Chapter 14.3, Problem 39E

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

a.    If F = 〈–y, x〉 and C is the circle of radius 4 centered at (1.0) oriented counterclockwise, then ${\displaystyle {\oint }_{C}F·dr=0}$.

b.    If F = 〈x, –y) and C is the circle of radius 4 centered at (1, 0) oriented counterclockwise, then ${\displaystyle {\oint }_{C}F·dr=0}$.

c.    A constant vector field is conservative on ¡².

d.    The vector field F = 〈f(x), g(y)〉 is conservative on ¡² (assume f and g are defined for all real numbers).

e.    Gradient fields are conservative.