Problem 9. (20 points) Let F be the vector field F(x, y, z) = (ey, xey...
Consider the vector field F(x, y) = (ey – ysin x + 2x, xey + cos x) (a) (4pts) Compute curl F. (b) (2pts) Is F conservative? Clearly indicate yes or no. (c) (8pts) Suppose C is the curve parameterized by r(t) = (t3 + 1, t– 2t) 0<t< 2 Compute ( F. dr.
Problem 4. (15 points each) Let F(x, y, z) = (0, x, y) G(x, y, z) = (2x, z, y) + (x, y, z) = (3y, 2x, z). (a) For each field, either find a scalar potential function or prove that none exists. (b) For each field, either find a vector potential function or prove that none exists. (c) Let F(t) = (2, 2t, t2). For which of these vector fields is ñ a flow line? Justify your answer.
Let F(x, y, z) be the gradient vector field of f(x, y, z) = exyz , let C be the curve of the intersection of the plane y + z = 2 and the cylinder x2 + y2 = 1, oriented counterclockwise, evaluate Sc F. dr. OT O -TT O None of the above. 00
(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...
F(x,y,z)= (y² +e",2xy +z sin y, -cos y) is a gradient vector field. Compute Sc F. dr where C=C UC2, C, is the 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).
Let F(x, y, z) = sin yi + (x cos y + cos z)j – ysin zk be a vector field in R3. (a) Verify that F is a conservative vector field. (b) Find a potential function f such that F = Vf. (C) Use the fundamental theorem of line integrals to evaluate ScF. dr along the curve C: r(t) = sin ti + tj + 2tk, 0 < t < A/2.
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) 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 =...
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(x, y, z) = ( 2xy tooz, xooz, ey sing) is a vector field in R. Find a scalar function f such that F=of. of this function doesn't exist, give reasons.