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

Inside sums equal their boundary totals

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

  1. Use the divergence theorem to find the flux of F(x, y, z) = (x, y, z) out of a cube of side 2.

    Answer: ______________

  2. Which pairing fits the divergence theorem?

    • A curve bounded by two endpoints
    • A solid ball bounded by a sphere
    • A surface bounded by another surface

  3. The field (x, y, z) has divergence 3 everywhere. A cube of side 2 has volume 8. Type the outward flux through the closed cube skin.

    Answer: ______________

  4. Which pairing fits Stokes theorem?

    • A surface with its rim curve
    • A solid with its center point
    • Two solids sharing a face

  5. Which theorem converts circulation around a closed space curve into flux of curl through a surface it bounds?

    • Stokes theorem
    • Divergence theorem
    • Chain rule
    • Mean value theorem

  6. Find the flux of F(x, y, z) = (2x, 2y, 2z) out of the unit cube.

    Answer: ______________

  7. How do you keep the rim direction and the cap normal consistent in Stokes?

    • Point the normal along the flow
    • Use the right hand rule
    • Always point the normal upward

  8. A Stokes surface is a crumpled mess, but a flat disk shares its rim. What should you do?

    • Parametrise the crumpled surface anyway
    • Give up, since the rim moved
    • Use the flat disk with the matching orientation

  9. A student computes a closed-skin flux by parametrising all six faces of a cube. The divergence is constant. What shortcut did they miss?

    • Looking inside first: flux is divergence times volume
    • Splitting each face into triangles
    • Reversing every normal at the end

  10. Find the flux of F(x, y, z) = (x, 0, 0) out of the box 1 by 2 by 3.

    Answer: ______________

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Answer key

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

Inside sums equal their boundary totals W1-mt_sRojZcrw-q-s1

  1. 24 · Divergence is 3 and the cube volume is 8, so the flux is 3 times 8 = 24.
  2. A solid ball bounded by a sphere · The theorem pairs a solid region with its closed outer skin.
  3. 24 · Constant divergence makes flux equal divergence times volume: 3 times 8.
  4. A surface with its rim curve · Stokes equates curl flux through a cap with circulation around its rim.
  5. Stokes theorem · Stokes turns circulation around a closed space curve into flux of curl through any capped surface.
  6. 6 · Divergence is 6 and the unit cube has volume 1, so the flux is 6.
  7. Use the right hand rule · Fingers curl with the rim and the thumb gives the matching normal.
  8. Use the flat disk with the matching orientation · Any cap with the same edge gives the same curl flux, so take the easy one.
  9. Looking inside first: flux is divergence times volume · Both theorems exist so you can skip nasty direct setups for an inside sum.
  10. 6 · Divergence is 1 and the box volume is 6, so the flux is 6.
Worksheet · LightMySky