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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

Intégrale curviligne de champs scalaire Champs vectoriels conservatifs

Intégrale curviligne de champs de vecteurs

(3)

Aujourd’hui, nous allons voir

Théorème fondamental des intégrales curvilignes.

Indépendance de chemin.

Déterminer si un champ de vecteur est conservatif.

(4)

Le théorème fondamental du calcul dit

Z b

a

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

<latexit sha1_base64="hW8Lo2bBFehfTBT9tsFEtsmDBHc=">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</latexit>

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.

(5)

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

<latexit sha1_base64="aek7iQWn9FheymNH/G6npsSGhyE=">AAADHHicjVLNTttAEP4wPwVaIKVHLlYjpHCJHECil0oILj1SqQEkQqP1egMrHNtaryNFVp6Ad+AduMKVW9Vrpb5Be+sjdHa6oLaoomt5/e03841nZicuUl3aKPo6FUzPzM49m19YfP5iaXml8XL1sMwrI1VX5mlujmNRqlRnqmu1TdVxYZQYxqk6ii/2nf1opEyp8+yDHRfqdCjOMj3QUlii+o31tz2d2b74GIe9gRGyTiZ1YieDVm+kZGhadmMjTGy/0YzaEa/wMeh40IRfB3njB3pIkEOiwhAKGSzhFAIlPSfoIEJB3Clq4gwhzXaFCRZJW5GXIg9B7AXtZ3Q68WxGZxezZLWkv6T0GlKGWCdNTn6GsPtbyPaKIzv2X7FrjulyG9M39rGGxFqcE/uU7t7zf3WuFosB3nANmmoqmHHVSR+l4q64zMPfqrIUoSDO4YTshrBk5X2fQ9aUXLvrrWD7N/Z0rDtL71vhu89SYcRRxw/Z13yHmuwF93JMyNLOt0Qj0fl7AB6Dw812Z6u9+X67ubvnh2Mea3iNFk3ADnbxDgfoUjaXuMYNboOr4C74FHz+5RpMec0r/LGCLz8BxpCpqQ==</latexit>

=<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>

(6)

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=">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</latexit> f (~r(b)) f (~r(a))

On vient donc de démontrer le théorème suivant:

(7)

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

<latexit sha1_base64="NE2iE9BnmvhWdjBYUEhFFyk3ShM=">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</latexit>

Et ce, peut importe le chemin!

(8)

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

<latexit sha1_base64="RgFCP1Mjh0AKp71EMljU+rfUJr8=">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</latexit>

C<latexit sha1_base64="kUl7v/GAtuJPJhWHc63rYP1AClY=">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</latexit> 2

C3

<latexit sha1_base64="5NOheiGUms+uYa3IMHHrmStTRhE=">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</latexit>

Z

C1 rf · d~r1

<latexit sha1_base64="AfocjkewEtzOSKzXZnWKYmE+KLc=">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</latexit>

= Z

C2 rf · d~r2

<latexit sha1_base64="PsidT7WIIoeetx9MeW9S1Jr92rc=">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</latexit>

= Z

C3 rf · d~r3

<latexit sha1_base64="/+Ec9RTLZK2H+AuBNJD9RJIG4Tg=">AAADInicjVLNahRBEP4y/iXxb9VjLo2LIAjLbDagl0AwF48R3CSQCUNPb2/SpOeHnp6FZclD5B18B6969SaehDxAvPkIVpcdUYNoD9Pz9Vf11VRVV9FY0/o0/bKUXLt+4+at5ZXV23fu3rvfe/Bwt607p/RY1bZ2+4VstTWVHnvjrd5vnJZlYfVecbId7Hsz7VpTV2/8vNGHpTyqzNQo6YnKe882M1P5fJGV0h8racV2PjrNKllYKaaZmtReTLKZVsLlo7zXTwcpL3EVDCPoI66duvcNGSaoodChhEYFT9hCoqXnAEOkaIg7xII4R8iwXeMUq6TtyEuThyT2hPYjOh1EtqJziNmyWtFfLL2OlAJPSFOTnyMc/ibY3nHkwP4t9oJjhtzm9C1irJJYj2Ni/6W79PxfXajFY4oXXIOhmhpmQnUqRum4KyFz8UtVniI0xAU8IbsjrFh52WfBmpZrD72VbL9gz8CGs4q+Hb7GLDVmHHX+M/sF36Ehe8O9nBPytPMt0UgM/xyAq2B3fTAcDdZfb/S3XsbhWMYaHuMpTcBzbOEVdjCmbM7wDu/xIXmbfEw+JZ9/uCZLUfMIv63k/DvbJqwr</latexit>

= f (B) f (A) A<latexit sha1_base64="KNDrGw5vbVzqPDG4CR7kclALd4w=">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</latexit>

