Let F(x, y, z) = (4x + y, 2y - z, -z + x). What is its divergence?

The field is F(x, y, z) = (x, y, z). Each partial rate is 1. Type its divergence.

At a point, the limiting net outward flux per unit volume is negative. What does this indicate?

• Negative divergence: net inward flux per unit volume
• Positive divergence: net outward flux per unit volume
• Zero divergence: equal inward and outward flux rates

The field is F(x, y, z) = (minus y, x, 0). What is its curl?

• (0, 0, 0)
• (1, 1, 0)
• (0, 0, 2)

A differentiable fluid velocity field has zero divergence at a point, but its velocity there is nonzero. What follows?

• The local net outward flux per unit volume is zero
• The velocity must vanish throughout a nearby region
• The curl must also vanish throughout a nearby region
• The fluid must follow a circular path through the point

For a differentiable fluid velocity field, what does the magnitude of curl measure at a point?

• The net outward flux per unit volume of the fluid
• Twice the local angular speed of the rotational part
• The magnitude of the velocity vector, regardless of rotation

Which field has nonzero curl and zero divergence?

• F = (x, y, z): radial outflow
• F = (1, 0, 0): constant flow
• F = (-y, x, 0): uniform spin

Let F(x, y, z) = (x squared times y, yz, z squared). What is its divergence at (2, 1, -1)?

F = (x, y, z) and G = (x - y, y + x, z) both have divergence 3. A student says their curls must be equal. Which calculation tests this?

• curl F = (0, 0, 0); curl G = (0, 0, 2)
• curl F = (0, 0, 2); curl G = (0, 0, 2)
• curl F = (0, 0, 0); curl G = (0, 0, 0)

Let F(x, y, z) = (a x + y, 2y + z, -5z + x), where a is constant. What value of a makes the divergence zero everywhere?