Final answer:
To determine if the vector field G has a potential function φ, one must calculate the curl of G and check if it satisfies the conditions for a conservative field. If the curl of G is zero, the field is conservative and has a potential function.
Step-by-step explanation:
The student asks whether there is a potential function φ associated with the vector field G = e^y i + [x e^y + ln(z)] j + zy k. To determine if a vector field has a potential function, it needs to be conservative, which means it must have a zero curl. Calculating the curl of G, ∇ x G, involves taking the partial derivatives of the components of G and checking if they meet the conditions for a conservative field: ∂N/∂x = ∂M/∂y, ∂P/∂y = ∂N/∂z, and ∂M/∂z = ∂P/∂x.
If these conditions are satisfied, G will be conservative and thus have a potential function φ. If not, G cannot be derived from a scalar potential, indicating that there is no associated potential function φ.