4. True or False? Explain your answer (otherwise you won’t get points). (a) (2 p

Question

4. True or False? Explain your answer (otherwise you won’t get points). (a) (2 points) A paddle wheel with its axis in direction 0, 1, 1 would not spin when put in the vector field F = 1, 1, 2 × x, y, z. (b) (2 points) Stokes’ Theorem relates flux of a vector field F across a surface to the values of F on the boundary of the surface. (c) (2 points) A vector field of the form F = a + f(x),b + g(y),c + h(z), where a,b,c are constants, has zero circulation on a closed curve. (d) (2 points) If a vector field has zero circulation on all simple closed smooth curves C in a region D, then F is conservative on D.

Explanation / Answer

(a) False , Because axis of rotation of paddle and vector field are not perpendicular.

(b) True, Becuse stoke's theorem of flux of vecor field gives acroos surface.

(c) True, Because Force field perpendicular to closed surface. So, workdone would be zero

(d) False , Because vector field closed surface curve C , F is not conservative on D.

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