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3.6 COORDONNÉES CYLINDRIQUE ET SPHÉRIQUE ET LE

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Texte intégral

(1)

cours 23

3.6 COORDONNÉES CYLINDRIQUE ET SPHÉRIQUE ET LE

JACOBIEN

(2)

Au dernier cours, nous avons vu

Intégrale triple Type I, II et III Volumes

Applications

(3)

Aujourd’hui, nous allons voir

Changement de variable en coordonnées cylindrique.

Changement de variable en coordonnées sphériques.

Jacobien

(4)

0<latexit sha1_base64="dkaGuWlheZF6wCwT9dojZUunNe4=">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</latexit>  r < 1 0<latexit sha1_base64="wDTYyotqJAl4B01VJqQZLrDeFh4=">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</latexit>  ✓ < 2⇡ 1<latexit sha1_base64="ZBBL2YTQJ1zLo7EL8mXxLTgABQ0=">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</latexit> < z < 1

(5)

V = r ✓ r z

<latexit sha1_base64="nphg3dJmcybNUW7VpeHIqX3HRzU=">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</latexit>

(6)

x = r cos ✓

<latexit sha1_base64="Bczx4ic1bLFnHKOdX/Cr5ys5f2A=">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</latexit>

y = r sin ✓

<latexit sha1_base64="uib263dzPl17frtEDNQsQsNdy2I=">AAADAnicjVJBT9RAFP4ooiuIrHD00rgh8bRplUQuJkQuHJeEBRJ2Y9oywIRuW6dTkmbjzf/AFa7cjFf/iP8AbvwEv3kORiVGp+n0m++97/W9Ny+tcl3bKPo2E8w+mHv4qPN4fuHJ4tOl7rPl3bpsTKaGWZmXZj9NapXrQg2ttrnar4xKJmmu9tLTTWffO1Om1mWxY9tKjSfJcaGPdJZYUuP2rRnVuhjZE2WT991e1I9khfdB7EEPfg3K7i1GOESJDA0mUChgiXMkqPkcIEaEitwYU3KGSItd4SPmqW3opeiRkD3lfszTgWcLnl3MWtQZ/5LzNVSGWKWmpJ8hdn8Lxd5IZMf+LfZUYrrcWn5TH2tC1uKE7L90d57/q3O1WBxhXWrQrKkSxlWX+SiNdMVlHv5SlWWEipzDh7Qb4kyUd30ORVNL7a63idivxdOx7px53wY3PkuFM4na/sx+Kneoaa+kly2R5S63xJGI/xyA+2D3VT9+3Y+313ob7/xwdPAcL/CSE/AGG9jCAENm8wHnuMBl8Cm4Cj4HX364BjNes4LfVvD1Oxf/oH8=</latexit>

z<latexit sha1_base64="+NlxIejqP/ix10XNTderII1WS04=">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</latexit> = z

= @x

@r dr + @x

@✓ d✓ + @x

@z dz

<latexit sha1_base64="qTGWlk0vtjngCMQQ27jEZ39wPDE=">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</latexit>

= @y

@r dr + @y

@✓ d✓ + @y

@z dz

<latexit sha1_base64="1123BllKi81n9ljBRqvgmmfN3Kg=">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</latexit>

= @z

@r dr + @z

@✓ d✓ + @z

@z dz

<latexit sha1_base64="O0vFUFEfCdzJKhbsHuJ0VkFHcfY=">AAADYHicjVJNb9QwEH3d8LGUj27hBpeIFRIS0ioBJLggVXDhWCS2rdStKifxtlazSeQ4lbar/Y2cuXKCGxeuwPPU5asq4Cj285t545mxs6Y0rUuS9yu96NLlK1f711av37h5a22wfnurrTub63Fel7XdyVSrS1PpsTOu1DuN1WqWlXo7O3rl7dvH2ramrt66eaP3ZuqgMlOTK0dqf2BeTKZW5YtJo6wzqoxPlj+xXRb20cX2iTvUTi2L0/UvjsTFyf5gmIwSGfF5kAYwRBib9eAzJihQI0eHGTQqOOISCi2/XaRI0JDbw4KcJTJi11hildqOXpoeiuwR5wPudgNbce9jtqLOeUrJ31IZ4wE1Nf0ssT8tFnsnkT17UeyFxPS5zblmIdaMrMMh2X/pzjz/V+drcZjiudRgWFMjjK8uD1E66YrPPP6lKscIDTmPC9otcS7Ksz7Hommldt9bJfaP4ulZv8+Db4dPIUuNY4k6/5H9Qu7Q0N5IL+dEjrPcEp9E+ucDOA+2Ho/SJ6P0zdPhxsvwOPq4h/t4yBfwDBt4jU2Mmc07fMFXfOt9iPrRWrR+6tpbCZo7+G1Ed78D3WDFAw==</latexit>

