Green's Theorem in the Plane · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Trading a rim walk for the inside

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

  1. Take P = 0 and Q = x, so dQ/dx minus dP/dy equals 1. Over the unit square the double integral is 1. What circulation does Green give?

    Answer: ______________

  2. Green's theorem equates a circulation integral to a double integral. Which boundary orientation is the positive one?

    • Counterclockwise
    • Clockwise
    • Either, the sign never changes
    • Straight out of the page

  3. What does Green's theorem equate the rim integral P dx plus Q dy to?

    • The double integral of dQ/dx minus dP/dy over the inside
    • The length of the boundary curve
    • The value of the field at the center

  4. Green's theorem can be used on any open curve, not just closed ones.

    Circle one:   True   False

  5. Which pair of P and Q lets a rim integral report the area of any region?

    • P = x and Q = y
    • P = 0 and Q = 0
    • P = minus y over 2 and Q = x over 2

  6. Area via half the integral of (x dy minus y dx): for a 3 by 3 square the boundary integral equals 18. What is the area?

    Answer: ______________

  7. Area from the boundary: for the rectangle 0 <= x <= 4, 0 <= y <= 2, the integral of x dy around the boundary equals 8. What is the area?

    Answer: ______________

  8. You compute a counterclockwise circulation of 5. What is the clockwise value for the same field and curve?

    • 5
    • 0
    • Minus 5

  9. A field blows up at one interior point, but a student applies Green's theorem anyway. What should they have done?

    • Walked the rim twice as fast
    • Checked smoothness first, since a spike breaks the theorem
    • Swapped P and Q in the answer

  10. For the field (minus y, x), what does dQ/dx minus dP/dy simplify to, and what does that buy you?

    • It is 2, turning circulation into twice the area
    • It is 0, turning circulation into zero
    • It is x, turning circulation into the center of mass
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Answer key

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

Trading a rim walk for the inside W1-mt_UDgivmPVEJ-s1

  1. 1 · The circulation equals the double integral, which is 1.
  2. Counterclockwise · Positive orientation keeps the region on the left, which is counterclockwise.
  3. The double integral of dQ/dx minus dP/dy over the inside · The edge total equals the summed swirl density over the enclosed region.
  4. False · The theorem needs a closed boundary that encloses a region.
  5. P = minus y over 2 and Q = x over 2 · That pair makes dQ/dx minus dP/dy equal 1, so the inside sum is the area.
  6. 9 · Half of 18 is 9, matching the 3 by 3 area.
  7. 8 · For this choice the boundary integral equals the area, so 8.
  8. Minus 5 · Flipping the direction negates every tangent, so the total flips sign.
  9. Checked smoothness first, since a spike breaks the theorem · The field must be smooth on the whole region, with no holes or spikes.
  10. It is 2, turning circulation into twice the area · dQ/dx is 1 and dP/dy is minus 1, so the difference is the constant 2.
Worksheet · LightMySky