Problem Set 12—Conservative Fields, Green’s Theorem

Purpose

This problem set mainly helps you use Green’s theorem, but it also further reinforces your understanding of conservative vector fields. When you finish this problem set, you should thus be able to (1) recognize conservative vector fields, (2) evaluate line integrals of conservative vector fields, and (3) use Green’s theorem to evaluate line integrals in any sort of vector field.

Background

Green’s theorem comes from section 16.4 of our textbook, and we discussed it in class on April 26 and April 27.

Section 16.3 of our textbook discusses conservative vector fields. We discussed this material in class on April 21.

Activity

Solve each of the following problems:

Problem 1

When we were discussing conservative vector fields and path independence in class, someone asked what sort of vector field would not have the path independence property. I speculated that maybe some simple constant vector field would lead to integrals dependent on the length of the path between two points, but then cautioned that maybe the ways paths of different lengths took different directions through the field would cancel out any effect of length. It turns out that this caution was a good one. Show that every constant vector field (i.e., every field of the form F(x,y,z) = ⟨ c₁, c₂, c₃ ⟩ where c₁, c₂, and c₃ are constants) is conservative.

Problem 2

A spaceship is spiraling away from a black hole along the path r(t) = ⟨ t cost, t sint, t ⟩ as t ranges from π to 3π. The gravitational field of the black hole is given by F(x,y,z) = ⟨ -x/(x²+y²+z²)^3/2, -y/(x²+y²+z²)^3/2, -z/(x²+y²+z²)^3/2 ⟩. How much work does the spaceship do flying along this path?

Problem 3

Exercise 2 in section 16.4 of our textbook (verify both forms of Green’s theorem for the field F(x,y) = ⟨ y, 0 ⟩ and the curve r(t) = ⟨ a cost, a sint ⟩, 0 ≤ t ≤ 2π).

Problem 4

Exercise 26 in section 16.4 of our textbook (use Green’s theorem to find the area of the ellipse r(t) = ⟨ a cost, b sint ⟩, 0 ≤ t ≤ 2π).

Problem 5

Exercise 22 in section 16.4 of our textbook (use Green’s theorem to evaluate the integral of 3y dx + 2x dy around the boundary of the region defined by 0 ≤ x ≤ π and 0 ≤ y ≤ sinx).

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as that will speed the process along.

Sign up for a meeting via Google calendar. If you worked in a group on this exercise, the whole group should schedule a single meeting with me. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.