvepmL/IZ6e9WcVfBv8DFlqjCXq5DL7qdyhob2SXk6IHHe5JY5E/98BuAq2l3r9V72lzdfttfUwHHNYxHN0OAHLWMNbbGDAbE7xEZ/wOfoQnUVfoq+/XaOZoHmKv1b07RzjMqOI</latexit> f (~r(b)) f (~r(a))

On vient donc de démontrer le théorème suivant:

(22)

Exemple

Le potentiel gravitationnel est

(23)

Exemple

Le potentiel gravitationnel est f (x, y, z) = mM G

px2 + y2 + z2

<latexit sha1_base64="t6DTouAIGH0vOQ08Km66W40HaOY=">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</latexit>

(24)

Exemple

Le potentiel gravitationnel est f (x, y, z) = mM G

px2 + y2 + z2

<latexit sha1_base64="t6DTouAIGH0vOQ08Km66W40HaOY=">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</latexit>

rf(x, y, z) =

xmM G

(x2 + y2 + z2) 32 , ymM G

(x2 + y2 + z2)32 , zmM G

(x2 + y2 + z2)32

<latexit sha1_base64="WM5csGju1HsRlPmZfyA7tNw4L7A=">AAADlHicjVJdaxQxFL3T8aOtH90q9MWX4CLs4rrMzgr1QaG0iL4oFdy20OmWTJrZhmY+yGTKzg7zU/xh/Qf65k/wJk1FLdJmSObk3HNucpPEhRSlDoILb8m/c/fe/eWV1QcPHz1e66w/2SvzSjE+YbnM1UFMSy5FxidaaMkPCsVpGku+H5/tmPj+OVelyLOvui74UUpnmUgEoxqp4863KKOxpCTpzQekHpBF/x2JJE9071WUKMqaefrpQ9v05tPwZY19MQ370+YyNG6bsG3bgVPWt1YublRGSsxOdf+40w2GgW3kOhg50AXXdvPOT4jgBHJgUEEKHDLQiCVQKPE7hBEEUCB3BA1yCpGwcQ4trKK3QhVHBUX2DMcZzg4dm+Hc5Cytm+EqErtCJ4EX6MlRpxCb1YiNVzazYf+Xu7E5zd5q/McuV4qshlNkb/JdKW/rM7VoSOCNrUFgTYVlTHXMZansqZidkz+q0pihQM7gE4wrxMw6r86ZWE9pazdnS238u1Ua1syZ01bww+2Sw7nNWv/efWPvUGC8sGdZI9I42lvCJzH69wFcB3vhcDQehl9ed7e23eNYhmfwHHr4AjZhCz7CLkyAeb7X90Jv7G/4b/0d//2ldMlznqfwV/M//wIIec+x</latexit>

(25)

Exemple

Le potentiel gravitationnel est f (x, y, z) = mM G

px2 + y2 + z2

<latexit sha1_base64="t6DTouAIGH0vOQ08Km66W40HaOY=">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</latexit>

rf(x, y, z) =

xmM G

(x2 + y2 + z2) 32 , ymM G

(x2 + y2 + z2)32 , zmM G

(x2 + y2 + z2)32

<latexit sha1_base64="WM5csGju1HsRlPmZfyA7tNw4L7A=">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</latexit>

= mM G

(x2 + y2 + z2) 32 (x, y, z)

<latexit sha1_base64="RD4x8JKjacHvYE/99mOvRhW7zWc=">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</latexit>

(26)

Exemple

Le potentiel gravitationnel est f (x, y, z) = mM G

px2 + y2 + z2

<latexit sha1_base64="t6DTouAIGH0vOQ08Km66W40HaOY=">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</latexit>

rf(x, y, z) =

xmM G

(x2 + y2 + z2) 32 , ymM G

(x2 + y2 + z2)32 , zmM G

(x2 + y2 + z2)32

<latexit sha1_base64="WM5csGju1HsRlPmZfyA7tNw4L7A=">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</latexit>

= mM G

(x2 + y2 + z2) 32 (x, y, z)

<latexit sha1_base64="RD4x8JKjacHvYE/99mOvRhW7zWc=">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</latexit>

= mM G k~xk3 ~x

<latexit sha1_base64="J/uALbRYiKmH1giUYY/LbsaIkh4=">AAADGHicjVI9T9xAEH04QAgf4ZKUaRxOkWg4+QCJNJEQKZIGiUgcIGGC7GWPrPCX1usTJ3N1/gP/gRbadChtuvyDpOMnZHZYEB9CsJbXb9/MG8/MTlwkqjRB8GfIezY8Mvp87MX4xOTUy+nGq9cbZV5pITsiT3K9FUelTFQmO0aZRG4VWkZpnMjN+OCTtW/2pC5Vnq2bfiF30mg/U10lIkPUbuPdR38u7OpI1Onq50EdHoU9KfzD8OjbwuAS7jaaQSvg5d8HbQeacGstb1wgxB5yCFRIIZHBEE4QoaRnG20EKIjbQU2cJqTYLjHAOGkr8pLkERF7QPs+nbYdm9HZxixZLegvCb2alD7ekyYnP03Y/s1ne8WRLftQ7Jpj2tz69I1drJRYg+/EPqa78nyqztZi0MUHrkFRTQUztjrholTcFZu5f6MqQxEK4izeI7smLFh51WefNSXXbnsbsf0ve1rWnoXzrfDPZSnR46j96+xrvkNF9oJ72SdkaOdbopFo3x2A+2BjvtVeaM1/XWwur7jhGMNbzGCWJmAJy/iCNXQomx84wSnOvGPvp3fu/bp09Yac5g1uLe/3f+KOqFE=</latexit>

