Chapter 14.5, Problem 39E

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

a.    For a function f of a single variable, if f′(x) = 0 for all x in the domain, then f is a constant function. If ▿ ·F = 0 for all points in the domain, then F is constant.

b.    If ▿ × F = 0, then F is constant.

c.    A vector field consisting of parallel vectors has zero curl.

d.    A vector field consisting of parallel vectors has zero divergence.

e.    curl F is orthogonal to F.