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

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

Advanced Engineering Mathematics
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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(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
Transcribed Image Text:(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.
Transcribed Image Text: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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