Problem Set 11—Vector Fields

Purpose

This problem set reinforces your understanding of vector fields and integrals over them.

Background

This exercise is based on material in sections 16.2 and 16.3 of our textbook. We will discuss this material in class between November 23 and December 2.

Activity

Solve each of the following problems:

Problem 1

Exercise 8c in section 16.2 of our textbook (find the line integral of ⟨ 0, 1/(x²+1), 0 ⟩ along a path from (0,0,0) to (1,1,0) and then from (1,1,0) to (1,1,1)).

Problem 2

Exercise 46 in section 16.2 of our textbook (show that the work done by a radial force of constant magnitude in moving a particle along a curve y = f(x) from point ( a, f(a) ) to point ( b, f(b) ) equals k[(b²+f(b)²)^1/2 - (a²+f(a)²)^1/2]).

Problem 3

Exercise 32 in section 16.2 of our textbook (find the circulation and flux of ⟨ x², y² ⟩ along the pair of curves r₁(t) = ⟨ acost, asint ⟩ for 0 ≤ t ≤ π and r₂(t) = ⟨ t, 0 ⟩ for -a ≤ t ≤ a).

Problem 4

Give the gradient field for f(x,y) = x³/3 + y³/3 (i.e., give an equation for the vector field that is the gradient of f(x,y)). Sketch this field, showing a few vectors in it near the origin. Explain in a sentence or two why the answer to this question helps you answer (or check your answer to) part of Problem 3 of this problem set.

Problem 5

Exercise 2 in section 16.3 of our textbook (determine whether the field ⟨ ysinz, xsinz, xycosz ⟩ is conservative or not).

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.