= 0 + 0 +<latexit sha1_base64="a4oJ7asbjZnfzUG3ac7L8ohXsN4=">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</latexit> dz

= cos<latexit sha1_base64="Vuxxp16f/czuYJwczayJr/rNDrA=">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</latexit> ✓dr r sin ✓d✓ + 0

= sin<latexit sha1_base64="TzmA4WXZWsXwkDEkusvwQ7jeU4o=">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</latexit> ✓dr + r cos ✓d✓ + 0 dx<latexit sha1_base64="+g3xtn74ltYAnc038Mt2fLW+Xls=">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</latexit>

dy

<latexit sha1_base64="uIRuQFBAddEOy+3b5MBhj6E36Kc=">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</latexit>

dz<latexit sha1_base64="o5GYwRACbQcjTYHfNwvhQIlldBs=">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</latexit>

(7)

x = r cos ✓

<latexit sha1_base64="Bczx4ic1bLFnHKOdX/Cr5ys5f2A=">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</latexit>

y<latexit sha1_base64="uib263dzPl17frtEDNQsQsNdy2I=">AAADAnicjVJBT9RAFP4ooiuIrHD00rgh8bRplUQuJkQuHJeEBRJ2Y9oywIRuW6dTkmbjzf/AFa7cjFf/iP8AbvwEv3kORiVGp+n0m++97/W9Ny+tcl3bKPo2E8w+mHv4qPN4fuHJ4tOl7rPl3bpsTKaGWZmXZj9NapXrQg2ttrnar4xKJmmu9tLTTWffO1Om1mWxY9tKjSfJcaGPdJZYUuP2rRnVuhjZE2WT991e1I9khfdB7EEPfg3K7i1GOESJDA0mUChgiXMkqPkcIEaEitwYU3KGSItd4SPmqW3opeiRkD3lfszTgWcLnl3MWtQZ/5LzNVSGWKWmpJ8hdn8Lxd5IZMf+LfZUYrrcWn5TH2tC1uKE7L90d57/q3O1WBxhXWrQrKkSxlWX+SiNdMVlHv5SlWWEipzDh7Qb4kyUd30ORVNL7a63idivxdOx7px53wY3PkuFM4na/sx+Kneoaa+kly2R5S63xJGI/xyA+2D3VT9+3Y+313ob7/xwdPAcL/CSE/AGG9jCAENm8wHnuMBl8Cm4Cj4HX364BjNes4LfVvD1Oxf/oH8=</latexit> = r sin ✓

z<latexit sha1_base64="+NlxIejqP/ix10XNTderII1WS04=">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</latexit> = z = 0 + 0 +<latexit sha1_base64="a4oJ7asbjZnfzUG3ac7L8ohXsN4=">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</latexit> dz

= cos<latexit sha1_base64="Vuxxp16f/czuYJwczayJr/rNDrA=">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</latexit> ✓dr r sin ✓d✓ + 0

= sin<latexit sha1_base64="TzmA4WXZWsXwkDEkusvwQ7jeU4o=">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</latexit> ✓dr + r cos ✓d✓ + 0 dx<latexit sha1_base64="+g3xtn74ltYAnc038Mt2fLW+Xls=">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</latexit>

dy<latexit sha1_base64="uIRuQFBAddEOy+3b5MBhj6E36Kc=">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</latexit>

dz<latexit sha1_base64="o5GYwRACbQcjTYHfNwvhQIlldBs=">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</latexit>

0

@ dx dy dz

1

A =

0

@ cos ✓ r sin ✓ 0 sin ✓ r cos ✓ 0

0 0 1

1 A

0

@ dr d✓

dz

1 A

<latexit sha1_base64="hSYIOwDyDMyOSl7o1O3ayf0UyVc=">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</latexit>

cos ✓ r sin ✓ 0 sin ✓ r cos ✓ 0

0 0 1

= r cos2 ✓ + r sin2

<latexit sha1_base64="mIoL2OpMptdl6kt05zNVplswGKg=">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</latexit>

=<latexit sha1_base64="qv9drQR1fVG1SAIUwWdiH+1EkLE=">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</latexit> r

dV<latexit sha1_base64="c4XPURXvHxOKiQS6kzVRia3mUnI=">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</latexit> = dx dy dz = r dr d✓ dz

(8)

Exemple

ZZZ

R

x2 + y2dV

<latexit sha1_base64="8Ic1dNEPYLFb+vtJMMymthO2btQ=">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</latexit>

x<latexit sha1_base64="Q9KYVLhqx+t/KOMAo/3CdkQ1Dr8=">AAAC/nicjVLLSh1BED2OefhKvJqlmyEXQRAuMzcBswmI2bg0kKsGH2FmbE3jvJjpkQwXwX/IVrfuglt/JX8Qd35CTlfa4INgepie06fq1FRVV1ymujZB8HPEG33y9NnzsfGJyakXL6c7M7PrddFUiRokRVpUm3FUq1TnamC0SdVmWakoi1O1ER9+sPaNI1XVusg/mbZUO1l0kOt9nUSG1Odvu/3Fdrf/PvzS6Qa9QJb/EIQOdOHWWtG5xjb2UCBBgwwKOQxxigg1ny2ECFCS28GQXEWkxa5wjAlqG3opekRkD7kf8LTl2JxnG7MWdcK/pHwrKn3MU1PQryK2f/PF3khky/4r9lBi2txafmMXKyNr8JXsY7obz//V2VoM9vFOatCsqRTGVpe4KI10xWbu36rKMEJJzuI92iviRJQ3ffZFU0vttreR2H+Jp2XtOXG+Da5clgpHErX9m/1Q7lDTXkovWyLDXW6JIxHeH4CHYL3fC9/0+h/fdpdX3HCMYQ6vscAJWMIyVrGGAbPJ8B2nOPNOvHPvh3fxx9UbcZpXuLO8y9/HFZ3C</latexit> 2 + y2 = 1 x<latexit sha1_base64="hynwE6t6MkSHOJZplh327W5ImUc=">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</latexit> 2 + y2 = 4

z<latexit sha1_base64="IM8tZkuT106Ak5/n3NvrAeIR8Nc=">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</latexit> = 0 z<latexit sha1_base64="7maU/DfJbThU4mxDm8uIXJiaYqU=">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</latexit> = 1 x<latexit sha1_base64="SvHMJQgfISuN0u73Y5zlZzDa5us=">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</latexit> = 0

x<latexit sha1_base64="UHKW50mrT1gu4GWXR3xaEmgIKlQ=">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</latexit> = y

Z 1

0

Z 2

1

Z 4

0

r2 rd✓drdz

<latexit sha1_base64="vbPOLwVN3hlc/qYZAMaC1KneJbs=">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</latexit>

=

Z 1

0

Z 2

1

⇡r3

4 drdz

<latexit sha1_base64="47BruyaxW13XWRM7ROEjm7X8t14=">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</latexit>

=

Z 1

0

⇡r4 16

2 1

dz

<latexit sha1_base64="6uTQXDhOKpC0ayYLm/wI8coxeHQ=">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</latexit>

=

Z 1

0

16⇡

16

16 dz

<latexit sha1_base64="5a+adZpR3K+QsHd91Po3BRJf7Gk=">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</latexit>

= 15⇡

16

<latexit sha1_base64="WOt9ErB0RuHCwveOf5zPUz90iSw=">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</latexit>

(9)

Exemple

Trouver le volume à l’intérieur d’une sphère de rayon 6 et le paraboloïde

z<latexit sha1_base64="KXfJTYVS2I2ObYUTHkG9Dwd+gs0=">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</latexit> = x2 + y2 x<latexit sha1_base64="x6D6YaGBE7cFYrY8jvQ3GOXSLpo=">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</latexit> 2 + y2 + z2 = 6

z<latexit sha1_base64="rAZjI6KvYU5oqXl8JnWt+3CN098=">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</latexit> 2 = 6 x2 y2 = (x<latexit sha1_base64="iDuNJt07f8V9w3RJVwNuSFuh+3Q=">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</latexit> 2 + y2)2 6<latexit sha1_base64="dKMgHRxZ5MTXr0qVydXfQBhpmNg=">AAADAnicjVLBThRBEH2MCisqrHL0MmFjggc2swtBLyYbuXDExAUSXMzM0GCH2Zmhp4dks+HGP3jVKzfClR/hD/TmJ/i6aAxCiPZkel6/qldTVV1JmenKRtHlRPDg4aPJqcbj6SdPn83MNp+/2KiK2qSqnxZZYbaSuFKZzlXfapuprdKoeJhkajM5WHX2zSNlKl3kH+2oVINhvJ/rPZ3GltRgZdHsdN8tcHu90/3cbEXtSFZ4F3Q8aMGv9aL5C5+wiwIpagyhkMMSZ4hR8dlGBxFKcgOMyRkiLXaFY0xTW9NL0SMme8B9n6dtz+Y8u5iVqFP+JeNrqAzxipqCfobY/S0Uey2RHXtf7LHEdLmN+E18rCFZiy9k/6W79vxfnavFYg9vpQbNmkphXHWpj1JLV1zm4Y2qLCOU5Bzepd0Qp6K87nMomkpqd72Nxf5DPB3rzqn3rfHTZ6lwJFFHf7Ifyx1q2kvp5YjIcpdb4kh0bg/AXbDRbXeW2t0Py63eez8cDbzEPBY4AW/QwxrW0Wc2h/iKb/genASnwVlwfuUaTHjNHP5awcVvm6SexQ==</latexit> r2 = (r2)2

r<latexit sha1_base64="LAwI/HNJSGopR1T5Mxc83pFUcco=">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</latexit> 4 + r2 6 = 0 = (r<latexit sha1_base64="O6ose4cDy2pjY7a0QdRrZdrUXeQ=">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</latexit> 2 + 3)(r2 2)

= (r2 + 3)(r p

2)(r + p 2)

<latexit sha1_base64="vER/xChWxiUroF47mmM614QkR7I=">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</latexit>

r = p 2

<latexit sha1_base64="cNNVVjAZK4u1uBFiPFOUlajuBKc=">AAAC/nicjVJNSxxBEH2OMX5GVz3mMrgInpbZNWAuAdFLjgquH6iEmbHVxvlKT4+wLEL+Q67m6k1yzV/JP4i3/ARfV9oQlZD0MD2vX9WrqaqupMp0baPo+0gw+mLs5fjE5NT0zKvZudb8wm5dNiZV/bTMSrOfxLXKdKH6VttM7VdGxXmSqb3kYtPZ9y6VqXVZ7NhBpY7z+KzQpzqNLakD8+6o/mhs2PvQakedSFb4HHQ9aMOvrbL1E0c4QYkUDXIoFLDEGWLUfA7RRYSK3DGG5AyRFrvCFaaobeil6BGTveB+xtOhZwueXcxa1Cn/kvE1VIZYpqaknyF2fwvF3khkx/4t9lBiutwG/CY+Vk7W4pzsv3QPnv+rc7VYnOKt1KBZUyWMqy71URrpiss8/KMqywgVOYdPaDfEqSgf+hyKppbaXW9jsf8QT8e6c+p9G9z5LBUuJergd/ZDuUNNeyW9HBBZ7nJLHInu0wF4DnZ7ne5qp7f9pr2+4YdjAq+xhBVOwBrW8R5b6DObHJ9xjS/Bp+AmuA2+/nINRrxmEY9W8O0eCk2ePw==</latexit>

Z 2⇡

0

Z p2

0

Z p6 r2

r2

r dzdrd✓

<latexit sha1_base64="IjTtUsUCpooMV4g9sqIUX5Sy/WQ=">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</latexit>

=

Z 2⇡

0

Z p2

0

zr

p6 r2

r2

drd✓

<latexit sha1_base64="JOooWPFJetBlHprSufIHb9DTkOI=">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</latexit>

(10)

Exemple

Trouver le volume à l’intérieur d’une sphère de rayon 6 et le paraboloïde

Z 2⇡

0

Z p2

0

Z p6 r2

r2

r dzdrd✓

<latexit sha1_base64="IjTtUsUCpooMV4g9sqIUX5Sy/WQ=">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</latexit>

=

Z 2⇡

0

Z p2

0

zr

p6 r2

r2

drd✓

<latexit sha1_base64="JOooWPFJetBlHprSufIHb9DTkOI=">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</latexit>

=

Z 2⇡

0

Z p2

0

rp

6 r2 r3 drd✓

<latexit sha1_base64="UgbWepb8NRi+3+rWPOQjQzBE/5c=">AAADM3icjVLLbtRAEKyYR0J4LeHIxWKFxIWV10GEA0hRuHAMEptEipOV7Z0ko3htMx5HWln+mfwD/5Ar3BA3xAEOSHwCNR0vAiIEY3lcU93V7u7ppMx0ZYPgw4J36fKVq4tL15av37h563bvzspWVdQmVaO0yAqzk8SVynSuRlbbTO2URsXTJFPbyfELZ98+UabSRf7azkq1N40Pc32g09iSGveePY90bsfBfhNGpW7nh6h6Y2wTtq05R08emf2w5bbqR/7ETCJ7pGw87vWDQSDLvwiGHeijW5tF7zsiTFAgRY0pFHJY4gwxKj67GCJASW4PDTlDpMWu0GKZ2ppeih4x2WPuhzztdmzOs4tZiTrlXzK+hkofD6gp6GeI3d98sdcS2bF/i91ITJfbjN+kizUla3FE9l+6uef/6lwtFgd4KjVo1lQK46pLuyi1dMVl7v9SlWWEkpzDE9oNcSrKeZ990VRSu+ttLPYv4ulYd0473xpfuywVTiTq7Gf2jdyhpr2UXs6ILHe5JY7E8M8BuAi2wsFwdRC+etxf3+iGYwn3cB8POQFrWMdLbGLEbE5xhnd47731PnqfvM/nrt5Cp7mL35b37QeBwLPA</latexit>

=

Z 2⇡

0

(6 r2) 32 2

r4 4

p2

0

d✓

<latexit sha1_base64="zy4BDCvboN1tp1CGZ4ybBWT+7RU=">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</latexit>

=

Z 2⇡

0

4 1 + (6) 32

2 d✓

<latexit sha1_base64="vwj/kDoME+sP1BDFf0EqSsxur9M=">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</latexit>

= 2⇡ (6) 32

2 5

!

(11)

Faites les exercices suivants

p.887 # 1, 2, 5, 7 à 11 p.887 # 1, 2, 5, 7 à 11

(12)

0<latexit sha1_base64="sUN/CzPBl9bqgUSfzcUsaodT16k=">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</latexit>  ⇢ < 1 0<latexit sha1_base64="wDTYyotqJAl4B01VJqQZLrDeFh4=">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</latexit>  ✓ < 2⇡ 0<latexit sha1_base64="8KdkNM1JLOOLHcCrIsptdk1jH38=">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</latexit>  < ⇡

(13)

z<latexit sha1_base64="LtZFiGoCJP0lVPln6IC1rV1qcjI=">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</latexit> = ⇢ cos x<latexit sha1_base64="h7CZYJcsATcfUfKNA6daZgeWp2s=">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</latexit> = ⇢ sin cos ✓

y<latexit sha1_base64="gptuxrNPBKyA03qoJvo4m9Djq8g=">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</latexit> = ⇢ sin sin ✓

<latexit sha1_base64="/7si2G3AVFJaSUqF9KspTu95/7M=">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</latexit> 2 = x2 + y2 + z2

(14)

V = ⇢ sin ✓ ⇢ ⇢

<latexit sha1_base64="nV/BWWpY7F2omBXwMj/XGC5pUmM=">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</latexit> = ⇢2 sin ✓ ⇢<latexit sha1_base64="lx3Wrred6Ua3KgeCYb9ldA4KIn0=">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</latexit>

(15)

dx<latexit sha1_base64="+g3xtn74ltYAnc038Mt2fLW+Xls=">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</latexit>

dy<latexit sha1_base64="uIRuQFBAddEOy+3b5MBhj6E36Kc=">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</latexit>

dz<latexit sha1_base64="o5GYwRACbQcjTYHfNwvhQIlldBs=">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</latexit>

x<latexit sha1_base64="V1nLO2Dvk6Nhg0mTU7jFZs9SbxE=">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</latexit> = ⇢ cos ✓ sin y<latexit sha1_base64="DIWHZp2fAG0C7ya8OOtwhK6JMUc=">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</latexit> = ⇢ sin ✓ sin z<latexit sha1_base64="uqae2QA0kPr80f7JP76Lmt6tGXk=">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</latexit> = ⇢ cos

