- Use the divergence theorem to compute the flux of the following vector fields outward
through the surface of the ball where x2 + y2 + z2
 1:
- F = (y2 2x, x + z, cos(y) + z).
- F = (sin(z y2), y2 + 2y + z4, x4 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 xy-plane. Compute the flux of F outward through the z
 0 portion
of the surface of the ball where x2 + y2 + z2 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
x2 + y2 + z2 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 y2 + z2 = 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 x2 + y2 + z2 1 is zero.
- Write down a vector field whose path integral is zero around all circles with z = 0 and
x2 + y2 = 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.
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