(rcos 0,‚r sin 0, 4)oriented upwards, where 0 C4 Arc with r = 1F. dr =do%3DCheck that the sum of these integrals agrees with your answer from Stokes' theorem.

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Answered: Verify Stokes' theorem for the helicoid…

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Verify Stokes' theorem for the helicoid Yr. Ø) = rsin 0, oriented upwards, where 0

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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Question

(rcos 0,
‚r sin 0, 4)
oriented upwards, where 0 <r <1 ,0<0<5
, and F is the vector field F = (4z, 5x2, 8y)
Verify Stokes' theorem for the helicoid Y(r, 0) =
First, compute the surface integral:
(curl F) · n dS
dr de
Compare that computation with the line integral on the boundary of Y. From the picture, notice that boundary consists of 4 curves. Parametrize each curve by restricting the domain of Y to an
appropriate subset.
Ci Straight line with 0 = 0
01 =
F. dr
dr
C2 Straight line with 0 = 4
02 =
F. dr =
dr
C3 Straight line with r = 0
03 =
F. dr =
de

C4 Arc with r = 1
F. dr =
do
%3D
Check that the sum of these integrals agrees with your answer from Stokes' theorem.

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