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1. Let F(x, y) = (1 — y, ½x³ +e³²), and let C with the position vector r be the portion of the parabola y = x² – 1 from (−1,0) to (1,0). Complete the following steps to determine the value of SF · dr. (a) Make a rough sketch of the curve C. Note that C is not closed, so we cannot apply Green's Theorem to calculate F. dr. Identify another curve C₂ that will make the combined curve Ỡ = CỤC₂ (i.e. C together with C₂) closed and positively oriented. Be sure to specify the orientation of C₂. (b) Use Green's Theorem to calculate ſ♂ F · dr. (c) Calculate fc, F. dr directly. (d) Now use the fact that ſ♂ F · . dr = ScF·dr + Sc₂ F·dr and your answers to the previous parts to calculate fF.dr.

1. Let F(x, y) = (1 — y, ½x³ +e³²), and let C with the position vector r be the portion of the parabola y = x² – 1 from (−1,0) to (1,0). Complete the following steps to determine the value of SF · dr. (a) Make a rough sketch of the curve C. Note that C is not closed, so we cannot apply Green's Theorem to calculate F. dr. Identify another curve C₂ that will make the combined curve Ỡ = CỤC₂ (i.e. C together with C₂) closed and positively oriented. Be sure to specify the orientation of C₂. (b) Use Green's Theorem to calculate ſ♂ F · dr. (c) Calculate fc, F. dr directly. (d) Now use the fact that ſ♂ F · . dr = ScF·dr + Sc₂ F·dr and your answers to the previous parts to calculate fF.dr.

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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1. Let F(x, y) = (1 − y, x³ +e³²), and let C with the position vector r be the portion of the parabola y = x² – 1 from (−1,0) to (1,0). Complete the following steps to determine the value of F · dr. (a) Make a rough sketch of the curve C. Note that C is not closed, so we cannot apply Green's Theorem to calculate F. dr. Identify another curve C₂ that will make the combined curve Ĉ = CỤC₂ (i.e. C together with C₂) closed and positively oriented. Be sure to specify the orientation of C₂. (b) Use Green's Theorem to calculate fF.dr. (c) Calculate ₂ F. dr directly. (d) Now use the fact that fF.dr = F·dr + Sc₂ F. dr and your answers to the previous parts to calculate fF.dr.
Transcribed Image Text:1. Let F(x, y) = (1 − y, x³ +e³²), and let C with the position vector r be the portion of the parabola y = x² – 1 from (−1,0) to (1,0). Complete the following steps to determine the value of F · dr. (a) Make a rough sketch of the curve C. Note that C is not closed, so we cannot apply Green's Theorem to calculate F. dr. Identify another curve C₂ that will make the combined curve Ĉ = CỤC₂ (i.e. C together with C₂) closed and positively oriented. Be sure to specify the orientation of C₂. (b) Use Green's Theorem to calculate fF.dr. (c) Calculate ₂ F. dr directly. (d) Now use the fact that fF.dr = F·dr + Sc₂ F. dr and your answers to the previous parts to calculate fF.dr.
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