Show that F = y2z sinh(2xz)i + 2y cosh2(xz)j + y2x sinh(2xz)k is conservative, and find a scalar potential φ such that F = −∇φ.

3. Given the following scalar potentials (V), calculate the solution for the gradient of V (VV), and plot the vector arrow representation of this vector field over the given limits. (a) V = 15 + r cos o, for 0 < r < 10, and 0 < $ < 2n. (b) V = 100 + xy, for –10 < x < 10,

•prove that the force is Conserved • then find the potential function

Consider the 3-dimensional force field ⃗ F = (x^2 − ze^y)⃗i + (y^3 − xze^y)⃗j + (z^4 − xe^y)⃗k:(a) Show that ⃗ F is conservative.(b) If ⃗ F is conservative, find the corresponding potential function f (x, y).(c) If an object travels on a path ⃗r (t), (t_0 < t < t_f ), does the work done bythe force field depend on the path taken? Find the work done on the object movingon the path ⃗r (t) by the force field ⃗ F , if ⃗r (t_0) = (3, 1, 2) and ⃗r (t_f ) = (6, 2, 5)

Please solve asap: 7. Find a potential function, and you don’t need to verify that they are conservative because they are (a) F = < -y , x > (b) F = < yz, xz, xy >