In this question we use two ways to calculate an integral and in doing so, verify the Generalized Stokes's Theorem. Integrate over the oriented boundary of the manifold Min R³ consisting of those points x satisfying Part 1. First we compute = ydx / dz, ²+590≤3. 4. Since M is a cylinder in R³, it's boundary consists of three parts, and we integrate each one separately. вм For the side surface S₁ of the cylinder, we parametrize it using = 3 cos, y = 3 sinu and z = z. So P₁ = (C ◉◉ [+] (2) [L <= dudz= For the disc S₂ at the top of the cylinder, we parametrize it using dudz= P₁ = Pr оо Op Op 4= $ B₂ Dr Ou Finally for the disc Sy at the bottom of the cylinder, we parametrize it using drdu= P₁ = (0 = др др drdu= Dr' Ou +2 Part 2. Now let's compute q= q= 4== -L- Now: M M оо do. To begin we parametrize M: O O O ㅁㅁㅁ dsdud:= drdu= p = (3 cosu, 3 sinu, z), u € [0,2], = = [0,3]. p = (rcosu,rsinu, 3), [0, 2π], r € [0,3]. p= 14 p=(rcosu, rsinu, 0), [0,2], † € [0,3], q= (scosu, 8sinu, ), 8€ [0,3], µ = [0,2], z ¤ [0,3].
In this question we use two ways to calculate an integral and in doing so, verify the Generalized Stokes's Theorem. Integrate over the oriented boundary of the manifold Min R³ consisting of those points x satisfying Part 1. First we compute = ydx / dz, ²+590≤3. 4. Since M is a cylinder in R³, it's boundary consists of three parts, and we integrate each one separately. вм For the side surface S₁ of the cylinder, we parametrize it using = 3 cos, y = 3 sinu and z = z. So P₁ = (C ◉◉ [+] (2) [L <= dudz= For the disc S₂ at the top of the cylinder, we parametrize it using dudz= P₁ = Pr оо Op Op 4= $ B₂ Dr Ou Finally for the disc Sy at the bottom of the cylinder, we parametrize it using drdu= P₁ = (0 = др др drdu= Dr' Ou +2 Part 2. Now let's compute q= q= 4== -L- Now: M M оо do. To begin we parametrize M: O O O ㅁㅁㅁ dsdud:= drdu= p = (3 cosu, 3 sinu, z), u € [0,2], = = [0,3]. p = (rcosu,rsinu, 3), [0, 2π], r € [0,3]. p= 14 p=(rcosu, rsinu, 0), [0,2], † € [0,3], q= (scosu, 8sinu, ), 8€ [0,3], µ = [0,2], z ¤ [0,3].
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![In this question we use two ways to calculate an integral and in doing so, verify the Generalized Stokes's Theorem.
Integrate
over the oriented boundary of the manifold Min R³ consisting of those points x satisfying
Part 1. First we compute
= ydx / dz,
²+590≤3.
4. Since M is a cylinder in R³, it's boundary consists of three parts, and we integrate each one separately.
вм
For the side surface S₁ of the cylinder, we parametrize it using = 3 cos, y = 3 sinu and z = z. So
P₁ =
(C
◉◉
[+] (2) [L
<=
dudz=
For the disc S₂ at the top of the cylinder, we parametrize it using
dudz=
P₁ =
Pr
оо
Op Op
4=
$
B₂
Dr Ou
Finally for the disc Sy at the bottom of the cylinder, we parametrize it using
drdu=
P₁ =
(0
=
др др
drdu=
Dr' Ou
+2
Part 2. Now let's compute
q=
q=
4==
-L-
Now:
M
M
оо
do. To begin we parametrize M:
O O O
ㅁㅁㅁ
dsdud:=
drdu=
p = (3 cosu, 3 sinu, z), u € [0,2], = = [0,3].
p = (rcosu,rsinu, 3), [0, 2π], r € [0,3].
p=
14
p=(rcosu, rsinu, 0), [0,2], † € [0,3],
q= (scosu, 8sinu, ), 8€ [0,3], µ = [0,2], z ¤ [0,3].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a3bc775-5cde-4b35-b474-a2df7a4a99fb%2Fb0f55ad0-cc7e-4e4a-9d38-275b8cecfe30%2Fyv3l32o_processed.png&w=3840&q=75)
Transcribed Image Text:In this question we use two ways to calculate an integral and in doing so, verify the Generalized Stokes's Theorem.
Integrate
over the oriented boundary of the manifold Min R³ consisting of those points x satisfying
Part 1. First we compute
= ydx / dz,
²+590≤3.
4. Since M is a cylinder in R³, it's boundary consists of three parts, and we integrate each one separately.
вм
For the side surface S₁ of the cylinder, we parametrize it using = 3 cos, y = 3 sinu and z = z. So
P₁ =
(C
◉◉
[+] (2) [L
<=
dudz=
For the disc S₂ at the top of the cylinder, we parametrize it using
dudz=
P₁ =
Pr
оо
Op Op
4=
$
B₂
Dr Ou
Finally for the disc Sy at the bottom of the cylinder, we parametrize it using
drdu=
P₁ =
(0
=
др др
drdu=
Dr' Ou
+2
Part 2. Now let's compute
q=
q=
4==
-L-
Now:
M
M
оо
do. To begin we parametrize M:
O O O
ㅁㅁㅁ
dsdud:=
drdu=
p = (3 cosu, 3 sinu, z), u € [0,2], = = [0,3].
p = (rcosu,rsinu, 3), [0, 2π], r € [0,3].
p=
14
p=(rcosu, rsinu, 0), [0,2], † € [0,3],
q= (scosu, 8sinu, ), 8€ [0,3], µ = [0,2], z ¤ [0,3].
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