cours 2
1.2 FONCTIONS
VECTORIELLES, COURBES
ET SURFACES
Au dernier cours, nous avons vu
Survol des cours calcul différentiel, calcul intégral et algèbre linéaire.
Introduction des idées que nous verrons cette session.
Aujourd’hui, nous allons voir
Fonctions vectorielles Comment les visualiser Limite
Surfaces paramétrées
On a déjà vu les fonctions de la forme f
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~ r
<latexit sha1_base64="oR7WNJ5BDcbw+7EE5fbfezzKsyA=">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</latexit><latexit sha1_base64="oR7WNJ5BDcbw+7EE5fbfezzKsyA=">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</latexit><latexit sha1_base64="oR7WNJ5BDcbw+7EE5fbfezzKsyA=">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</latexit><latexit sha1_base64="oR7WNJ5BDcbw+7EE5fbfezzKsyA=">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</latexit>: R ! R
2x
<latexit sha1_base64="RImMqKPne7idc0aUvLEVQmpY+54=">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</latexit><latexit sha1_base64="RImMqKPne7idc0aUvLEVQmpY+54=">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</latexit><latexit sha1_base64="RImMqKPne7idc0aUvLEVQmpY+54=">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</latexit><latexit sha1_base64="RImMqKPne7idc0aUvLEVQmpY+54=">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</latexit>7 ! f (x)
t 7 ! ~ r(t)
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= (f (t), g (t))
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3= (f
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<latexit sha1_base64="DtbUPBUzViBqsXWyjQer/0c00Go=">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</latexit><latexit sha1_base64="DtbUPBUzViBqsXWyjQer/0c00Go=">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</latexit><latexit sha1_base64="DtbUPBUzViBqsXWyjQer/0c00Go=">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</latexit><latexit sha1_base64="DtbUPBUzViBqsXWyjQer/0c00Go=">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</latexit>
~ r
<latexit sha1_base64="MraIf2ilsw5HB2bxstJJC8cWqAE=">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</latexit><latexit sha1_base64="MraIf2ilsw5HB2bxstJJC8cWqAE=">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</latexit><latexit sha1_base64="MraIf2ilsw5HB2bxstJJC8cWqAE=">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</latexit><latexit sha1_base64="MraIf2ilsw5HB2bxstJJC8cWqAE=">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</latexit>: R ! R
n= (f
1(t), f
2(t), . . . , f
n(t))
<latexit sha1_base64="nTmCyDOBC1s9B+nYe3HUqBL3XyA=">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</latexit><latexit sha1_base64="nTmCyDOBC1s9B+nYe3HUqBL3XyA=">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</latexit><latexit sha1_base64="nTmCyDOBC1s9B+nYe3HUqBL3XyA=">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</latexit><latexit sha1_base64="nTmCyDOBC1s9B+nYe3HUqBL3XyA=">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</latexit>
t 7 ! ~ r (t)
<latexit sha1_base64="DtbUPBUzViBqsXWyjQer/0c00Go=">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</latexit><latexit sha1_base64="DtbUPBUzViBqsXWyjQer/0c00Go=">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</latexit><latexit sha1_base64="DtbUPBUzViBqsXWyjQer/0c00Go=">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</latexit><latexit sha1_base64="DtbUPBUzViBqsXWyjQer/0c00Go=">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</latexit>
Aujourd’hui nous allons explorer les fonctions vectorielles.
