2. a) Find a potential of the vector field f(x, y) = (a2 +2xy - y2,...
The vector field + (x,y) = (ycos (xy) + 2e*, xcos(xy) + 2ycos (y2)) is conservative. Find f(x,y) such that F = Of. f=sin(x?y) + ye* + 2ys in (y? a. f = 2sin(xy) +2e* + sin(y? O b. 2 f=sin(xy) +2e* + sin(y?) OC f=sin(x+y) + 2ye* + sin(y) O d. sin (2) f=sin(x) + 2ye* + e. 2
(a) Given the vector field F = (0,22 + 2xy) = ui + (x2 + 2xy)j Find u for 7 to be conservative and find the potential, if it exists (b) Given u= (e? – zły, xy + y) = (e– r’y)i + (xy + y); Evaluate I= dos u dr where is the circle with radius r = 1 and center at the origin.
F(x, y,z)=(y2 +e", 2xy + z sin y, cos y) is a gradient vector field. Compute Sc F. dr where C=GUC,, C işthe curve y = x^, z = 0 from (0,0,0) to (1,1,0) and C, is the straight line from (1,1,0) to (2,2,3)
3. F =< y2 + x-3, 2xy + e? -y + 2, ye? +2z -4> (1) Prove or disprove that F is conservative. (ii) If F is conservative find the potential function f.
D Question 11 12 pts to Consider the vector field F (x, y, z) =< 2x – yz, 2y – az,2z – xy>. a) (3) Is this vector field conservative? Justify your answer. b) (9) Find the amount of work done by this vector field in moving a particle along the curve (t) =< 3cost, cos’t, cos” (2t) > from t = 0 tot = 1
(6) Fundamental Theorem of Line Integrals F = <M,N> = < 2xy, x² + y2 > (6a) Show that F is a Conservative Vector Field. (6b) Find the Potential Function f(x,y) for the Vector Field F. (60) Evaluate W = | Mdx + Ndy from (5,0) to (0,4) over the path C: È + K3 = 1 с
F(x, y,z) = (y2 +e", 2xy + z sin y, cos y) is a gradient vector field. Compute Sc F. dr where C=GUC,, C işthe curve y = x^, z = 0 from (0,0,0) to (1,1,0) and C, is the straight line from (1,1,0) to (2,2,3).
(1 point) Determine whether the vector field is conservative and, if so, find the general potential function. F = (cos z, 2y!}, -x sin z) Q= +c Note: if the vector field is not conservative, write "DNE". (1 point) Show F(x, y) = (8xy + 4)i + (12x+y2 + 2e2y)j is conservative by finding a potential function f for F, and use f to compute SF F. dr, where is the curve given by r(t) = (2 sinº 1)i +...
Q6 [10+1+3=14 Marks] Let F be a force field given by F(x, y) = y2 sin(xy?) i + 2xy sin(xy?)j. (a) Show that F. dr is exact by finding a potential function f. (b) Is I = S, y2 sin(xy2) dx + 2xy sin(xy?) dy independent of path C? Justify your answer. (c) Use I to find the work done by the force field F that moves a body along any curve from (0,0) to (5,1).
5. Let F (y”, 2xy + €35, 3yes-). Find the curl V F. Is the vector field F conservative? If so, find a potential function, and use the Fundamental Theorem of Line Integrals (FTLI) to evaluate the vector line integral ScF. dr along any path from (0,0,0) to (1,1,1). 6. Compute the Curl x F = Q. - P, of the vector field F = (x4, xy), and use Green's theorem to evaluate the circulation (flow, work) $ex* dx +...