Chapter 17.3, Problem 57E

a.

To determine

Whether the statement “If $F=\langle -y,x\rangle$ and C is the circle of radius 4 at $\left(1,0\right)$ oriented counterclockwise, then ${\displaystyle \underset{C}{\oint }F\cdot dr}=0$” is true or false.

b.

To determine

Whether the statement “If $F=\langle x,-y\rangle$ and C is the circle of radius 4 at $\left(1,0\right)$ oriented counterclockwise, then ${\displaystyle \underset{C}{\oint }F\cdot dr}=0$” is true or false.

c.

To determine

Whether the statement “A constant vector field is conservative on ${ℝ}^2$” is true or false.

d.

To determine

Whether the statement “The vector field $F=\langle f\left(x\right),g\left(y\right)\rangle$ is conservative on ${ℝ}^2$” is true or false.

e.

To determine

Whether the statement “Gradient fields are conservative’’ is true or false.