Math 223 Conservative Fields

Purpose

This exercise develops your ability to work with conservative vector fields and line integrals in them. 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
Outcome 12. Use technological tools such as computer algebra systems or graphing calculators for visualization and calculation of multivariable calculus concepts.

Background

This exercise is based on material in section 5.3 in our textbook, with some supporting information in sections 5.1 and 5.2. We covered the main material in classes on November 22 and 28, and the most important supporting material between November 16 and 21.

This exercise also asks you to plot vector fields with Mathematica. We discussed that in classes on November 15 and 16.

Activity

Solve each of the following problems.

Problem 1

For each of the following vector fields…

Give the value of the vector field at the origin
Use Mathematica to plot the vector field in a small region centered on the origin (you can decide for yourself what “small” should mean here)
Find a potential function for the field, or show that no potential function exists for it.

Vector Field A

\vec{F} (x, y) = ⟨ y - 1, x + 1 ⟩

Vector Field B

\vec{G} (x, y, z) = ⟨ x, y, z ⟩

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) = ⟨ \frac{- x}{x^{2} + y^{2}}, \frac{- y}{x^{2} + y^{2}} ⟩

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, i.e., that the flux of $\vec{V} (x, y)$ across circle $C$ does not depend on the radius of $C$ .

Part D

Physically, the flux of water across a circle around the drain is the volume of water crossing the circle per unit of time. If that volume per unit time remains constant as the circles get smaller, then the speed at which the water flows must increase (since the same volume of water is crossing a shorter circumference in the same time, and so the volume of water per unit circumference per unit time has to increase). Show that the speed represented by $\vec{V} (x, y)$ does indeed increase as distance from the origin decreases.

Part E

Use Mathematica to plot $\vec{V} (x, y)$ near the origin to check that it behaves as implied by the context of the question and Parts C and D. In particular, check that the flow is everywhere towards the origin (the drain), and the magnitude of the field is larger near the origin than farther away from it.

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. 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.