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F(x,y,z)= (y² +e",2xy +z sin y, -cos y) is a gradient vector field. Compute Sc 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)
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 Consider the vector field F- Mi+ Nj Pk defined by: F- ysinzi+sinjry cos z k. Compute the line integral ScF dr over a unit circle. Compute the line integral ysin z dr+ r sin z dy + ry cos zdz (0,0,0) #3 Use Green's Theorem to evaluate the line integral along the given positively orientated curve C. e2*t d e" dy, C is the triangle with vertices (0,0), (1,0), and (1,1) #3 Consider the vector field F- Mi+ Nj...
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
Show that vector field F(x,y) = 2x cos yi + (1 - zsiny) is a gradient field and then find the function f(x,y) such that F = VS. Use it to evaluate line integral SF. dr where the curve C is the arc of the circle 12 + y2 = 4 from (2,0) to (0,2)
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.
Evaluate the line integral f F dr for the vector field F(x, y, z) curve C parametrised by Vf (x, y, z) along the with tE [0, 2 r() -(Vt sin(2πt), t cos (2πi), ?) , Evaluate the line integral f F dr for the vector field F(x, y, z) curve C parametrised by Vf (x, y, z) along the with tE [0, 2 r() -(Vt sin(2πt), t cos (2πi), ?) ,
Consider the vector field F (x, y, z) = <y?, z2, x?>. Compute the curl (F). Use Stokes' Theorem to evaluate S. F. dr where C is the triangle (0,0,0), (1,0,0), and (0, 1, 1) oriented counter-clockwise when viewed from above.
Problem 9. (20 points) Let F be the vector field F(x, y, z) = (ey, xey + e*, ye*). (a) (5 points) Compute V F(x, y, z). (b) (10 points) Find a potential function for F or explain why none exists. (c) (5 points) Find ScF. dr, where C is the curve consisting of the line segments from (0,0,0) to (1,2,0), from (1,2,0) to (1,2, 1), and from (1, 2, 1) to (1,2,2).
3. 8p] Show that the force field F(x,y, z) sin y, x cos y + cos z, -y sin z) is conservative and use this fact to evaluate the work done by F in moving a particle with unit mass along the curve C with parametrization r(t (sin t, t, 2t), 0 <t<T/2. 4. 8p] A thin wire has the shape of a helix x = sin t, 0 < t < 27r. If the t, y = cos t,...