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1.4 LONGUEUR D’ARC ET ABSCISSE CURVILIGNE

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Texte intégral

(1)

cours 4

1.4 LONGUEUR

D’ARC ET ABSCISSE

CURVILIGNE

(2)

Au dernier cours, nous avons vu

Dérivées de fonctions vectorielles.

Règles de dérivation.

Vecteur tangent unitaire.

(3)

Aujourd’hui, nous allons voir

Intégrale d’une fonction vectorielle La longueur d’une courbe

L’abscisse curviligne Repère de Frenet

(4)

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OP<latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit> ~ = OA~ + kd~

OP<latexit sha1_base64="2Awddmdbya/AE8AnkIfNJhpxk24=">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</latexit><latexit sha1_base64="2Awddmdbya/AE8AnkIfNJhpxk24=">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</latexit><latexit sha1_base64="2Awddmdbya/AE8AnkIfNJhpxk24=">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</latexit><latexit sha1_base64="2Awddmdbya/AE8AnkIfNJhpxk24=">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</latexit> ~ = ~r(a) + k~r 0(a)

(5)

Exemple

~r(t) = (cos t, sin t, t)

<latexit sha1_base64="jwisrcU9OPLyYxt5s5F2mO2Uf7M=">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</latexit><latexit sha1_base64="jwisrcU9OPLyYxt5s5F2mO2Uf7M=">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</latexit><latexit sha1_base64="jwisrcU9OPLyYxt5s5F2mO2Uf7M=">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</latexit><latexit sha1_base64="jwisrcU9OPLyYxt5s5F2mO2Uf7M=">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</latexit>

Hélice

~r 0(t) = ( sin t, cos t, 1)

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L’équation de la droite tangente en t = 2

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(x, y, z) = ~r ⇣ ⇡ 2

⌘ + k~r 0 ⇣ ⇡ 2

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=

p2 2 ,

p2

2 , ⇡ 2

!

+ k

p2 2 ,

p2

2 , 1

!

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= ⇣

cos ⇡

2 , sin ⇡

2 , ⇡ 2

⌘ + k ⇣

sin ⇡

2 , cos ⇡

2 , 1⌘

<latexit sha1_base64="dFR2+N2ME/e7IQ1+eoYLPIwfVVo=">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</latexit><latexit sha1_base64="dFR2+N2ME/e7IQ1+eoYLPIwfVVo=">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</latexit><latexit sha1_base64="dFR2+N2ME/e7IQ1+eoYLPIwfVVo=">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</latexit><latexit sha1_base64="dFR2+N2ME/e7IQ1+eoYLPIwfVVo=">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</latexit>

(6)

Faites les exercices suivants

p. 706 # 21 à 29

(7)

Bien que l’utilité de l’intégrale définie d’une fonction vectorielle soit douteuse, on peut quand même la définir.

Z b

a

~r(t) dt =

Z b a

f (t) dt,

Z b

a

g(t) dt,

Z b

a

h(t) dt

!

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Z

~r(t) dt =

✓Z

f (t) dt, Z

g(t) dt, Z

h(t) dt

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Par contre, l’intégrale indéfinie s’avère utile pour trouver des primitives.

(8)

Exemple

Z

~r(t) dt

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=

✓Z

cos t dt, Z

sin t dt, Z

t dt

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

=

sin t + C1, cos t + C2, t2

2 + C3

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

=

sin t, cos t, t2 2

+ (C1, C2, C3)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

=

sin t, cos t, t2 2

+ C~

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

~r(t) = (cos t, sin t, t)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(9)

Faites les exercices suivants

p. 707 # 33 à 38

(10)

Il y a une distinction qui est importante à souligner entre courbe et fonction vectorielle.

Si la fonction vectorielle est continue, l’image de la fonction est une courbe. Mais comme on l’a déjà vu, il peut exister plusieurs fonctions

dont l’image est la même courbe.

Dans ce cas, on dit que la fonction est une paramétrisation de la courbe . ~r<latexit sha1_base64="Gv0DUiu5qRBwdVnomJoEOjcq7AY=">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</latexit><latexit sha1_base64="Gv0DUiu5qRBwdVnomJoEOjcq7AY=">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</latexit><latexit sha1_base64="Gv0DUiu5qRBwdVnomJoEOjcq7AY=">AAACzHicjVHLSsNAFD2Nr1pfVZdugkWom5KKoMuiG1dSwdZKWySZTuvQvJhMCqV06w+41e8S/0D/wjtjCmoRnZDkzLn3nJl7rxf7IlGO85qzFhaXllfyq4W19Y3NreL2TjOJUsl4g0V+JFuem3BfhLyhhPJ5K5bcDTyf33jDcx2/GXGZiCi8VuOYdwN3EIq+YK4i6rYz4syWZXV4Vyw5Fccsex5UM1BCtupR8QUd9BCBIUUAjhCKsA8XCT1tVOEgJq6LCXGSkDBxjikKpE0pi1OGS+yQvgPatTM2pL32TIya0Sk+vZKUNg5IE1GeJKxPs008Nc6a/c17Yjz13cb09zKvgFiFe2L/0s0y/6vTtSj0cWpqEFRTbBhdHctcUtMVfXP7S1WKHGLiNO5RXBJmRjnrs200iald99Y18TeTqVm9Z1luind9Sxpw9ec450HzqFIlfHVcqp1lo85jD/so0zxPUMMF6miQd4BHPOHZurSUNbGmn6lWLtPs4tuyHj4AhiCSRg==</latexit><latexit sha1_base64="Gv0DUiu5qRBwdVnomJoEOjcq7AY=">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</latexit> (t)