B<latexit sha1_base64="KhKy6DrKy96Y6rnk29HP8EqVNrM=">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</latexit>

(9)

Par contre si le champ de vecteur n’est pas conservatif, en général, Z

C1

F~ · d~r1 6= Z

C2

F~ · d~r2

<latexit sha1_base64="7XHZs31zVSePmOgSwgwvMcqjt6o=">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</latexit>

Si une intégrale a la propriété que Z

C1

F~ · d~r1 = Z

C2

F~ · d~r2

<latexit sha1_base64="L7Jfj7gObTlBRwDoNQnuB55kqqU=">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</latexit>

peut importe le chemin reliant les mêmes points de départ et d’arrivée alors, on dit que l’intégrale est indépendante de chemin.

(10)

Faites les exercices suivants

p. 932 # 1 et 2 p. 932 # 1 et 2

(11)

Lorsqu’une intégrale curviligne est évaluée sur une courbe dont le point de départ est le même que le point d’arrivée (une courbe

fermée).

on écrit souvent I

C

F~ · d~r

<latexit sha1_base64="uYymes9/XowtrH/Jhc4Vy8ckG9U=">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</latexit>

C

F~ · d~r

<latexit sha1_base64="93Aj9QjlmeMNKxy/M3xE6AcTi8k=">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</latexit>

C

F~ · d~r

<latexit sha1_base64="HhkT6YulAULL1P1PsbW37SX9Roc=">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</latexit>

ou

si on veut préciser les sens du parcours

(12)

Théorème

Preuve:

est indépendante de chemin sur un domaine

˛

C

F~ · d~r = 0

<latexit sha1_base64="injFA0Imd9Eub3XGi2JjJxomlb4=">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</latexit>

ˆ

C

F~ · d~r

<latexit sha1_base64="BV5kOQh1+IykAKSzzg4U7qEtkiM=">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</latexit> D<latexit sha1_base64="AKHw3Z+FV4xDzyVtOGEFpEtnO8I=">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</latexit>

()<latexit sha1_base64="6PtMM0iY/xUGEQTvp3+MWmVHgUc=">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</latexit> pour tout chemin fermé sur D<latexit sha1_base64="AKHw3Z+FV4xDzyVtOGEFpEtnO8I=">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</latexit>

C1

<latexit sha1_base64="RgFCP1Mjh0AKp71EMljU+rfUJr8=">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</latexit>

C<latexit sha1_base64="kUl7v/GAtuJPJhWHc63rYP1AClY=">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</latexit> 2

A<latexit sha1_base64="KNDrGw5vbVzqPDG4CR7kclALd4w=">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</latexit>

B<latexit sha1_base64="KhKy6DrKy96Y6rnk29HP8EqVNrM=">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</latexit>

˛

C

F~ · d~r = ˆ

C1

F~ · d~r + ˆ

C2

F~ · d~r

<latexit sha1_base64="ZrM2kctnNds/snz0OkEOZZalo/A=">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</latexit>

= ˆ

C1

F~ · d~r

ˆ

C2

F~ · d~r

<latexit sha1_base64="ht8MklXRGekqJjXg27STvH5al/U=">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</latexit>

= 0<latexit sha1_base64="oC1QjWxamNkjnzgaGg+r2S6XF7o=">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</latexit>

(13)

Exemple

F~ = (cos y, sin x)

<latexit sha1_base64="K1GQaejyHqerwWvIYD6/DO2YfiA=">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</latexit>

˛

C

F~ · d~r 6= 0

<latexit sha1_base64="ECFQfVidCMt/uGGy749Mxai5wAE=">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</latexit>

F<latexit sha1_base64="5lr/EGtVSwCK3+c7+fNo+zyS1yk=">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</latexit> ~ 6= rf donc

(14)
(15)

Faites les exercices suivants

p. 933 # 23 et 24 p. 933 # 25 et 26

(16)

A<latexit sha1_base64="f6ULgkZCfX1NBJ2KX5jXK6i/414=">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</latexit>

B<latexit sha1_base64="mdImvZWddRKGeZnsh52gKptxjvg=">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</latexit>

Région ouverte et connexe

A

<latexit sha1_base64="f6ULgkZCfX1NBJ2KX5jXK6i/414=">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</latexit>

B<latexit sha1_base64="mdImvZWddRKGeZnsh52gKptxjvg=">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</latexit>

Pas connexe Pas ouverte

(17)

Soit un champ de vecteur continue dans une région ouverte et connexe . Si

Théorème

Preuve:

F<latexit sha1_base64="nZL5d/qZ6NMn/4A0UoqlFrUo5bA=">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</latexit> ~

D<latexit sha1_base64="AKHw3Z+FV4xDzyVtOGEFpEtnO8I=">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</latexit>

