Σ96596LLN6)H.4.fs7.8.6.144f12where C is the line segment from (1,0) to (-1,3).ロis a conservative vector field. Then find a function f such that F = V f, and use it to evaluate the integralGusF drShow thatF(x, y) = (ye + sin y) î + (e + x cos y) ĵQuestion 5%3DIf the linear density is a constant k, find the mass and center of mass of the wire.

Answered: Image /qna-images/answer/36fed810-f02f-4772-8fc5-dbfb08fa0a36.jpg

Answered: Σ 96 5 96 LL N 6) H. 4. fs 7. 8. 6. 144…

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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Consider the vector field F(x, y) =⟨2xy3,1 + 3x2y2⟩inR2. (a) Show thatFis conservative. (b) Find a functionfsuch that∇f=F. (c) Evaluate the line integral∫CF•drwhereCis given byr(t) =⟨t+ 1,2−t2⟩when0≤t≤2.
1. Consider the vector field F(x, y) =⟨2xy3,1 + 3x2y2⟩ in R2. (a) Show that F is conservative. (b) Find a function f such that ∇f=F. (c) Evaluate the line integral ∫CF•dr where C is given by r(t) =⟨t+ 1,2−t2⟩ when 0≤t≤2.
Consider I = ∫CF⋅dr, where (img17) is a conservative vector field and curve C is parameterized by:α (t): = ((2 − cos (5t)) cost, (2 − cos (5t)) synt, sin (5t)) with 0≤t≤π. We have that the value of I is equal to: (img18)
1).Evaluate ∮CF•dr, whereF=⟨e^x−y^3,cosy+x^3⟩, and C is a circle of radius 2, centered at the origin, traversed once counterclockwise. 2).Let F(x, y, z) =⟨3x^2y+az, x^3,3x+ 3z^2⟩ be a vector field. For what values of a is F conservative?
Consider the vector field F=(x^2+y^2,10xy). Compute the line integrals ∫c_1 F⋅dr and ∫c_2 F⋅dr, where c_1 (t)=(t, t^2) and c_2 (t)=(t, t) for 0≤t≤1. Can you decide from your answers whether or not F is a gradient vector field? Why or why not?
1) Consider the conservative vector field given by: F(x, y) = (exy3 + 2e2xy, e2x + 3exy2) A potential function that generates the vector field F corresponds to: A) f(x, y) = exy + exy3 B) f(x, y) = 3exy2 +(e2x/2)+(exy4)/4 C) f(x, y) = e2xy + exy3 D) f(x, y) = exy + e2xy3 2) Consider the vector field F(x, y, z) = (y - z sinx, x, 2z + cosx). The work that performs the F field to displace a body, from point A (3π, −1, 1) to point B (π, 2, 0) corresponds approximately to: A) 28, 45 JB) 32, 42 JC) 15, 71 JD) 13, 72 J
A vector field is given by F = ((1 + xy)exy, x2exy). Compute the integral of the vector field over the curve r(t) = (cos(t), 2sin(t)), where t ranges from zero to π/2.

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