When calculating F dr is obtained as a result: A) - 16.884 B) - 11.959 C) 11.959 D) 16.884
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- 1-6 Sketch the vector field F by drawing a diagram as in figure 3. F(x,y)=(xy)i+xj1) Consider the conservative vector field given by: F(x, y) = (exy3 + 2e2xy, e2x + 3exy2) A potential function that generates the vector field F corresponds to: A) f(x, y) = exy + exy3 B) f(x, y) = 3exy2 +(e2x/2)+(exy4)/4 C) f(x, y) = e2xy + exy3 D) f(x, y) = exy + e2xy3 2) Consider the vector field F(x, y, z) = (y - z sinx, x, 2z + cosx). The work that performs the F field to displace a body, from point A (3π, −1, 1) to point B (π, 2, 0) corresponds approximately to: A) 28, 45 JB) 32, 42 JC) 15, 71 JD) 13, 72 J1).Evaluate ∮CF•dr, whereF=⟨e^x−y^3,cosy+x^3⟩, and C is a circle of radius 2, centered at the origin, traversed once counterclockwise. 2).Let F(x, y, z) =⟨3x^2y+az, x^3,3x+ 3z^2⟩ be a vector field. For what values of a is F conservative?
- Which of the following statements are true for all vector fields, and which are true only for conservative vector fields? (a) The line integral along a path from P to Q does not depend on which path is chosen. (b) The line integral over an oriented curve C does not depend on how C is parametrized. (c) The line integral around a closed curve is zero. (d) The line integral changes sign if the orientation is reversed. (e ) The line integral is equal to the difference of a potential function at the two endpoints. (f) The line integral is equal to the integral of the tangential component along the curve. (g) The cross partial derivatives of the components are equal. 3Let the vector field F (x, y) = (3x2y-2+ 2xy-1)i + (-2x3y-3 − x2y-2)j, be a conservative field. Which of the following scalar fields is a potential function? Note: the answer options are in image 16. Consider F = ⟨x, xy, xyz⟩. Does there exist a vector field G such that curl G = F? Why or why not?
- A=(2x-3y)ax+(2xy-y2)ay Prove the Stokes theorem for the triangular linear path in the shape that is under the effect of the vector field.Consider I = ∫CF⋅dr, where (img17) is a conservative vector field and curve C is parameterized by:α (t): = ((2 − cos (5t)) cost, (2 − cos (5t)) synt, sin (5t)) with 0≤t≤π. We have that the value of I is equal to: (img18)7. (a) Determine the value of k for which (u, v, w) is an orthogonal coordinate system ifx = −(u2 + kv2), y = uv and z = w.(b) Given that F and G are vector fields with G a vector potential of F, prove that G is notunique.(c) Show that a vector field F =(4uv − θ3/√u2 + v2, 2u2/√u2 + v2, (lnθ − 3uθ2/uv), defined in paraboloidalcoordinate system (x = uv cos θ, y = uv sin θ, z =1/2(u2 − v2) is irrotational and hence find its scalar potential.
- 6. (a) Show that a vector field F = (exsiny − yz, excos y − xz, z − xy) is irrotational and hence find its scalar field.(b) For a vector field F =(Rz, 1/R, eR − z2) in cylidrical polar coordinates, show that it is solenoidal and hence find its vector potential G = (G1, 0, G3). (PLEASE ANSWER MY OTHER THREE QUESTION)Consider the following vector field. F(x, y, z) = 9ex sin(y), 8eysin(z), 2ezsin(x) (a) Find the curl of the vector field.1) Of the following vector fields, the one that is conservative corresponds to: (See the answers in the images for question 1) 2) If f is a potential function for the vector field F = (−2y + 2xyz, −2x + x2z, x2y + 8z), then the value of f(3, −4, 1) is: A) −12B) 14C) −8D) 10