Please show steps for rate! 6) F(x, y) = (2xy² +2x)i+(2x²y+2y)j is a conservative vector field with potentialfunction f (x, y) = x² +x²y² + y². Find [F.dr where C is the curve that travelsalong the ellipse9=1 from (0,2) to (3,0) then along the line segments from4(3,0) to (4,1) followed by (4,1) to (5,0).

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Answered: F(x, y)=(2xy² +2x)i+(2x*y+2y)j is a…

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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Let F(x, y) = ⟨xy^2+2x+1, x^2y − 3y⟩ be a vector field. Let C be the curve thatconsists of two line segments C1 and C2, where C1 has vertices (−1, 0) and (0, 1)and C2 has vertices (0, 1) and (1, 0). C is oriented clockwise.(a) Sketch C accurately.(b) Find the work done by F by moving a particle along the path C in the givendirection.
We are given a vector field and a parametric curve
Sketch the vector field F(x, y) = x3, the line segment from (1, 3) to (9, 3), and the line segment from (3,5) to (3,9). (a) Calculate the line integral of the vector field F along the line segment from (1, 3) to (9, 3). (b) Calculate the line integral of the vector field F along the line segment from (3, 5) to (3, 9).
Find the curl of the vector field F(x,y,z)=x^2yi-2y^3zj-3zk. A. curl F = 2y3i + x2k B. curl F = -2y3i - x2k C. curl F = 2y3i - x2k D. curl F = -2y3i + k
f(x,y) = 2x3+4y3-3x2-120x-192y+5 Is f(x,y) a curve, vector field, or surface?
Suppose C is the curve from (0, 0) to (2, 0) to (2, 3) to (0, 3) to (0, 0). Find the work done by the vector fieldF(x, y) = <x^(3)−2y^(2), x + cos(√y)> on a particle moving along C.
Consider vector field F(x, y) = (axy - y³)i + (4y + 3x² + bxy²)j. (i) Determine the value of constant a and b so F would be conservative vector field. (ii) Determine its potential function. (iii) Then, find the work done by F to move one particle from (1,0) to point (0,1).
The gradient vector field of f(x,y)=y(2x2 -y3 ) is given by: O1. (2xy)i +(x2 -3y² )i O II. (4xy)i -(2x2 -3y? )i O II (4xy)i +(4x2 -3y² )i OV. (4xy)i +(2x2 -3y² )i
4. Given the 2-D vector field : G(x,y) = (2y)î + (x)ĵ and the triangular curve T : line segments from point A(0, 0) to point B(4,0) to point D(4,2) and back to point A(0,0). 1 (а) Sketch the curve, and the vector field at the 9 points making a square from (-2,2) to (2, –2). -x- -3: -2 -1 1 2 3 -1 -2 -3 -y Compute W = $. G· dr parametrically. (Note : You will need to set up and evaluate three line integrals.) (b) (c) Apply Green's Theorem to compute W = f, G· dr.
2. For the vector field F (x, y, z) = (x²y, xyz, −x²y²) F(x, y, z)=(x2y,xyz,-x2 y 2 ), find: (a) div (F)div (F) (b) curl (F)curl(F) (c) div (curl (F))
Compute the vector assigned to the pointP = (−3, 5) by the vectorfield:(a) F(x, y) = (xy,y − x)(b) F(x, y) = (4, 8)(c) F(x, y) = 3x+y , log2(x + y)
Sketch the vector field F(x, y)=2i-j and calculate the line integral of Palong the line segment from (-1,-5) to (-3, 4). -19

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F(x, y)=(2xy² +2x)i+(2x*y+2y)j is a conservative vector field with potential function f(x, y)=x²+x°y° +y°. Find (F.dr where C is the curve that travels x? along the ellipse 9 y² =1 from (0,2) to (3,0) then along the line segments from 4 (3,0) to (4,1) followed by (4,1) to (5,0).