Problem Set 11—Vector Fields

Purpose

This problem set helps you understand vector fields and line integrals of them. When you finish this problem set, you should be able to (1) use vector fields to describe a physical situation, (2) plot vector fields with muPad, (3) evaluate line integrals of vector fields, (4) use line integrals to find such physical quantities as work, circulation, or flux in vector fields, (5) show that vector fields are or aren’t conservative, (6) find potential functions for conservative vector fields, and (7) use potential functions to evaluate line integrals.

Background

This problem set is based on material in sections 16.2 and 16.3 of our textbook. We discussed this material in class between April 13 and April 21.

Activity

Solve each of the following problems:

Problem 1

Exercise 44a in section 16.2 of our textbook (find a vector field F such that at point (x,y), F(x,y) points towards the origin and |F(x,y)| equals the distance from the origin to (x,y); see textbook for further description). Also plot this field with muPad.

Problem 2

Exercise 10b in section 16.2 of our textbook (find the integral of ⟨ xy, yz, xz ⟩ along the curve r(t) = ⟨ t, t², t⁴ ⟩, 0 ≤ t ≤ 1).

Problem 3

Exercise 30 in section 16.2 of our textbook, for field F₁ (find the flux of F₁(x,y) = ⟨ 2x, -3y ⟩ across a circle; see textbook for details).

Problem 4

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]). Also use muPad to plot the vector field for the “radial force of constant magnitude,” assuming some convenient value, e.g., 1, for the constant k (but do not assume that k has this convenient value in the first part of the question).

Problem 5

Exercise 2 in section 16.3 of our textbook (determine whether F(x,y,z) = ⟨ ysinz, xsinz, xycosz ⟩ is conservative).

Problem 6

Exercise 18 in section 16.3 of our textbook (find and use a potential function to evaluate the integral from (0,2,1) to (1,π/2,2) of 2cosydx + (1/y - 2xsiny)dy + (1/z)dz).

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.