Q. On what questions can we use the Fundamental Theorem of Line Integrals or knowledge from section 17.3?
There’s some confusion about whether we’re allowed to use the Fundamental Theorem of Line Integrals for homework. You are allowed to use it only for the textbook questions from 17.3. It’s such a great, useful theorem, that you’ll now be tempted to use it for questions in 17.2, but for now, you’ll learn more by doing it “the hard way”. In particular, Extra Problem #3 and the Bonus Problem should be done without it (see the two posts immediately before this one for these problems).
Q. For #43, can we use the fact that the force of gravity function is conservative? [For the bonus?]
You could conceivably get an answer to #43 in at least 3 fundamentally different ways:
1) You can use a parametrisation of the helix;
2) You can directly argue why the parametrisation of the helix doesn’t matter in this case; and
3) you can use conservation of the graviational force and the big fundamental theorem of line integrals.
You learn the most by doing it all three ways, but my intention for this assignment was: full marks for (1); bonus marks for (2); and not allowed to do (3). The solutions manual does (1), and I suggest you do this first. Then, you’ll see that the calculation simplifies in certain ways. What I am looking for as (2) is to explain in a down-to-earth way (nothing about gradients or conservation) why one expects the simplification in the calculation and can therefore exploit it to avoid using the parametrisation of the helix at all.