(f) Find the constants \( a, b, c \), so that \[ \vec{F}=(x+2 y+a z) \vec{i}+(b x-3 y-z) \vec{j}+(4 x+c y+2 z) \vec{k} \] is irrotational and hence find function \( \phi \) such that \( \vec{F}=\nabla \phi \). (g) Show that the vector field \( \vec{F}=\left(e^{x} \cos y+y z\right) \vec{i}+\left(x z-e^{x} \sin y\right) \vec{j}+(x y+z) \vec{k} \) is conservative by (3 marks) finding a potential for it.

(f) The constants \( a, b, c \) are \( a = 4 \), \( b = 2 \), and \( c = -1 \). The potential function \( \phi \) is \( \phi = \frac{x^2}{2} + 2xy - \frac{az^2}{2} + C \), \( \phi = \frac{by^2}{2} - 3y^2 + \frac{z^2}{2} + C \), and \( \phi = 2xz + \frac{cy^2}{2} + 2z^2 + C \), where \( C \) is a constant. (g) The vector field \( \vec{F} \) is conservative, and a potential function for it is given by \( \phi = \frac{x^2}{2} + 2xy - \frac{az^2}{2} + C \), \( \phi = \frac{by^2}{2} - 3y^2 + \frac{z^2}{2} + C \), and \( \phi = 2xz + \frac{cy^2}{2} + 2z^2 + C \).