how can i solve this problem?

In quantum mechanics, we learned that the wavefunction (t,x) can be multiplied by a spacetime dependent phase e W(t x) a) while not changing the physics. Here q is the charge of the particle and I(t, x) is the phase angle. This does not change the physics since in quantum mechanics the probability is given by ~*w. However, because of this arbitrary phase, the derivative 0,7 in some ways is no longer uniquely defined as the field a(x + ) and w(x) at nearby points can have arbitrary different phases and thus their difference can be arbitrary while not changing any physics. But there is a good way to define, or generalize, the derivative of the field. That is, we define the covariant derivative D,,, where Dyw = (Op igAy)w. (11) Here A,,(t,x) is a dual vector field. We require that when 7(x) undergoes a phase change w + wy! _ etal (t@) gy, (12) the dual vector field A,, also undergoes a transformation of the form such that D,,y transforms in the same way as 7. That is, we want D,,~ transforms as Diy cP) Dap. (14) (b) (10 pts) Based on the requirement that Duh Diab! = (Ou ig Ai,) (ev) (15) should eventually has the form of Eq. (14), find how the dual vector field A,, should transform. That is, find what 6A, should be so that we get Eq. (14)