C

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On peut voir la courbe comme une piste de course, et la paramétrisation comme un coureur la parcourant.

(11)

On a que est un vecteur qui est tangent à la courbe, dont la norme donne la vitesse à laquelle la paramétrisation la

parcoure.

~r 0(t)

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k<latexit sha1_base64="cB76g6OmJVPxOYtgUNsR9b3Hp2s=">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</latexit><latexit sha1_base64="cB76g6OmJVPxOYtgUNsR9b3Hp2s=">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</latexit><latexit sha1_base64="cB76g6OmJVPxOYtgUNsR9b3Hp2s=">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</latexit><latexit sha1_base64="cB76g6OmJVPxOYtgUNsR9b3Hp2s=">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</latexit> ~r 0(t)k

On a donc que la direction de la dérivée dépend de la courbe tandis que sa longueur et son sens dépendent de la paramétrisation.

Lorsqu’on veut étudier une courbe, on ne veut pas travailler avec des paramétrisations qui reviennent sur leurs pas.

Dans un premier temps, on pourrait dire qu’une bonne paramétrisation est une paramétrisation telle que

a<latexit sha1_base64="Lw1ClOPKvQ7xh0YoosRtPqvSxas=">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</latexit><latexit sha1_base64="Lw1ClOPKvQ7xh0YoosRtPqvSxas=">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</latexit><latexit sha1_base64="Lw1ClOPKvQ7xh0YoosRtPqvSxas=">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</latexit><latexit sha1_base64="Lw1ClOPKvQ7xh0YoosRtPqvSxas=">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</latexit> 6= b =) ~r(a) 6= ~r(b)

(12)

Mais ceci nous empêcherait d’étudier des courbes qui ont des croisements.

Une meilleure approche serait d’éviter que change de sens.~r<latexit sha1_base64="eAt0M0ifxsM4SkbCWf55fewz8+k=">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</latexit><latexit sha1_base64="eAt0M0ifxsM4SkbCWf55fewz8+k=">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</latexit><latexit sha1_base64="eAt0M0ifxsM4SkbCWf55fewz8+k=">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</latexit> 0(t)

Une façon de s’en assurer serait d’utiliser des paramétrisations telles que

k~r 0(t)k 6= 0

<latexit sha1_base64="j5hNH84+hdXoE9Sggs/h3XRCe7g=">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</latexit><latexit sha1_base64="j5hNH84+hdXoE9Sggs/h3XRCe7g=">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</latexit><latexit sha1_base64="j5hNH84+hdXoE9Sggs/h3XRCe7g=">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</latexit><latexit sha1_base64="j5hNH84+hdXoE9Sggs/h3XRCe7g=">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</latexit>

(13)

k<latexit sha1_base64="wiljrlObqAHe62srfSx8wAwjDWc=">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</latexit><latexit sha1_base64="wiljrlObqAHe62srfSx8wAwjDWc=">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</latexit><latexit sha1_base64="wiljrlObqAHe62srfSx8wAwjDWc=">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</latexit><latexit sha1_base64="wiljrlObqAHe62srfSx8wAwjDWc=">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</latexit> ~r(a + h) ~r(a)k Xn

k=1<latexit sha1_base64="LavRB1LeU8dl4iMaepdDSKFuN9c=">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</latexit><latexit sha1_base64="LavRB1LeU8dl4iMaepdDSKFuN9c=">AAACz3icjVHLSsNAFD2Nr1pfVZdugkVwVRIRdCMU3bhswT6g1ZKk0zo0LyYTpZSKW3/Arf6V+Af6F94ZU1CL6IQkZ86958zce93Y54m0rNecMTe/sLiUXy6srK6tbxQ3txpJlAqP1b3Ij0TLdRLm85DVJZc+a8WCOYHrs6Y7PFPx5g0TCY/CCzmK2WXgDELe554jiep0kjTojocn9uQq7BZLVtnSy5wFdgZKyFY1Kr6ggx4ieEgRgCGEJOzDQUJPGzYsxMRdYkycIMR1nGGCAmlTymKU4RA7pO+Adu2MDWmvPBOt9ugUn15BShN7pIkoTxBWp5k6nmpnxf7mPdae6m4j+ruZV0CsxDWxf+mmmf/VqVok+jjWNXCqKdaMqs7LXFLdFXVz80tVkhxi4hTuUVwQ9rRy2mdTaxJdu+qto+NvOlOxau9luSne1S1pwPbPcc6CxkHZJlw7LFVOs1HnsYNd7NM8j1DBOaqok3eMRzzh2agZt8adcf+ZauQyzTa+LePhAyPIlCA=</latexit><latexit sha1_base64="LavRB1LeU8dl4iMaepdDSKFuN9c=">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</latexit><latexit sha1_base64="LavRB1LeU8dl4iMaepdDSKFuN9c=">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</latexit>

