Verify Stokes' theorem for the helicoid (r, 0) = (r cos 0, r sin 0, 20) oriented upwards, where 0 ≤r≤ 1,0 ≤ 0≤, and F is the vector field F = (4z, 3x², 5y). First, compute the surface integral: $= [[(cu (curl F) n dS -110 $1 = Compare that computation with the line integral on the boundary of V. From the picture, notice that boundary consists of 4 curves. Parametrize each curve by restricting the domain of to an appropriate subset. C₁ Straight line with = 0 = √₁. F. dr [ dr de dr
Verify Stokes' theorem for the helicoid (r, 0) = (r cos 0, r sin 0, 20) oriented upwards, where 0 ≤r≤ 1,0 ≤ 0≤, and F is the vector field F = (4z, 3x², 5y). First, compute the surface integral: $= [[(cu (curl F) n dS -110 $1 = Compare that computation with the line integral on the boundary of V. From the picture, notice that boundary consists of 4 curves. Parametrize each curve by restricting the domain of to an appropriate subset. C₁ Straight line with = 0 = √₁. F. dr [ dr de dr
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please help with this!
![1.0
0.0
0.5
0.0
0.0
Φ =
$1
05
Verify Stokes' theorem for the helicoid (r, 0) = (r cos 0, r sin 0,20 oriented upwards, where 0 < r ≤ 1,0 ≤ 0 ≤<, and F is the vector field F = (4z, 3x², 5y).
First, compute the surface integral:
0.5
=
// (CU
=
1.0
=JJ
1.0
(curl F). n dS
Compare that computation with the line integral on the boundary of V. From the picture, notice that boundary consists of 4 curves. Parametrize each curve by restricting the domain of to an appropriate
subset.
C₁ Straight line with = 0
dr de
[10
[ F. dr ]
C₁
dr](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e87940a-4964-4b75-8957-b270f7cb8a14%2F3ab62234-ba6d-4199-87f6-1d8ea10e2aaf%2F6ttrea_processed.png&w=3840&q=75)
Transcribed Image Text:1.0
0.0
0.5
0.0
0.0
Φ =
$1
05
Verify Stokes' theorem for the helicoid (r, 0) = (r cos 0, r sin 0,20 oriented upwards, where 0 < r ≤ 1,0 ≤ 0 ≤<, and F is the vector field F = (4z, 3x², 5y).
First, compute the surface integral:
0.5
=
// (CU
=
1.0
=JJ
1.0
(curl F). n dS
Compare that computation with the line integral on the boundary of V. From the picture, notice that boundary consists of 4 curves. Parametrize each curve by restricting the domain of to an appropriate
subset.
C₁ Straight line with = 0
dr de
[10
[ F. dr ]
C₁
dr
Transcribed Image Text:C2 Straight line with =
Φ2
=
Ja
C₂
$3
F. dr =
C3 Straight line with r = 0
=
T
2
»-[x-+-=[D®
F. dr =
C3
C4 Arc with r = 1
dr
do
«-[x«-[D•
ΦΑ = F. dr =
C4
do
Check that the sum of these integrals agrees with your answer from Stokes' theorem.
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