Line Integrals and Conservative Fields

When the path stops mattering.

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

What this lesson teaches

The student computes work along a path and identifies conservative fields.

A line integral adds a field along a path, and for a force that total is work.
In a conservative field the path does not matter, only the endpoints.
Conservative fields are gradient fields, and every closed loop in one gives zero.

Straightforward practice. Get the method working first.

5 problems

A potential f = x² + y. Work from (1, 1) to (3, 2)?
Answer 9
Why 9.
What makes a field conservative?
Answer The line integral depends only on the endpoints, not the route.
Why Path independence.
A potential f = x². Work from (2, 0) to (5, 0)?
Answer 21
Why 25 − 4.
A potential f = 3y. Work from (0, 1) to (0, 6)?
Answer 15
Why 18 − 3.
A closed loop in a conservative field. What is the work?
Answer 0
Why You returned to the start.

The same ideas with more to keep track of.

3 problems

Is every gradient field conservative? 1 yes, 0 no.
Answer 1
Why They are the same class of field.
A force perpendicular to the motion throughout. How much work does it do?
Answer 0
Why The dot product is zero.
A potential f = x + y. Work from (0, 0) to (3, 4)?
Answer 7
Why 7 − 0.

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.
Answer True for conservative fields: Every closed loop gives zero, The path taken does not matter, A potential function exists · Not true in general: Every loop gives a positive total
Why Three of these are equivalent statements.
A potential f = 2xy. Work from (1, 1) to (2, 3)?
Answer 10
Why 12 − 2.
The Potential: A conservative field has potential f = xy. What is the work from (1, 1) to (4, 5)?
Answer 19
Why 19.
The Loop: A closed loop walked in a conservative field. How much work is done?
Answer 0
Why 0.