 Use the divergence theorem to compute the flux of the following vector fields outward
through the surface of the ball where x^{2} + y^{2} + z^{2} 1:
 F = (y^{2} 2x, x + z, cos(y) + z).
 F = (sin(z y^{2}), y^{2} + 2y + z^{4}, x^{4} 1).
 Suppose that F is a vector field in space with div(F) = 3 and that is, at all points where z
= 0, tangent to the xyplane. Compute the flux of F outward through the z 0 portion
of the surface of the ball where x^{2} + y^{2} + z^{2} 1.
 Write down a vector field with vanishing divergence and with flux equal to through the
z 0 portion of the surface of the ball where
x^{2} + y^{2} + z^{2} 1.
 Write down a vector field whose curl is equal to (1, 0, 0). Exhibit such a vector field
whose path integral around the circle where x = 0 and y^{2} + z^{2} = 1 is equal to
2 ; or else
explain why there aren't any.
 Exhibit a vector field, F, with the following property: Whenever C is a circle on which
y is constant (so parallel to the xz plane), then the path integral of F around the
circumference of C is equal to the square of Cıs radius.
 Exhibit a vector field, F, with the following property: Whenever C is a circle on which y
is constant (so parallel to the xz plane), then the path integral of F around the
circumference of C is equals the constant value of y times the square of Cıs radius.
 Write down a vector field whose components are not constant, but that has zero curl and
zero divergence.
 Write down a vector field whose divergence is not everywhere zero, but whose flux
through the surface of the ball where x^{2} + y^{2} + z^{2} 1 is zero.
 Write down a vector field whose path integral is zero around all circles with z = 0 and
x^{2} + y^{2} = constant, but whose curl has component along (0,0,1) which is not everywhere
zero.
 Explain why Green's theorem is a special case of Stokes' theorem.
