SUNY Geneseo Department of Mathematics

Last Problem Set! Conservative Vector Fields

Math 223
Spring 2023
Prof. Doug Baldwin

Complete by Thursday, May 4
Grade by Wednesday, May 10

Purpose

This exercise develops your understanding of conservative (aka gradient) vector fields and potential functions. It therefore contributes to the following learning outcomes for this course:

Background

This exercise focuses on recognizing conservative vector fields and finding their potential functions. These topics are covered in sections 5.1 and 5.3, respectively, of our textbook. We discussed, or will discuss, them in class on April 28 and May 1.

Activity

Solve each of the following problems.

Problem 1

Determine whether each of the following vector fields is conservative or not. For each that is conservative, give a potential function for it.

Vector Field A

\[\vec{F}(x,y) = \langle 2x - y, 2xy \rangle\]

Vector Field B

\[\vec{G}(x,y,z) = \langle x, y, z \rangle\]

Problem 2

Imagine a large flat-bottom kitchen sink with a drain in the center. Water flows in a thin sheet over the bottom of this sink towards the drain. Because the water flows in a thin sheet, it can be thought of as 2 dimensional, and its velocity can be described by the vector field

\[\vec{V}(x,y) = \left\langle \frac{-x}{x^2+y^2}, \frac{-y}{x^2+y^2} \right\rangle\]

This equation is relative to a coordinate system whose origin is at the center of the drain.

Part A

Is this vector field conservative? If so, give its potential function.

Part B

Show that the circulation of \(\vec{V}(x,y)\) around any circle centered at the origin is 0, regardless of that circle’s radius.

Part C

Show that the flux of this field across any circle around the drain is constant; in particular, that the flux of \(\vec{V}(x,y)\) across circle \(C\) does not depend on the radius of \(C\).

Follow-Up

I will grade this exercise during an individual meeting with you. That meeting should happen on or before the “Grade By” date above, and should ordinarily last half an hour. During the meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Sign up for the meeting via Google calendar. Please have a written solution to the exercise ready to share with me during your meeting, as that will speed the process along.