Is the below statement True or False? The vector field F(x,y) =<xy?, x?y) is conservative. True False
7. The vector field F =< 3x2z In y + ze+2 +20, - 3y?, x° In y + ce2 +423 > is conservative. Find a potential function f(x, y, z) such that F=Vf. Y
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
D Question 11 12 pts to Consider the vector field F (x, y, z) =< 2x – yz, 2y – az,2z – xy>. a) (3) Is this vector field conservative? Justify your answer. b) (9) Find the amount of work done by this vector field in moving a particle along the curve (t) =< 3cost, cos’t, cos” (2t) > from t = 0 tot = 1
Let Ě =< 2x + 2,3y+z, 6x + 6y > be a vector field in R3. Evaluate the following surface integral directly: xdA || i-dš= $ 8. (XFL) S Where S is the part of the plane 2x + 3y + z = 6 in the first octant (with upward orientation). SHOW ALL OF YOUR WORK!
6. (4) (a) Is F(x, y, z) = <e'siny, e cosx, esiny > a conservative vector field? Justify. (4) (b) Is F incompressible? Explain. Is it irrotational? Explain. (8) (c) The vector field F(x,y,z)= < 6xy+ e?, 6yx²+zcos(y), sin(y)+xe?> is conservative. Find the potential function f. That is, the function f such that Vf= F. Use a process. Don't guess and check.
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)
3. Consider the vector field F(x,y) = (27x D = {(1,y): 0 < rº + y2 <2}. +ya) defined on the region D where a) Directly compute SF. Tds using the definition of the line integral, where C is the unit circle oriented counterclockwise. b). Use Theorem 3.3 (Test for Conservative Vector Fields) from the text to determine if F is conservative. Is your answer consistent with part a)? If not, what is the source of the discrepancy?
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.
Let F(XYZ) = <2y27, 4xyz, 2xy2> be a vector field. (a) Knowing that F is conservative, find a function f such that F = vfand 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, 1.8, 2.3). y2 + x4z2 + 2x4(x3 + y2 + 24)1/2 = K Kis a constant .- Answer: