1) 
a) The divergence of F is 1,
so according to the Divergence Theorem, the flux is equal to 4/3.
b) The divergence of F is 2, so the flux is equal to 8/3. 
2)  Since F is tangent to the xy plane at z = 0, its flux is zero through the disk where
x^{2} + y^{2} 1 and
z = 0. This means that the flux of F through the surface made by joining this disk along its
boundary to the boundary of the top half of the ball is equal to the flux of F just through the top
half of the ball. With this point understood, the Divergence Theorem asserts that the flux in
question is equal to three time the volume of the top half of the ball, thus 2. 
3)  F = (0, 0, 1) has this property. 
4)  F = (0, 0, y) has curl equal to (1, 0, 0). There is no vector field with the given curl and path
integral around the circle having absolute value 2. Indeed, according to Stokes theorem, any
vector field with curl equal to (1, 0, 0) must have path integral on this circle equal to ±. 
5)  F = (z/, 0, 0) has this property. This can be proved using Stokes theorem. 
6)  F = (zy/, 0, 0) has this property. 
7)  F = (0, z, y) has this property. 
8)  F = (x^{2}, 0, 0) has this property. 
9)  F = (0, x^{2}, 0) has this property. 
10)  Interpret a vector v = (f(x,y), g(x,y)) in the plane as the vector F=(f(x, y),g(x, y),0) in R^{3}.
Then, curl(F) = (0, 0, g  f) and so if R is a region in the xy plane,
and C is its boundary curve oriented to be traversed in the counterclockwise direction, then Stokes theorem says that
C F·dx = R curl(F)·k dxdy,
where k = (0,0,1). Using the expression just provided for curl(F) turns curl(F)·k into
g  f which is the
correct integrand for Green's theorem.
