Unformatted text preview: 23. Although y x2 -y + y2 = x x2 x + y2 for all (x, y) = (0, 0), Theorem 1 does not imply that curves in 2 . The set consisting of points in and the vector field -yi + xj F= 2 x + y2 x dy - y d x is zero for all closed x 2 + y2 except the origin is not simply connected, is not conservative on any domain in 2 that contains the origin in its interior. (See Example 5.) However, the integral will be 0 for any closed curve that does not contain the origin in its interior. (An example is the curve in Exercise 22(c).) ...
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- Spring '12
- Topology, Manifold, General topology