Verify Stokes' theorem for the helicoid (r, 0) = (r cos 0, r sin 0, 2) oriented upwards, where 0 < r ≤ 1,0 ≤ 0 ≤, and F is the vector field F = (8z, 7z², 8y). First, compute the surface integral: = ff (₁₁ =14/3 $1= 1 (curlF)-n dS plz 1 0 -J₁² $2= 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 F-dr ¹ = /F·dr = Sa $4= 0 C₂ Straight line with = 0 1 2= √₁²² 0 F-dr = C3 Straight line with r = 0 C4 Arc with r = 1 = 14/3 F-dr = 1 0 pi/ 16/pi'sintheta- 16/pi costheta + 14r^2costheta 0 pi/ 0 0 0 0 dr dr de dr de de Check that the sum of these integrals agrees with your answer from Stokes' theorem.
Verify Stokes' theorem for the helicoid (r, 0) = (r cos 0, r sin 0, 2) oriented upwards, where 0 < r ≤ 1,0 ≤ 0 ≤, and F is the vector field F = (8z, 7z², 8y). First, compute the surface integral: = ff (₁₁ =14/3 $1= 1 (curlF)-n dS plz 1 0 -J₁² $2= 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 F-dr ¹ = /F·dr = Sa $4= 0 C₂ Straight line with = 0 1 2= √₁²² 0 F-dr = C3 Straight line with r = 0 C4 Arc with r = 1 = 14/3 F-dr = 1 0 pi/ 16/pi'sintheta- 16/pi costheta + 14r^2costheta 0 pi/ 0 0 0 0 dr dr de dr de de Check that the sum of these integrals agrees with your answer from Stokes' theorem.
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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Solve correctly all parts
Transcribed Image Text:Verify Stokes' theorem for the helicoid V (r, 6) = (r cos , r sin 0,20) oriented upwards, where 0 ≤ r ≤ 1,0 ≤ 0 ≤ , and F is the vector field F = (8z, 7x², 8y).
6
First, compute the surface integral:
$=
ff (₁₁
(curlF). n dS
pi/ 1
= 14/3
$₁ =
0
=√₁.
$2=
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
E
0
F.dr
0
0
C₂ Straight line with 0 = 1/
F.dr =
JC₂
0
0
C4 Arc with r = 1
1
14/3
0
C3 Straight line with r = 0
-----P
$3= F.dr =
pi/
0
16/pi*sintheta- 16/pi costheta + 14r^2costheta
pi/
----
F.dr =
=
0
0
0
0
dr
dr
do
dr do
do
Check that the sum of these integrals agrees with your answer from Stokes' theorem.
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Step 1: Note the consequence of Stokes' Theorem
VIEWStep 2: Evaluate Curl
VIEWStep 3: Evaluate Curl integral
VIEWStep 4: Evaluate line integral along C1
VIEWStep 5: Evaluate line integral along C2
VIEWStep 6: Evaluate line integral along C3
VIEWStep 7: Evaluate line integral along C4
VIEWStep 8: Verify Stokes' Theorem
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