Divergence and Curl · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Outflow at a point, spin at a point

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

  1. Let F(x, y, z) = (x, y, z). What is the divergence of F?

    Answer: ______________

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

    Answer: ______________

  3. At some point the arrows bunch tightly together. What does that show?

    • Negative divergence, a sink
    • Positive divergence, a source
    • Zero field everywhere

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

    • (0, 0, 0)
    • (1, 1, 0)
    • (0, 0, 2)

  5. At a point the divergence of a flow field is positive. What does that say physically?

    • More fluid leaves than enters: a source
    • More fluid enters than leaves: a sink
    • The fluid spins rapidly with no outflow
    • The fluid is completely still

  6. The curl at a point is a long vector. What does its length say?

    • Nothing about rotation
    • Fast local rotation there
    • Strong outflow there

  7. Which field has curl but no divergence?

    • Radial outflow like (x, y, z)
    • A constant field like (1, 0, 0)
    • Uniform rotation like (minus y, x, 0)

  8. Let F(x, y, z) = (x squared, y squared, z squared). What is the divergence at (1, 2, 3)?

    Answer: ______________

  9. A student sees nonzero divergence and concludes the field must also curl. What is the error?

    • Outflow and spin are independent, so one never implies the other
    • Divergence is always zero in three dimensions
    • Curl can only be measured with integrals

  10. Let F(x, y, z) = (3x, 2y, z). What is the divergence of F?

    Answer: ______________

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Answer key

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

Outflow at a point, spin at a point W1-mt_JbjW8WKOPE-s1

  1. 3 · Each term contributes 1: 1 + 1 + 1 = 3.
  2. 3 · Divergence adds the three outward rates: 1 plus 1 plus 1.
  3. Negative divergence, a sink · Bunching arrows mean net inflow, which is negative divergence.
  4. (0, 0, 2) · Only the k component survives: 1 minus minus 1 is 2.
  5. More fluid leaves than enters: a source · Positive divergence means net outflow, the signature of a source.
  6. Fast local rotation there · Curl size is twice the rotation rate, so long means fast spin.
  7. Uniform rotation like (minus y, x, 0) · Pure spin loops without spreading, so curl is nonzero and divergence is zero.
  8. 12 · Divergence is 2x + 2y + 2z, which at (1, 2, 3) is 2 + 4 + 6 = 12.
  9. Outflow and spin are independent, so one never implies the other · Radial outflow diverges with no spin, and pure rotation spins with no outflow.
  10. 6 · Divergence is 3 + 2 + 1 = 6 at every point.
Worksheet · LightMySky