

(1 point) Consider the vector field F(x, y, z) = (2z + 3y)i + (2z + 3x)j + (2y + 2x)k. a) Find a function f such that F = Vf and f(0,0,0) = 0. f(x, y, z) = b) Suppose C is any curve from (0,0,0) to (1,1,1). Use part a) to compute the line integral / F. dr. (1 point) Verify that F = V and evaluate the line integral of F over the given path: F =...
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1 point) Use Stokes' Theorem to evaluate F dr where Fx,y,z)-(5x +y-.y +2.2z +) and C is the triangle with vertices (3,0,0), (0,3,0), and (0,0,3) oriented counterclockwise as viewed from above. Since the triangle is oriented counterclockwise as viewed from above the surface we attach to the triangle is oriented upwards curl F = |...
(1 point) Use Stokes' Theorem to evaluate / (2xyi + zj+ 3yk) dr where C is the intersection of the plane x z 8 and the cylinder x2 y9oriented counterclockwise as viewed from above. Since the ellipse is oriented counterclockwise as viewed from above the surface we attach is oriented upwards curl(2xyi+zj +3yk)- 2,0,-2x The easiest surface to attach to this curve is the interior of the cylinder that lies on the plane x + z-8. Using this surface in...
8) Find the circulation of F =(6x+5 y,4y+3z, 2x+1z) around a square of side 7, centered at (1,2,1), lying in the plane 4x+1y+6z = 12 , and oriented clockwise when viewed from the origin
8) Find the circulation of F =(6x+5 y,4y+3z, 2x+1z) around a square of side 7, centered at (1,2,1), lying in the plane 4x+1y+6z = 12 , and oriented clockwise when viewed from the origin
1 Help Entering Answers 1 point) Use Stokes' Theorem to evaluateF.dr where F(x,y,z) 6yzi 3xzj +3e k and C is the circy4,z 5 oriented counterclockwise as viewed from above Since the circle is oriented counterclockwise as viewed from above the surface we attach to the circle is oriented upwards The easiest surface to attach to this curve is the disk x2 + y2 < 4, z-5. Using this surface in Stokes' Theorem evaluate the following. F-dr = where sqrt(4-xA2) sqrt(4-x^2)...
Help Entering Answers 1 point) Verify that Stokes' Theorem is true for the vector field F that lies above the plane z1, oriented upwards. 2yzi 3yj +xk and the surface S the part of the paraboloid z 5-x2-y To verify Stokes' Theorem we will compute the expression on each side. First computecurl F dS curl F0,3+2y,-2 Edy dx curl F dS- where x2 = curl F ds- Now compute F.dr The boundary curve C of the surface S can be...
Problem 6 Using Stokes' Theorem, we equate F dr curl F dA. Find curl F- PreviousS us Problem ListNext Noting that the surface is given by (1 point) Calculate the circulation, Fdr7in z - 16-x2 - y2, find two ways, directly and using Stokes' Theorem. dA The vector field F = 6y1-6y and C is the boundary of S, the part of the surface dy dx With R giving the region in the xy-plane enclosed by the surface, this gives...
4. Use Stokes' Theorem to evaluate F dr. F(x,y,z)-(3z,4x, 2y); C is the circle x2 + y2 4 in the xy-plane with a counterclockwise orientation looking down the positive z-axis. az az F dr-JI, (curl F) n ds and VGy, 1) Hint: use ax' dy
Construct and evaluate a surface integral that represents the work done by the vector field F(x, y, z)-(x, 2z, 3y), around the triangular section of the plane 2x + y + z traversed counterclockwise. 8 in the first octant
Construct and evaluate a surface integral that represents the work done by the vector field F(x, y, z)-(x, 2z, 3y), around the triangular section of the plane 2x + y + z traversed counterclockwise. 8 in the first octant
(1 point) Let F (72+72) i + (2y +62 + 6 sin(y*)) 3+ (2x + 6y + 2e=") R. (a) Find curl F. curl F = <0,0,0> (b) What does your answer to part (a) tell you about Sc F. dr where is the circle (2 – 30)2 + (y - 35)2 = 1 in the ty-plane, oriented clockwise? SF. dr = 0 (c) If C is any closed curve, what can you say about SF. dr? SoF dr =...