Chapter 16.7, Problem 27E

(a)

To determine

Show that the circulation of the field $F=2xi+2yj+2zk$ around the boundary of any smooth orientable surface is zero by using $\nabla \times \nabla f=0$ and Stoke’s theorem.

(b)

To determine

Show that the circulation of the field $F=\nabla \left(x{y}^2{z}^3\right)$ around the boundary of any smooth orientable surface is zero by using $\nabla \times \nabla f=0$ and Stoke’s theorem.

(c)

To determine

Show that the circulation of the field $F=\nabla \times \left(xi+yj+zk\right)$ around the boundary of any smooth orientable surface is zero by using $\nabla \times \nabla f=0$ and Stoke’s theorem.

(d)

To determine

Show that the circulation of the field $F=\nabla f$ around the boundary of any smooth orientable surface is zero by using $\nabla \times \nabla f=0$ and Stoke’s theorem.