nlim!1

<latexit sha1_base64="8k9vAGrZIBJMAdQRYAhqE27eFTs=">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</latexit><latexit sha1_base64="8k9vAGrZIBJMAdQRYAhqE27eFTs=">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</latexit><latexit sha1_base64="8k9vAGrZIBJMAdQRYAhqE27eFTs=">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</latexit><latexit sha1_base64="8k9vAGrZIBJMAdQRYAhqE27eFTs=">AAAC1nicjVHLSsNAFD2Nr1pfqS7dBIvgqiQi6LLoxmUF+4CmlCSd1qF5kUyUUupO3PoDbvWTxD/Qv/DOmIJaRCckOXPuPWfm3uvGPk+Fab4WtIXFpeWV4mppbX1jc0svbzfTKEs81vAiP0rarpMyn4esIbjwWTtOmBO4Pmu5ozMZb12zJOVReCnGMesGzjDkA+45gqieXrZ9HvQmoS0im4cDMZ729IpZNdUy5oGVgwryVY/0F9joI4KHDAEYQgjCPhyk9HRgwURMXBcT4hJCXMUZpiiRNqMsRhkOsSP6DmnXydmQ9tIzVWqPTvHpTUhpYJ80EeUlhOVphopnylmyv3lPlKe825j+bu4VECtwRexfulnmf3WyFoEBTlQNnGqKFSOr83KXTHVF3tz4UpUgh5g4ifsUTwh7Sjnrs6E0qapd9tZR8TeVKVm59/LcDO/yljRg6+c450HzsGoRvjiq1E7zURexiz0c0DyPUcM56miQ9w0e8YRnra3danfa/WeqVsg1O/i2tIcPdYuW/g==</latexit>

k~r(a + h) ~r(a)k

h h

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k~r(a + t) ~r(a)k

t t

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k~r 0(a)k t

<latexit sha1_base64="hpAkArfMpC3LHybIrEcgeGB6Lg8=">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</latexit><latexit sha1_base64="hpAkArfMpC3LHybIrEcgeGB6Lg8=">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</latexit><latexit sha1_base64="hpAkArfMpC3LHybIrEcgeGB6Lg8=">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</latexit><latexit sha1_base64="hpAkArfMpC3LHybIrEcgeGB6Lg8=">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</latexit> =

Z b

a k~r 0(t)k t

<latexit sha1_base64="rJa2Oe2PX5iNen8EC1eCMxFP+CE=">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</latexit><latexit sha1_base64="rJa2Oe2PX5iNen8EC1eCMxFP+CE=">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</latexit><latexit sha1_base64="rJa2Oe2PX5iNen8EC1eCMxFP+CE=">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</latexit><latexit sha1_base64="rJa2Oe2PX5iNen8EC1eCMxFP+CE=">AAAC5nicjVHLSsNAFD3G97vq0k20iLopqQi6EURduFSwKjS1TKbTdjBNwmQiiLp2507c+gNu9VPEP9C/8M4YQS2iE5Kce+49Z+bODZJQptrzXnqc3r7+gcGh4ZHRsfGJycLU9GEaZ4qLCo/DWB0HLBWhjERFSx2K40QJ1glCcRScbpv80ZlQqYyjA32eiFqHtSLZlJxpouqFuQ1fRrrOTgL/0j8T3FW+u7iklynaEaFmrq4Xil7Js8vtBuUcFJGvvbjwDB8NxODI0IFABE04BENKTxVleEiIq+GCOEVI2rzAFUZIm1GVoApG7Cl9WxRVczai2HimVs1pl5BeRUoXC6SJqU4RNru5Np9ZZ8P+5n1hPc3Zzukf5F4dYjXaxP6l+6z8r870otHEuu1BUk+JZUx3PHfJ7K2Yk7tfutLkkBBncIPyijC3ys97dq0mtb2bu2U2/2orDWtintdmeDOnpAGXf46zGxyulMqE91eLm1v5qIcwi3ks0TzXsIld7KFC3td4wCOenLZz49w6dx+lTk+umcG35dy/A7Gzm/g=</latexit>

(14)

=

Z b

a k~r 0(t)k t

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Longueur de la courbe

=

Z b

a

pf 0(t)2 + g0(t)2 + h0(t)2 t

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