Let F(XYZ) = <2y27, 4xyz, 2xy2> be a vector field. (a) Knowing that F is conservative,...
Let F(x,y,z) = <2y2z, 4xyz, 2xy2> be a vector field. (a) Knowing that F is conservative, find a function f such that F = Vf and f(1,2,1)= 8. (b) Using the result of part(a), evaluate the line integral of F along the following curve C from (0, 0, 0) to (3.9, 1.4, 2.6). y2 + x4z3 + 2xy(x3 + y4 + 24)1/3 = K ; K is a constant Answer: Next page
Please make it simple and clear to understand 3. A vector field is given by (a) Show that the vector field r is conservative. Then find a scalar potential function f(r,y,) such that r - gradf and f(0,0,0) 0 (b) By the result of (a) the following line integral is path independent. Using the scalar potential obtained in (a) evaluate the integral from (0,0,2) (where-y-0) to (4,2,3) (where -1,y 0,2) 4.2,3) J(0,0,2) 3. A vector field is given by (a)...
Problem #7: Let R = r \ {(0,0,0)) and F is a vector field defined on R satisfying curl(F) = 0. Which of the following statements are correct? [2 marks] (1) All vector fields on R are conservative. (ii) All vector fields on Rare not conservative. (iii) There exists a differentiable function / such that F - Vf. (iv) The line integral of Falong any path which goes from (1,1,1) to (-2,3,-5) and does not pass through the origin, yields...
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 +...
1. (20 points) Identify if the following vector fields are conservative. If there exists a vector field that is conservative, you must also find a potential function for that field. (a) F(x,y,z) = (x3 – xy +z)i + 2 (b) F(x,y,z) = (y+z)i + (x+z)j + (x+y)k (& +y +y-22) i + (- y2)k
8 points each 1. F is a conservative vector field. Evaluate ScF. dr where F =< 2xy3-4, 3x2y224, 4x^y323 > and C is the curve beginning at (3, 0, 5) and ending at (3, 2, -1)
Let F = (2,1,1) be a vector field in Rº a.) Show that F is a conservative vector field. b.) Find the potential function of F. In other words, find a scalar function f(x, y, z) such that ✓ f = 7. Please show all steps. c.) Let C be any smooth curve starting at (1,1,1) and ending at (e, e, 1). Compute (Fdi. С
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = Vf. (If the vector field is not conservative, enter DNE.) F(x, y, z) = 4xyi + (2x2 + 10yz)j + 5y2k f(x, y, z) =
7. Is F(x, y) =< 2x, 6y? > a conservative vector field?
(1 point) Determine whether the vector field is conservative and, if so, find the general potential function. F = (cos z, 2y!}, -x sin z) Q= +c Note: if the vector field is not conservative, write "DNE". (1 point) Show F(x, y) = (8xy + 4)i + (12x+y2 + 2e2y)j is conservative by finding a potential function f for F, and use f to compute SF F. dr, where is the curve given by r(t) = (2 sinº 1)i +...