(27)

Exemple

Le potentiel gravitationnel est f (x, y, z) = mM G

px2 + y2 + z2

<latexit sha1_base64="t6DTouAIGH0vOQ08Km66W40HaOY=">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</latexit>

rf(x, y, z) =

xmM G

(x2 + y2 + z2) 32 , ymM G

(x2 + y2 + z2)32 , zmM G

(x2 + y2 + z2)32

<latexit sha1_base64="WM5csGju1HsRlPmZfyA7tNw4L7A=">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</latexit>

= mM G

(x2 + y2 + z2) 32 (x, y, z)

<latexit sha1_base64="RD4x8JKjacHvYE/99mOvRhW7zWc=">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</latexit>

= mM G k~xk3 ~x

<latexit sha1_base64="J/uALbRYiKmH1giUYY/LbsaIkh4=">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</latexit>

=<latexit sha1_base64="d16Phv2K3nhqIYx4KHZEGCvBt5I=">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</latexit> F~

(28)

Exemple

Le potentiel gravitationnel est f (x, y, z) = mM G

px2 + y2 + z2

<latexit sha1_base64="t6DTouAIGH0vOQ08Km66W40HaOY=">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</latexit>

rf(x, y, z) =

xmM G

(x2 + y2 + z2) 32 , ymM G

(x2 + y2 + z2)32 , zmM G

(x2 + y2 + z2)32

<latexit sha1_base64="WM5csGju1HsRlPmZfyA7tNw4L7A=">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</latexit>

= mM G

(x2 + y2 + z2) 32 (x, y, z)

<latexit sha1_base64="RD4x8JKjacHvYE/99mOvRhW7zWc=">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</latexit>

= mM G k~xk3 ~x

<latexit sha1_base64="J/uALbRYiKmH1giUYY/LbsaIkh4=">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</latexit>

=<latexit sha1_base64="d16Phv2K3nhqIYx4KHZEGCvBt5I=">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</latexit> F~

Le travail effectué par une particule se déplaçant dans le champ gravitationnel du point (1,1,1) au point (2,3,4) est

(29)

Exemple

Le potentiel gravitationnel est f (x, y, z) = mM G

px2 + y2 + z2

<latexit sha1_base64="t6DTouAIGH0vOQ08Km66W40HaOY=">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</latexit>

rf(x, y, z) =

xmM G

(x2 + y2 + z2) 32 , ymM G

(x2 + y2 + z2)32 , zmM G

(x2 + y2 + z2)32

<latexit sha1_base64="WM5csGju1HsRlPmZfyA7tNw4L7A=">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</latexit>

= mM G

(x2 + y2 + z2) 32 (x, y, z)

<latexit sha1_base64="RD4x8JKjacHvYE/99mOvRhW7zWc=">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</latexit>

= mM G k~xk3 ~x

<latexit sha1_base64="J/uALbRYiKmH1giUYY/LbsaIkh4=">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</latexit>

=<latexit sha1_base64="d16Phv2K3nhqIYx4KHZEGCvBt5I=">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</latexit> F~

Le travail effectué par une particule se déplaçant dans le champ gravitationnel du point (1,1,1) au point (2,3,4) est

Z

C rf · d~r = mM G p29

mM Gp 3

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(30)

Exemple

Le potentiel gravitationnel est f (x, y, z) = mM G

px2 + y2 + z2

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rf(x, y, z) =

xmM G

(x2 + y2 + z2) 32 , ymM G

(x2 + y2 + z2)32 , zmM G

(x2 + y2 + z2)32

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= mM G

(x2 + y2 + z2) 32 (x, y, z)

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= mM G k~xk3 ~x

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=<latexit sha1_base64="d16Phv2K3nhqIYx4KHZEGCvBt5I=">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</latexit> F~

Le travail effectué par une particule se déplaçant dans le champ gravitationnel du point (1,1,1) au point (2,3,4) est

Z

C rf · d~r = mM G p29

mM Gp 3

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Et ce, peut importe le chemin!

(31)

Ce théorème simplifie beaucoup les choses, car l’intégrale curviligne ne dépend pas du chemin, mais seulement du point de départ et du

point d’arrivée!

(32)

Ce théorème simplifie beaucoup les choses, car l’intégrale curviligne ne dépend pas du chemin, mais seulement du point de départ et du

point d’arrivée!

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(33)

Ce théorème simplifie beaucoup les choses, car l’intégrale curviligne ne dépend pas du chemin, mais seulement du point de départ et du

point d’arrivée!

C1

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(34)

Ce théorème simplifie beaucoup les choses, car l’intégrale curviligne ne dépend pas du chemin, mais seulement du point de départ et du

point d’arrivée!

C1

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Z

C1 rf · d~r1

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(35)

Ce théorème simplifie beaucoup les choses, car l’intégrale curviligne ne dépend pas du chemin, mais seulement du point de départ et du

point d’arrivée!

C1

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Z

C1 rf · d~r1

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