= @x

@⇢ d⇢ + @x

@ d + @x

@✓ d✓

<latexit sha1_base64="EDEXGJ5020urq5wG6p9Z+Ll4ZSY=">AAADa3icjVJdS9xAFL3Z9EO3H27rm/YhdBEKhSXRgn0pSPvSRwuuCkZkMjvrDmaTYTIRl2V/aPsL2odCf0LPXMdqK7adkMmZc8+5uXdmClPqxqXp56gT37v/4OHScvfR4ydPV3rPnu83dWulGsq6rO1hIRpV6koNnXalOjRWiWlRqoPi7IOPH5wr2+i62nMzo46n4rTSYy2FA3XSc+/ysRVynhthnRZlcrG4xrmd1IuRn1/frcrNREOE+W8iN1FOQMbfk14/HaQ8ktsgC6BPYezWvR+U04hqktTSlBRV5IBLEtTgOaKMUjLgjmkOzgJpjitaUBfeFioFhQB7hvkUq6PAVlj7nA27Jf5S4rVwJrQBTw2dBfZ/SzjecmbP3pV7zjl9bTN8i5BrCtbRBOy/fFfK//X5XhyN6S33oNGTYcZ3J0OWlnfFV57c6MohgwHn8QhxCyzZebXPCXsa7t3vreD4V1Z61q9l0Lb0LVSp6Jyzzn5VP+cz1Igb3ssZkMPMp4Qrkf15AW6D/c1BtjXIPr3p77wPl2OJ1uklvcIN2KYd+ki7NEQ1XyKKlqNu53u8Gq/FLy6lnSh4Vum3EW/8BIqYxvo=</latexit>

= @y

@⇢ d⇢ + @y

@ d + @y

@✓ d✓

<latexit sha1_base64="iLBxqZyV2ld5TbpW19Aut2NNkCA=">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</latexit>

= @z

@⇢ d⇢ + @z

@ d + @z

@✓ d✓

<latexit sha1_base64="H5LLOzAC4wO3nMiack2w67CAmOs=">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</latexit>

= cos<latexit sha1_base64="jI/OIEDjUVoN3iFG6zwTvdrAL6c=">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</latexit> ✓ sin d⇢ + ⇢ cos ✓ cos d ⇢ sin ✓ sin d✓

= sin<latexit sha1_base64="XKmPEjD4KW5nN6Uctj0Z7Pczw18=">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</latexit> ✓ sin d⇢ + ⇢ sin ✓ cos d + ⇢ cos ✓ sin d✓

= cos<latexit sha1_base64="4rTVrOl0qboHMUV37QLn91+9xa8=">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</latexit> d⇢ ⇢ sin d + 0d✓

(16)

dx<latexit sha1_base64="+g3xtn74ltYAnc038Mt2fLW+Xls=">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</latexit>

dy<latexit sha1_base64="uIRuQFBAddEOy+3b5MBhj6E36Kc=">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</latexit>

dz<latexit sha1_base64="o5GYwRACbQcjTYHfNwvhQIlldBs=">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</latexit>

= cos<latexit sha1_base64="jI/OIEDjUVoN3iFG6zwTvdrAL6c=">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</latexit> ✓ sin d⇢ + ⇢ cos ✓ cos d ⇢ sin ✓ sin d✓

= sin<latexit sha1_base64="XKmPEjD4KW5nN6Uctj0Z7Pczw18=">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</latexit> ✓ sin d⇢ + ⇢ sin ✓ cos d + ⇢ cos ✓ sin d✓

= cos<latexit sha1_base64="4rTVrOl0qboHMUV37QLn91+9xa8=">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</latexit> d⇢ ⇢ sin d + 0d✓

0

@ dx dy dz

1

A =

0

@ cos ✓ sin ⇢ cos ✓ cos ⇢ sin ✓ sin sin ✓ sin ⇢ sin ✓ cos ⇢ cos ✓ sin

cos ⇢ sin 0

1 A

0

@ d⇢

d d✓

1 A

<latexit sha1_base64="TrqPPwz/l22qB798Kcfrqs5gm+o=">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</latexit>

cos ✓ sin ⇢ cos ✓ cos ⇢ sin ✓ sin sin ✓ sin ⇢ sin ✓ cos ⇢ cos ✓ sin

cos ⇢ sin 0

<latexit sha1_base64="LG0QEDR/cpLTHUQO3cDBh62N30M=">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</latexit>

