On standard axes, Mia sketches F(x, y) = (0, -2) with every arrow pointing up. Does her sketch match the field?

For F(x, y) = (-2y, x + 1), which vector is attached to the point (2, -3)?

• (2, -3)
• (-6, 3)
• (6, 3)
• (6, 2)

What does a vector field assign to each point?

• A scalar with no direction there
• A vector assigned to that point
• A curve passing through that point

A force field is dotted with each small displacement along a path. What does the line integral of these contributions measure?

• The total length of the curved path
• The mass of the moving object
• The work done by the force

A wire is r(t) = (3t, 4t), 0 <= t <= 2, with coordinates in centimetres. Its linear density is f(x, y) = 1 + x grams per centimetre. Find its mass in grams.

For the force field F(x, y) = (y, x), compute the work along r(t) = (t, t²), 0 <= t <= 1. This follows y = x² from (0, 0) to (1, 1).

You evaluate ∫C F dot dr, then reverse the same traversal while keeping F fixed. What happens to the integral?

• It is multiplied by -1
• It remains unchanged for every field
• It becomes zero for every field

For the constant field F(x, y) = (3, -2), evaluate ∫C F dot dr along the straight segment from (1, -1) to (5, 2).

A student uses ∫C |F| ds to calculate work done by a force field. Why can this differ from ∫C F dot dr?

• Arc length reverses sign along the path
• It loses the signed tangential force
• Vector work ignores the force magnitude

A velocity-field sketch shows nonzero arrows tangent to concentric circles, all in the same sense around their centre. Which flow pattern does this describe?

• Outward flow from the centre
• No motion around the centre
• Flow around the centre