Line Integrals and Conservative Fields

When the path stops mattering.

12 problems · about 24 minutes · N-VM.B.4

NameDate

Figure — use these to answer the problems

This is a gradient field, so it is conservative. Any loop you walk in it returns exactly zero work.

strength1

start x2

start y1

Figure 1

Straightforward practice. Get the method working first.

5 problems

A potential f = x² + y. Work from (1, 1) to (3, 2)?
Answer
2
What makes a field conservative?
- The line integral depends only on the endpoints, not the route.
- The field has the same value everywhere.
A potential f = x². Work from (2, 0) to (5, 0)?
Answer
A potential f = 3y. Work from (0, 1) to (0, 6)?
Answer
A closed loop in a conservative field. What is the work?
Answer

The same ideas with more to keep track of.

3 problems

Is every gradient field conservative? 1 yes, 0 no.
Answer
A force perpendicular to the motion throughout. How much work does it do?
Answer
A potential f = x + y. Work from (0, 0) to (3, 4)?
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each statement by whether it holds for a conservative field.
Write each item under the heading it belongs to: Every closed loop gives zero · The path taken does not matter · A potential function exists · Every loop gives a positive total
True for conservative fields
Not true in general
A potential f = 2xy. Work from (1, 1) to (2, 3)?
Answer
The Potential
A conservative field has potential f = xy. What is the work from (1, 1) to (4, 5)?
Answer
The Loop
A closed loop walked in a conservative field. How much work is done?
Answer