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Unformatted text preview: 1 Example 1. Let F be a conservative vector ﬁeld in the plane. In the Figure, suppose that in the integral of F along AOF is 3, along OF is 2, and along AB is −5. Compute the integral of F along the path BOEF.
y A F O B E x C D Solution. Since F is conservative, the line integral depends only on the endpoints. Thus, we can take the path BAOF instead. The integral along BAOF = (integral along BA) + (integral along AOF) = −(−5) + 3 = 8. ♦ Example 2. Suppose that the kinetic energy of a particle that moves in a circular path under the inﬂuence a force ﬁeld F according to Newton’s second law (that is, F = ma) increases after the particle makes one circuit. Can the force ﬁeld governing the particle’s motion be conservative? Solution. No. If the force ﬁeld F(r) is conservative, then F · ds = (Energy at start) − (Energy at ﬁnish) = 0,
C which is a contradiction. ...
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