Problem Set 12—Vector Fields & Green’s Theorem

Purpose

This problem set reinforces your ability to calculate with conservative vector fields and use Green’s Theorem.

Background

This exercise is based on material in sections 16.3 and 16.4 of our textbook. We will discuss this material in class between December 2 and December 9.

Activity

Solve each of the following problems:

Problem 1

Exercise 18 in section 16.3 of our textbook (evaluate the integral from (0,2,1) to (1,π/2,2) of 2cosydx + (1/y-2xsiny)dy + 1/zdz).

Problem 2

Exercise 30, all parts, in section 16.3 of our textbook (find the work done by ⟨ e^yz, xze^yz+zcosy, xye^yz+siny ⟩ over various paths from (1,0,1) to (1,π/2,0)).

Problem 3

Exercise 6 in section 16.4 of our textbook (use Green’s Theorem to find the circulation and flux of ⟨ x²+4y, x+y² ⟩ around the square bounded by x=0, x=1, y=0, y=1).

Problem 4

Exercise 22 in section 16.4 of our textbook (use Green’s Theorem to evaluate the line integral of 3ydx + 2xdy around a region bounded by half a sine wave—see textbook for details).

Problem 5

Exercise 20 in section 16.4 of our textbook (use Green’s Theorem to find the work done by ⟨ 4x-2y, 2x-4y ⟩ around a circle of radius 2 centered at (2,2)).

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.