=
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>y
Exemple ~ r
<latexit sha1_base64="BVCvstB1sCZXMbNmadWulxfxFyI=">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</latexit><latexit sha1_base64="BVCvstB1sCZXMbNmadWulxfxFyI=">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</latexit><latexit sha1_base64="BVCvstB1sCZXMbNmadWulxfxFyI=">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</latexit><latexit sha1_base64="BVCvstB1sCZXMbNmadWulxfxFyI=">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</latexit>(t) = (1, 3) + t(1, 1)
= (1 + t, 3 t)
<latexit sha1_base64="+/ax3IRNAE+dxMi1GhntadaL8HY=">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</latexit><latexit sha1_base64="+/ax3IRNAE+dxMi1GhntadaL8HY=">AAACzXicjVHLSsNAFD2Nr/quunQTLEJFLYkKuhGKbtxZwT6wFkmm0xqaJiGZCKXq1h9wq78l/oH+hXfGKahFdEKSM+fec2buvW7ke4mwrNeMMTY+MTmVnZ6ZnZtfWMwtLVeTMI0Zr7DQD+O66yTc9wJeEZ7weT2KudNzfV5zu8cyXrvhceKFwbnoR7zZczqB1/aYI4i6OCzYm2Jrd1tsXOXyVtFSyxwFtgZ56FUOcy+4RAshGFL0wBFAEPbhIKGnARsWIuKaGBAXE/JUnOMOM6RNKYtThkNsl74d2jU0G9BeeiZKzegUn96YlCbWSRNSXkxYnmaqeKqcJfub90B5yrv16e9qrx6xAtfE/qUbZv5XJ2sRaONA1eBRTZFiZHVMu6SqK/Lm5peqBDlExEnconhMmCnlsM+m0iSqdtlbR8XfVKZk5Z7p3BTv8pY0YPvnOEdBdadoEz7by5eO9KizWMUaCjTPfZRwgjIq5B3gEU94Nk6N1Lg17j9TjYzWrODbMh4+AEvZkb0=</latexit><latexit sha1_base64="+/ax3IRNAE+dxMi1GhntadaL8HY=">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</latexit><latexit sha1_base64="+/ax3IRNAE+dxMi1GhntadaL8HY=">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</latexit>
Exemple ~ r
<latexit sha1_base64="UZXlysglZc9vyNMHBcbXzlJPfcg=">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</latexit><latexit sha1_base64="UZXlysglZc9vyNMHBcbXzlJPfcg=">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</latexit><latexit sha1_base64="UZXlysglZc9vyNMHBcbXzlJPfcg=">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</latexit><latexit sha1_base64="UZXlysglZc9vyNMHBcbXzlJPfcg=">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</latexit>(t) = (t, t
2)
x
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y
<latexit sha1_base64="+2yAOIuk+XkBM6MDf0mnE6tnKzs=">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</latexit><latexit sha1_base64="+2yAOIuk+XkBM6MDf0mnE6tnKzs=">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</latexit><latexit sha1_base64="+2yAOIuk+XkBM6MDf0mnE6tnKzs=">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</latexit><latexit sha1_base64="+2yAOIuk+XkBM6MDf0mnE6tnKzs=">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</latexit>= t
2=
<latexit sha1_base64="vK/puGmV2wZB5o0AZIi9FZrBiJo=">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</latexit><latexit sha1_base64="vK/puGmV2wZB5o0AZIi9FZrBiJo=">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</latexit><latexit sha1_base64="vK/puGmV2wZB5o0AZIi9FZrBiJo=">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</latexit><latexit sha1_base64="vK/puGmV2wZB5o0AZIi9FZrBiJo=">AAACx3icjVHLSsNAFD2Nr1pfVZdugkVwVZIi6EYoutFdBfuAWiWZTtvQJBOSSWkpLvwBt/pn4h/oX3hnTEEtohOSnDn3njNz73Uj30ukZb3mjIXFpeWV/GphbX1jc6u4vdNIRBozXmfCF3HLdRLueyGvS0/6vBXF3Alcnzfd4bmKN0c8TjwRXstJxDuB0w+9nsccqajT8W3lrliyypZe5jywM1BCtmqi+IIbdCHAkCIARwhJ2IeDhJ42bFiIiOtgSlxMyNNxjnsUSJtSFqcMh9ghffu0a2dsSHvlmWg1o1N8emNSmjggjaC8mLA6zdTxVDsr9jfvqfZUd5vQ3828AmIlBsT+pZtl/lenapHo4UTX4FFNkWZUdSxzSXVX1M3NL1VJcoiIU7hL8Zgw08pZn02tSXTtqreOjr/pTMWqPctyU7yrW9KA7Z/jnAeNStkmfHVUqp5lo85jD/s4pHkeo4oL1FAn7wEe8YRn49IQxsgYf6YauUyzi2/LePgAxWeQaA==</latexit>x
2Exemple Cercle x
<latexit sha1_base64="vYMsS494V6/CXRl/NOeRbxFK310=">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</latexit><latexit sha1_base64="vYMsS494V6/CXRl/NOeRbxFK310=">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</latexit><latexit sha1_base64="vYMsS494V6/CXRl/NOeRbxFK310=">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</latexit><latexit sha1_base64="vYMsS494V6/CXRl/NOeRbxFK310=">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</latexit> 2+ y
2= 1
~ r (t) = (cos t, sin t)
<latexit sha1_base64="F06eQZ+MAHBdNgU3bpoyGUI5be8=">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</latexit><latexit sha1_base64="F06eQZ+MAHBdNgU3bpoyGUI5be8=">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</latexit><latexit sha1_base64="F06eQZ+MAHBdNgU3bpoyGUI5be8=">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</latexit><latexit sha1_base64="F06eQZ+MAHBdNgU3bpoyGUI5be8=">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</latexit>
