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:
- Outcome 9. Recognize gradient vector fields
- Outcome 10. Find potential functions
- Outcome 11. Evaluate line integrals directly and by the fundamental theorem.
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
Vector Field B
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
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
Part C
Show that the flux of this field across any circle around the drain is constant; in
particular, that the flux of
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.