3. (14 points) Let F = (xy, y’, yz) and let C be the line segment...
Consider F and C below. F(x, y, z) = yz i + xz j + (xy + 10z) k C is the line segment from (3, 0, -3) to (4, 4, 1) (a) Find a function f such that F = Vf. f(x, y, z) = (b) Use part (a) to evaluate [s vf. dr along the given curve C.
F.df, where F(x, y, z) = (yz, uz, xy), and C is ANY smooth path from (0,0,0) to 11. a) Evaluate (2, -1, -2). b) If a particle sat at (0,0,0), give a possible physical interpretation of the line integral you com puted.
2. (a) Let i. Show that F is cnservative in R i. Let C denote the path 1+cost,2+sint,3), 0StS 4 Evaluate F. dr Justify your answer. iii. Find a function y: R3-+ R such that F iv. Evaluate F.dr where「is the path y =r', z = 0, from (0.0.0) to (2.8.0) followed by the line segment from (2,8,0) to (1,1,2) 22 marks)
2. (a) Let i. Show that F is cnservative in R i. Let C denote the path 1+cost,2+sint,3),...
Question Help LetF=v(xy?) and let C be the path in the xy-plane from (-44) to (4.4) that consists of the line segment from (-44) to (0,0) followed by the line segment from (0,0) to (4.4). Evaluate SA Fodrin two ways a) Find parametrizations for the segments triat make up C C-v*(4-440stst C12(0) = (4) Costs С a) Find parametrizations for the segments that make up C and evaluate the integral b) Use 1(xy] =x?y? as a potential function for F
Use to fundamental theorem of line integrals to evaluate F dr for 6. F(xy) = (2xy,x2 -y) over the path C from the point (2, 0) to (0, 2)
Use to fundamental theorem of line integrals to evaluate F dr for 6. F(xy) = (2xy,x2 -y) over the path C from the point (2, 0) to (0, 2)
Please help solve the following question with steps. Thank
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6. Compute JF . T ds where F (-y,z) and (a) C is the line segment from (1,0) to (0,0) followed by the line segment from (0,0) to (0, 1) (b) C is the line segment from (1,0) to (0, 1) (c) C is the part of the unit circle in the first quadrant, moving from
6. Compute JF . T ds where F (-y,z) and (a) C is the...
Evaluate ScF. dr where F(x, y) = xy?i + xyºj and C is the polygonal path from (0,0) to (1,0) to (0,1) to (0,0) Select one: O a. 30 1 O b. 35 c. 110 O d. - 3
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).
→ (1 point) Let Vf-6xe-r sin(5y) +1 5e* cos(Sy) j. Find the change inf between (0,0) and (1, n/2) in two ways. (a) First, find the change by computing the line integral c Vf di, where C is a curve connecting (0,0) and (1, π/2) The simplest curve is the line segment joining these points. Parameterize it: with 0 t 1, K) = dt Note that this isn't a very pleasant integral to evaluate by hand (though we could easily...
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 +...