Conservative Fields and Path Independence · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Fields that only care about endpoints

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

  1. For F = (P, Q), what is the quick test for a conservative field?

    • Check whether P equals Q
    • Check whether dQ/dx equals dP/dy
    • Check whether P plus Q equals zero

  2. Ana says the line integral of a conservative field around any closed loop is zero. Is Ana right?

    Circle one:   True   False

  3. The field F = (2x, 2y) has potential f(x, y) = x squared + y squared. What is f(2, 1)?

    Answer: ______________

  4. What makes a field conservative?

    • It is the gradient of some potential function
    • It has the same arrows everywhere
    • It is zero at every point

  5. The potential is f(x, y) = x squared times y plus y. Type the integral of its gradient field along any curve from (0, 0) to (1, 1).

    Answer: ______________

  6. For the field above, you integrate P = 2xy with respect to x and get x squared times y plus g(y). What fixes g(y)?

    • Matching the y derivative against Q
    • Setting g(y) to zero always
    • Integrating P with respect to y instead

  7. The field is F = (2x, 2y). Which function is its potential?

    • x squared + y squared
    • 2x + 2y
    • x squared times y
    • 2xy

  8. Is the field (2xy, x squared + 1) conservative?

    • No, because the components look different
    • Yes, because both cross partials equal 2x
    • No, because it contains an x squared term

  9. A student finds a potential for a field whose cross partials differ, then evaluates f(end) minus f(start). What went wrong?

    • The subtraction step needs a longer curve
    • Nothing, the method works for every field
    • The field is not conservative, so the potential step was invalid

  10. Two hikers walk different trails between the same two camps in a gravity field. How does the work compare?

    • Equal, because only the endpoints matter
    • Bigger for the longer trail
    • Bigger for the steeper trail
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Answer key

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

Fields that only care about endpoints W1-mt_5djKpI5F33-s1

  1. Check whether dQ/dx equals dP/dy · Equal cross partials signal a conservative field on nice regions.
  2. True · Ana is right: start and end coincide, so f(end) minus f(start) = 0.
  3. 5 · f(2, 1) = 4 + 1 = 5.
  4. It is the gradient of some potential function · A conservative field is a gradient in disguise, F = grad f.
  5. 2 · The integral is f(1, 1) minus f(0, 0), which is 2 minus 0.
  6. Matching the y derivative against Q · Differentiating with respect to y and comparing with Q pins down the leftover piece.
  7. x squared + y squared · Only x squared + y squared has gradient (2x, 2y).
  8. Yes, because both cross partials equal 2x · dQ/dx is 2x and dP/dy is 2x, so the mixed partial condition holds.
  9. The field is not conservative, so the potential step was invalid · Skipping the test is the standard blunder: without it there is no potential at all.
  10. Equal, because only the endpoints matter · Gravity is conservative, so the route drops out and only the camps count.
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