3. Let (a) Show that F is conservative in R3. (b) Let T denote the triangular path with vertices ...
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),...
9. (10pts) Answer true or false: (a) The domain of f(x,y) = In(1-z?-уг) + Vi-z?-уг is the unit ball {(z, y): x2 + y2 1} . (b) The direction of the maximum rate of increase of g(x, y, 2yz at the point (1,1,1) is 2,1,1 (c) For F2y,2r3y1>, F-dr is independent of path in the plane. (d) × (▽ . F) makes sense. (e) ▽f.dr =4 where f(x, y, z) = zyz and C is the line segment starting at...
(1) Integrate f(x, y,z)+Vy - z2 over the straight line segment path from (0,0,0) to (1,1,1) (2) Consider the field F (2xyz+2,x2z, x2y), (a) (b) (c) Show that the field is conservative. Find a potential function for the field. Find the work the field does on an object that follows the path consisting of the line segment from (0,0,0) to (1,2,2), followed by the line segment from (1,2,2) to (2,4,3) Find the work done by the field ß-(x, 3y,-5z) along...
Let S CR be the tetrahedron having vertices (0,0,0), (0,1,1), (1,2,3), and (-1,0,1). Let f: R3 → R be the function defined by f(x, y, z) = 1 - 2y + 32. Using the change of variables theorem, rewrite Ss f as an integral over a 3-rectangle, then use Fubini's theorem to evaluate the integral (8 points).
1. Let F-yi(xr +6g) j + 2z k and (a) Which of these two fields (if any) are conservative on R3? Give detailed (b) Find potential functions for the fields that are conservative (c) Calculate the line integrals F dr and G dr where C is the arc of the reasoning r2 4 curve formed by the intersection of the plane z = 4 and the surface--+92 in the first octant, oriented anti-clockwise when view from above. 1. Let F-yi(xr...
Let F-_y i + (z + 6y) j+2z k and 1. (a) Which of these two fields (if any) are conservative on R3? Give detailed reasoning. (b) Find potential functions for the fields that are conservative (c) Calculate the line integralsF dr and G dr where C is the arc of the curve formed by the intersection of the plane4 and the surface+ in the first octant, oriented anti-clockwise when view from above. Let F-_y i + (z + 6y)...
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 +...
q4 please thanks (1) Let A - (0,0), B- (1,1) and consider the veetor field f(r, y,z)vi+aj. Evaluate the line integral J f.dr )along the parabola y from A to B and (i)along the straight line from A to B. Is the vector field f conservative? (2) For the vector feld f # 22(r1+ gd) + (x2 + y2)k use the definition of line integral to (3) You are given that the vector field f in Q2 is conservative. Find...
QB(27pts)(a). Evaluate the circulation ofF(xy)-<x,y+x> on the curve r(t)=<2cost, 2sinp, foross2n (b) Evaluate J F.dr, where C is a piecewise smooth path from (1,0) to (2,1) and F- (e'cos x)i +(e'sinx)j [Hint: Test F for conservative (c). Use green theorem to express the line integral as a double integral and then evaluate. where C is the circle x+y-4 with counterclockwise orientation. (d(Bonus10 pts) Consider the vector field Foxyz) a. Find curl F y, ,z> F.dr where C is the curve...
Consider the vector field F(x, ) (4x3y -6ry3,2rdy - 9x2y +5y*) along the curve C given by r(t)(tsin(rt), 2t +cos(xl)), -2ss 0 To show that F is conservative we need to check a) b) We wish to find a potential for F. Let r,y be that potential, then Use the first component of F to find an expression for ф(x, y)-Po(x,y) + g(y), where ф(x,y) in the form: Differentiate ф(x,y) with respect to y and determine g(y) e Using the...