Chapter 17.1, Problem 53E

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a.    The vector field F = 〈3x², 1〉 is a gradient field for both ${\phi }_1\left(x,y\right)=x^3+y$ and ${\phi }_2\left(x,y\right)=y+x^3+100$.

b.    The vector field $F=\frac{\langle y,x\rangle }{\sqrt{x^2+y^2}}$ is constant in direction and magnitude on the unit circle.

c.    The vector field $F=\frac{\langle y,x\rangle }{\sqrt{x^2+y^2}}$ is neither a radial field nor a rotation field.