ˆ

C

F~ · d~r

<latexit sha1_base64="nR8D35XvNU2kgDnd6aqgh65gA3k=">AAADQ3icjVLLShxBFD3Txkd8ZdRlNk0GIYsw9GhAlxIhZGkgo4ItQ3VNqYX9orp6YBj8Ar8mW13nI/yDiDvJNpBbN6X4wEc13X3q3Htu1X0kZaorG0UXjWDszfjE5NTb6ZnZufl3zYXF7aqojVRdWaSF2U1EpVKdq67VNlW7pVEiS1K1kxxvOvvOQJlKF/kPOyzVfiYOc32gpbBE9ZrLsc5tbxRnwh5JkYabJ/FAyfBrLPuFjT/1eWd6zVbUjniFj0HHgxb82ioWGmOI0UcBiRoZFHJYwikEKnr20EGEkrh9jIgzhDTbFU4wTdqavBR5CGKP6XtIuz3P5rR3MStWSzolpdeQMsQyaQryM4TdaSHba47s2Kdijzimu9uQ/omPlRFrcUTsS7obz9fr3P011+X5jC0OsM6ZOu+SGVcD6c+quXYuv/BO7pYilMQ53Ce7ISxZedONkDUVV8h1QLD9N3s61u2l961x6XNRGHDU4W2OI+60JnvJFR8SsvTlXtLkdB7OyWOwvdLurLZXvn9ubXzxMzSF9/iAjzQna9jAN2yhS7c5xU+c4Tz4FVwF18Gf/65Bw2uWcG8Ff/8BpkmxiA==</latexit>

est indépendante de chemin, alors F<latexit sha1_base64="nZL5d/qZ6NMn/4A0UoqlFrUo5bA=">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</latexit> ~ est conservatif.

(a, b)<latexit sha1_base64="GWih0N7Pz9cyjoWXgJZQzW+26r4=">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</latexit>

(x, y<latexit sha1_base64="sJpRei3Glk4g/qbvlgAy0ArWezc=">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</latexit> ) f (x, y) =

ˆ (x,y) (a,b)

F~ · d~r

<latexit sha1_base64="+LnIfhxTZuuwTwCYbNlCLrAYjpo=">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</latexit>

(x<latexit sha1_base64="wl/91h9IVXWjRBLdVWQrlAkjayE=">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</latexit> 1, y)

C1

<latexit sha1_base64="2BUWUsRxoMefrYeBNYLU21ooyP8=">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</latexit>

C2

<latexit sha1_base64="K4TyzRg/8GRQEb0+g4tvLtN+buo=">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</latexit>

= ˆ

C1

F~ · d~r + ˆ

C2

F~ · d~r

<latexit sha1_base64="Nv7MdEIPx+8CUlT8XW5onmEqASA=">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</latexit>

=

ˆ (x1,y) (a,b)

F~ · d~r + ˆ

C2

F~ · d~r

<latexit sha1_base64="8dFSzZUpxzOjbocadnzLscrOLjk=">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</latexit>

= @

@x ˆ

C2

P dx + Q dy

<latexit sha1_base64="1o+1F2GUjcHrMCPf0Nco7t1bWBk=">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</latexit>

= @

@x ˆ

C2

P dx

<latexit sha1_base64="n5kQ/3e28J8A1Ffp1STijo88sVY=">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</latexit>

=<latexit sha1_base64="8jQzVtktx0UpsaDZnYrDw6m0S+g=">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</latexit> P

= 0 + @

@x ˆ

C2

F~ · d~r

<latexit sha1_base64="BtYBq+KKgm/NAgmoVSvR0TdzDik=">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</latexit>

@f

@x

<latexit sha1_base64="2OnUg5CDI8OGW+q6V+xIk7So2/A=">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</latexit>

(18)

Soit un champ de vecteur continue dans une région ouverte et connexe . Si

Théorème

Preuve:

F<latexit sha1_base64="nZL5d/qZ6NMn/4A0UoqlFrUo5bA=">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</latexit> ~

D<latexit sha1_base64="AKHw3Z+FV4xDzyVtOGEFpEtnO8I=">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</latexit>

ˆ

C

F~ · d~r

<latexit sha1_base64="nR8D35XvNU2kgDnd6aqgh65gA3k=">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</latexit>

est indépendante de chemin, alors F<latexit sha1_base64="nZL5d/qZ6NMn/4A0UoqlFrUo5bA=">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</latexit> ~ est conservatif.

