A line integral adds up a field along a path. When the field is a force, the total is the work done travelling that path.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Only the useful part counts
At each step, the dot product keeps only the part of the field pointing along the motion. A sideways force does no work.
Usually the path matters
Two different routes between the same endpoints generally give different totals. Where you went matters, not just where you ended.
Conservative fields
Some fields are special: the total depends only on the endpoints. Those are called conservative, and they are exactly the gradient fields.
The Fundamental Theorem again
For a conservative field with potential f, the line integral is simply f at the end minus f at the start.
The loop test
In a conservative field, any closed loop gives zero. Gravity gives back on the way down exactly what it took on the way up.
Integrating along a path
A line integral accumulates a field's effect along a curve. For a force field it computes work done, which is the interpretation that motivates the definition.
Direction matters
Reversing the path negates the line integral of a vector field. Orientation is part of the specification, unlike arc length, which is unaffected by direction.
Conservative fields
A field is conservative if it is the gradient of a potential function. Then the line integral depends only on the endpoints, and the path between them is irrelevant.
Testing and consequences
A conservative field has zero curl, and its integral around any closed loop is zero. That is why gravitational potential energy depends on height alone and not on the route taken.
Step 2: Try It Yourself
Tap and try it out.
- Released from(2, 1)
These arrows point straight uphill on a bowl, always away from the lowest point and perpendicular to the contours.
Step 3: Watch an Example
One step at a time.
Watch Diego Use a Potential
Diego finds the work done by a conservative field with potential f = x² + y², moving from (1, 0) to (3, 4).
- Step 1
He notices the field is conservative, so the path can be ignored entirely.
Step 4: Your Turn
Practice makes it stick.
The Potential
Problem 1 of 2
A conservative field has potential f = xy. What is the work from (1, 1) to (4, 5)?
The Loop
Problem 2 of 2
A closed loop walked in a conservative field. How much work is done?
Walk the Path
1 of 8
A potential f = x². Work from (2, 0) to (5, 0)?
2 of 8
A potential f = 3y. Work from (0, 1) to (0, 6)?
3 of 8
A closed loop in a conservative field. What is the work?
4 of 8
Is every gradient field conservative? 1 yes, 0 no.
5 of 8
A force perpendicular to the motion throughout. How much work does it do?
6 of 8
A potential f = x + y. Work from (0, 0) to (3, 4)?
7 of 8
Sort each statement by whether it holds for a conservative field.
Tap something to move it.
- Empty
- Empty
8 of 8
A potential f = 2xy. Work from (1, 1) to (2, 3)?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A potential f = x² + y. Work from (1, 1) to (3, 2)?
Question 2 of 2
What makes a field conservative?
What You Learned
- 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.