= cos ⇢2 cos2 ✓ cos sin + ⇢2 sin2 ✓ cos sin

<latexit sha1_base64="5rihRaqkpmAJXdR1VQju6fn2I98=">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</latexit>

+⇢ sin ⇢ cos2 ✓ sin2 + ⇢ sin2 ✓ sin2

(17)

cos ✓ sin ⇢ cos ✓ cos ⇢ sin ✓ sin sin ✓ sin ⇢ sin ✓ cos ⇢ cos ✓ sin

cos ⇢ sin 0

<latexit sha1_base64="LG0QEDR/cpLTHUQO3cDBh62N30M=">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</latexit>

= cos ⇢2 cos2 ✓ cos sin + ⇢2 sin2 ✓ cos sin

<latexit sha1_base64="5rihRaqkpmAJXdR1VQju6fn2I98=">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</latexit>

+⇢ sin ⇢ cos2 ✓ sin2 + ⇢ sin2 ✓ sin2

<latexit sha1_base64="RM1jTTEcnT6pNnP4mUCiaSwPVCo=">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</latexit>

= cos ⇢2 cos sin + ⇢ sin ⇢ sin2

<latexit sha1_base64="sAmesFtdZ2AB7mWPUH4dkwa9FVU=">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</latexit>

= ⇢2 sin cos2 + sin2

<latexit sha1_base64="9lqfRkdn+dWRp4nNumojxgkEGEQ=">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</latexit>

=<latexit sha1_base64="ERCVJliI/hxpsnjc5GI68Vt/L3I=">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</latexit>2 sin

dV<latexit sha1_base64="Ld8mjN3bBtfMUBhrLll8CxKM4dY=">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</latexit> = ⇢2 sin d⇢ d d✓

(18)

Exemple

Calculer le volume d’une sphère de rayon R<latexit sha1_base64="kkzp9mIddd/fGOE4QbZZP5Z9+ao=">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</latexit>

V =

Z

0

Z 2⇡

0

Z R

0

2 sin d⇢d✓d

<latexit sha1_base64="LnGrkZMq0Sgav0JxscaO4apbKPw=">AAADPHicjVJNaxRBEH2Z+BHj1xqPXgYXwdMyuwp6EYIS8JiIuwlkkmVmtpNtMjvT9PQEliW/yP/gf/AYvXowiODJs6/LjqhBtIfpfv2qXnVVdeem1I1LktOlaPnS5StXV66tXr9x89btzp21UVO3tlDDoi5ru5NnjSp1pYZOu1LtGKuyWV6q7fzohbdvHyvb6Lp67eZG7c2yw0of6CJzpMadjdGzVFdunOynRge0GBCfhM2rOLXTen+QNrpKzVSn8cQTnN1UuYwryXGnm/QSGfFF0A+gizA2685XpJigRoEWMyhUcMQlMjT8dtFHAkNuDwtylkiLXeEEq9S29FL0yMgecT7kbjewFfc+ZiPqgqeU/C2VMR5QU9PPEvvTYrG3Etmzf4u9kJg+tznXPMSakXWYkv2X7tzzf3W+FocDPJUaNGsywvjqihClla74zONfqnKMYMh5PKHdEheiPO9zLJpGave9zcT+STw96/dF8G1xFrJUOJao85/ZL+QONe1Gejkncpzllvgk+n8+gItgNOj1H/UGW4+768/D41jBPdzHQ76AJ1jHS2xiyGze4B3e40P0NvoYfY6+/HCNloLmLn4b0bfvKSi3Cg==</latexit>

=

Z 0

Z 2⇡

0

R3

3 sin d✓d

<latexit sha1_base64="JEiCjIWvweNy5Z7uoGsqckAkpbo=">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</latexit>

=

Z 0

2⇡R3

3 sin d

<latexit sha1_base64="gZiDZZjH3hxyNmvSfZpA7ddx8JU=">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</latexit>

= 2⇡R3

3 ( cos ⇡ + cos 0)

<latexit sha1_base64="EaQyC82MbPwYHtV9IvVbDMiBFDc=">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</latexit>

= 4⇡R3 3

<latexit sha1_base64="TVBz+944vHqo7NNjslU/LYIVf38=">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</latexit>

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