(a, b)<latexit sha1_base64="GWih0N7Pz9cyjoWXgJZQzW+26r4=">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</latexit>

(x, y<latexit sha1_base64="sJpRei3Glk4g/qbvlgAy0ArWezc=">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</latexit> ) f (x, y) =

ˆ (x,y) (a,b)

F~ · d~r

<latexit sha1_base64="+LnIfhxTZuuwTwCYbNlCLrAYjpo=">AAADT3icjVLLattAFD2W+0jSl5MuuxE1BRuMkZJCuymEFEKWCdRJIEqNNB6nQ2RJjEamQvg7+jXZpuss+wfpF3RXcucyKUlDHyMknTn3njtzH0mRqtIEwbeW1753/8HDpeWVR4+fPH3WWV3bL/NKCzkSeZrrwyQuZaoyOTLKpPKw0DKeJak8SE7fW/vBXOpS5dkHUxfyeBafZGqqRGyIGnfCae/zwK/77yKVmXHTiwdJf/GxIbLuL6K5FP52JCa58aPBhLd63OkGw4CXfxeEDnTh1m6+2mojwgQ5BCrMIJHBEE4Ro6TnCCECFMQdoyFOE1Jsl1hghbQVeUnyiIk9pe8J7Y4cm9HexixZLeiUlF5NSh+vSJOTnyZsT/PZXnFky/4pdsMx7d1q+icu1oxYg0/E/kt37fn/Ont/xXX5e8YGU7zlTK13wYytgXBnVVw7m59/I3dDEQriLJ6QXRMWrLzuhs+akitkOxCz/ZI9LWv3wvlW+O5ykZhz1PpXjg13WpG94IrXhAx9uZc0OeHvc3IX7K8Pw43h+t7r7uaWm6ElvMBL9GhO3mATO9jFiG7zBWc4x1fvwvvh/Ww7V6/lwHPcWu3lK0mVsr4=</latexit>

=<latexit sha1_base64="8jQzVtktx0UpsaDZnYrDw6m0S+g=">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</latexit> P

@f

@x

<latexit sha1_base64="2OnUg5CDI8OGW+q6V+xIk7So2/A=">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</latexit>

C3

<latexit sha1_base64="NxdYyjRlGraaKGJzxd7sxEqmZCk=">AAADKHicjVK7btRAFD3r8AjhkQcljcUKiWple1fZ0EWkSRkkNomUjaLxZJKM4pfscaTVit+ghZqvSRelpYK/4MzFi6AIMJbtM+fec2fuI60y27gouu0FS/fuP3i4/Gjl8ZOnz1bX1jf2m7KttZnoMivrw1Q1JrOFmTjrMnNY1UblaWYO0ssdbz+4MnVjy+K9m1XmOFfnhT2zWjlS02mu3IVWWbhzMjxZ60eDN1ubyWgzjAZRNI6T2INkPBqOwpiMX310a69c7y1hilOU0GiRw6CAI86g0PA5QowIFbljzMnVRFbsBh+wQm1LL0MPRfaS33Pujjq24N7HbESteUrGt6YyxCtqSvrVxP60UOytRPbsXbHnEtPfbcZ/2sXKyTpckP2XbuH5/zp/fyt1+XvGDmfYkky9dyWMr4Huzmqldj6/8LfcHSNU5Dw+pb0m1qJcdCMUTSMV8h1QYv8mnp71e935tvje5WJwJVFnv3KcS6ct7ZVUfEbk+JVecnIW4xHeDfaTQTwcJO9G/e233Qwt4wVe4jXnZIxt7GIPE96mwkd8wufgS3Ad3AS3P12DXqd5jj9W8PUHGYmnFQ==</latexit>

C4

<latexit sha1_base64="WmOOxBQn5yqlEq9o2LOP/3r7WGo=">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</latexit>

(x, y<latexit sha1_base64="zAjDEdgo1S+1y8Z30Osa8oboaOM=">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</latexit> 1)

= ˆ

C3

F~ · d~r + ˆ

C4

F~ · d~r

<latexit sha1_base64="IJpXVm/anKC8gU+VdD1LzQTVnCI=">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</latexit>

=

ˆ (x,y1)

(a,b)

F~ · d~r + ˆ

C4

F~ · d~r

<latexit sha1_base64="zD0KXQ2mXy9uA2QJ3lVt950MLAg=">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</latexit>

@f

@y

<latexit sha1_base64="KJj7fwok1ncr2A1nBDV3yMd8x9I=">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</latexit>

= 0 + @

@y ˆ

C4

F~ · d~r

<latexit sha1_base64="nR6dBHcpSPbSyamjIYEoTr9DDoQ=">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</latexit>

= @

@y ˆ

C4

Q dy

<latexit sha1_base64="EMzqI7REnx7/L2Hwk0+G7RjTNKo=">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</latexit>

= @

@y ˆ

C4

P dx + Q dy

<latexit sha1_base64="k4vvlV1+eVTG5EkKtMZGjfAccRs=">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</latexit>

=<latexit sha1_base64="fpErMz9jiZZHfOv77onnVKQcg0E=">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</latexit> Q

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