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We consider the work done by
a gravitational force, which

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was path-independent.

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Now let's generalize that to
any force between two points,

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i, and another point,
f, that if the work done

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going from the initial
point to the final point--

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this is path-independent--
then we call this

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force a conservative force.

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There are many
conservative forces

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that we'll analyze
in this class.

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Examples will be the inverse
square, electric force,

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or the gravitational force.

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We've already seen
gravitational force.

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Near the surface of the
Earth is conservative.

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Spring forces are other
examples of conservative forces.

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But let's look a
little bit in detail.

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Suppose we call
this path 1 and we

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consider a second path, path 2.

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Then a property of
conservative forces

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is that the work done is
independent of that path.

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So if we total up the
work from the initial

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to final on path 1 of f dot ds.

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And then we go from
the final point

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to the initial point
on path 2 of f dot ds.

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Then because these intervals
are independent of the path,

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and all we've done is shifted
the endpoints, we get 0.

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And that's the statement.

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And we'll write w conservative
here for conservative forces.

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So we're going to call
this force conservative

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if it's path-independent.

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So what we have here
is the statement

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that the work done by
a conservative force.

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Now what we'll do here,
is indicate a circle

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by a closed path.

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The work done by conservative
force on a closed path is zero.

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What does a closed
path look like?

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Well, remember, what we did
was we went from 1, but then

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instead of going
in this direction,

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we came back on path
to, from the final,

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to the initial point.

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And if the force
is conservative,

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the work done around
a closed path is 0.