Vector Fields and Line Integrals · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Arrows everywhere, added along a path

Mathematics · Calculus & Analysis · ages 20-21
Name ______________________   Date ____________

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

    Circle one:   True   False

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

    • (2, -3)
    • (-6, 3)
    • (6, 3)
    • (6, 2)

  3. 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

  4. 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

  5. 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.

    Answer: ______________

  6. 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).

    Answer: ______________

  7. 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

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

    Answer: ______________

  9. 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

  10. 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
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Arrows everywhere, added along a path W1-mt_VZxKEmsVgO-s1

  1. False · The zero horizontal component gives no sideways direction. The negative vertical component points down.
  2. (6, 3) · At x = 2 and y = -3, the components are -2 × (-3) = 6 and 2+1 = 3.
  3. A vector assigned to that point · A vector field assigns one vector to each point. The field need not represent flow.
  4. The work done by the force · Force dotted with displacement gives a signed work contribution. The line integral accumulates these contributions along the path.
  5. 40 · The curve derivative has magnitude 5. The density on the wire is 1+3t, so integrate 5+15t from 0 to 2 to get 40 grams.
  6. 1 · The field on the curve is (t²,t), and the curve derivative is (1,2t). Their dot product is 3t², whose integral from 0 to 1 is 1.
  7. It is multiplied by -1 · Reversal negates every displacement and therefore negates the integral. An original value of zero remains zero.
  8. 6 · The segment's displacement is (4,3). Its dot product with (3,-2) is 12-6 = 6.
  9. It loses the signed tangential force · Work uses the signed tangential component of force. Force magnitude alone does not distinguish aligned, opposing and perpendicular force.
  10. Flow around the centre · Nonzero velocity arrows tangent to the circles describe rotational flow around their common centre.
Worksheet · LightMySky