For each of the following force functions, determine whether or not it represents a conservative force. If it is conservative, find the potential energy function U(~r):

a) F~(x, y, z) = 2ax(z³ + y³ )ˆi + (2ay(z³ + y³ ) + 3ay² (x² + y² ))ˆj + 3az² (x² + y² ) ˆk

b) F~(r, φ, z) = ar² cos(φ)ˆr + ar² sin(φ)φˆ + 2az²ˆk

c) F~(r, θ, φ) = −2ar sin(θ) cos(φ)ˆr − ar cos(θ) cos(φ) ˆθ + ar sin(θ) sin(φ)φˆ.

In each case a is assumed to have dimensions as to make the expressions dimensionally correct.