Chapter 17.7, Problem 29E

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

a.    A paddle wheel with its axis in the direction 〈0.1, –1〉 would not spin when put in the vector field F = 〈1, 1, 2〉 × 〈x, y, z〉.

b.    Stokes’ Theorem relates the flux of a vector field F across a surface to the values of F on the boundary of the surface.

c.    A vector field of the form F = 〈a + f(x), b + g(y), c + h(z)〉, where a, b, and c are constants, has zero circulation on a closed curve.

d.    If a vector field has zero circulation on all simple closed smooth curves C in a region D, then F is conservative on D.