x=J_y=K z=L Suppose G(J, K,L)= < 2 cos L, 6KL + eK,3K² - 2J sin L > • Then show: Gis conservative; find a potential function for G. By the Fundamental Theorem for Line Integrals: Then, evaluate TT Tr SdP such that A is path defined by P (r) = < −5r + 4, 2rer−¹, COS A 3 > for r = [0,1]

x=J_y=K z=L Suppose G(J, K,L)= < 2 cos L, 6KL + eK,3K² - 2J sin L > • Then show: Gis conservative; find a potential function for G. By the Fundamental Theorem for Line Integrals: Then, evaluate TT Tr SdP such that A is path defined by P (r) = < −5r + 4, 2rer−¹, COS A 3 > for r = [0,1]

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

x=J_y=K_z=L
Suppose G(J, K,L)= < 2 cos L, 6KL + eK,3K² – 2J sin L >
Then show: Gis conservative; find a potential function for G.
By the Fundamental Theorem for Line Integrals: Then, evaluate
SdP such that A is path defined by P (r) = < −5r + 4, 2rer−1,
COS
A
πr
> for r = [0,1]

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Follow-up Questions

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Follow-up Question

Please show integration process. Is this the correct combination?

Equating corresponding components,
fx = 2 cos z
fy = 6yz + ey
f₂ = 3y² - 2x sin z
Integrating, we have,
f = 2x cos z + C₁ (y, z)
f = 3y²z+ e + C₂ (x, z)
f = 3y²z+ 2x cos z + C3 (x, y)
Combining all these equations,
f (x, y, z) = 2x cos z + 3y²z + ey

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