4. Use the Fundamental Theorem for Conservative Vector Fields to compute F. dr. where F= <3y2...
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)
(6) Fundamental Theorem of Line Integrals F = <M,N> = < 2xy, x² + y2 > (6a) Show that F is a Conservative Vector Field. (6b) Find the Potential Function f(x,y) for the Vector Field F. (60) Evaluate W = | Mdx + Ndy from (5,0) to (0,4) over the path C: È + K3 = 1 с
1. One of the two vector fields listed below is conservative. The other one is not conservative. (a) Determine which one of these fields is conservative. Label the conservative field F and and find a potential function f for it. Label the other field G and prove that G is NOT conservative. (b) Use the fundamental theorem of line integrals to compute SCF . dr, where C is the curve parameterized by (c) Compute Jc G-dr, where C is the...
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?
Question 10 Compute the flux of the vector fields F(x, y, z) =< x, y2,1 > across the portion of the plane r+y+z=1 on the first octant, with orientation pointing toward the positive x direction. (Do not use Stokes' theorem)
1. One of the two vector fields listed below is conservative. The other one is not conservative. (a) Determine which one of these fields is conservative. Label the conservative field F and and find a potential function f for it. Label the other field G and prove that G is NOT conservative. (b) Use the fundamental theorem of line integrals to compute SCF . dr, where C is the curve parameterized by (c) Compute Jc G-dr, where C is the...
1. Let F(x,y,z) =< 32, 5x, – 2y >. Use Stokes's Theorem to evaluate the integral Scurl F.ds, where S is the part of the paraboloid z = x² + y2 that lies below the plane z = 4 with upward- pointing normal vector.
(1 point) Find the maximum and minimum values of the function f(x, y) = 3x² – 18xy + 3y2 + 6 on the disk x2 + y2 < 16. Maximum = Minimum =
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: