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 outward flux of F(x, y, z) = (y, x, 4z) through the closed surface of a 2 by 2 by 5 box.

    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. The fundamental theorem of calculus says the integral of f prime over [a, b] equals f(b) minus f(a). In the shared pattern of the four theorems, what plays the role of the boundary?

    • The two endpoints a and b
    • The whole interval from a to b
    • The derivative f prime itself
    • The area under the graph of f

  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
    • Use the disk, oriented against the rim
    • Use the disk, oriented to match the rim

  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?

    • Flux equals divergence times volume
    • Split each face into two triangles first
    • Reverse every normal before adding up

  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. 80 · The divergence is 0 + 0 + 4 = 4 and the box volume is 20, so the flux is 4 times 20 = 80.
  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. The two endpoints a and b · The region is the interval, and its boundary is its two endpoints, where f is evaluated.
  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 disk, oriented to match the rim · Any cap with the same rim and a matching orientation gives the same curl flux, so use the disk. Orienting it against the rim would flip the sign.
  9. Flux equals divergence times volume · The divergence theorem trades the six-face surface setup for one integral of divergence inside, and constant divergence makes that divergence times volume.
  10. 6 · Divergence is 1 and the box volume is 6, so the flux is 6.
Worksheet · LightMySky