Exemple Folium de Descartes
~ r (t) =
✓ 3t
1 + t
3, 3t
21 + t
3◆
<latexit sha1_base64="bdZk2L2r4W9d6n8RbBJVxI05B7g=">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</latexit><latexit sha1_base64="bdZk2L2r4W9d6n8RbBJVxI05B7g=">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</latexit><latexit sha1_base64="bdZk2L2r4W9d6n8RbBJVxI05B7g=">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</latexit><latexit sha1_base64="bdZk2L2r4W9d6n8RbBJVxI05B7g=">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</latexit>
x
3+ y
3= 3xy
<latexit sha1_base64="otr/EBYIF3HrPLSb21lyjA04a6Q=">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</latexit><latexit sha1_base64="otr/EBYIF3HrPLSb21lyjA04a6Q=">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</latexit><latexit sha1_base64="otr/EBYIF3HrPLSb21lyjA04a6Q=">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</latexit><latexit sha1_base64="otr/EBYIF3HrPLSb21lyjA04a6Q=">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</latexit>
Exemple ~ r
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>(t) = (1 + t, 1 t, 1 + t)
d ~ = (1, 1, 1)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
La droite de vecteur directeur
OA ~ = (1, 1, 1)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
passant par le point
Exemple ~ r(t) = ✓ cos t, sin t, t puisque cos
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> 2t + sin
2t = 1
4
◆
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
la courbe est sur le cylindre x
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> 2+ y
2= 1
Faites les exercices suivants
p.700 # 5, 7, 9, 10, 12 et 16
Une façon pour aider à visualiser une courbe dans l’espace est de regarder ses projections sur les plans de coordonnés.
x
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>y
<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
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>x
<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
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>y
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>✓
cos t + 2, t 2
◆
<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 + 2, sin t + 2, t 2
◆
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
(cos t + 2,sin t + 2)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
✓
sint + 2, t 2
◆
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Exemple
x
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
y
<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
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>x
<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
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>y
<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) = t, t
2, t
3<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
t, t
2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
t
2, t
3<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
t, t
3<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
x
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>y
<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
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>x
<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
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>y
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>(cos t + 2,sin 3t + 2)
<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 + 2, sin t + 2, sin 3t + 2)
<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 + 2,sin 3t + 2)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
(cos t + 2, sin t + 2)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
On peut construire certaines courbes en regardant l’intersection de deux surfaces.
z
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>= 1 x y x
2+ y
2= 1
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Le cylindre
x + y + z = 1
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Le plan
~ r (t) = (cos t, sin t, 1 cos t sin 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>
= 1
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>cos t sin t y = sin 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>
x = cos 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>
![Sur la forme des courbes [voir pdf]](data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==)