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MIT OpenCourseWare 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: .
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V6. Multiply-connected Regions; Topology In Section V5, we called a region D of the plane simply-connected if it had no holes in it. This is a typical example of what would be called in mathematics a topological property, that is, a property that can be described without using measurement. For a curve, such properties as having length 3, or being a circle, or a line, or a triangle - these are not topological properties, since they involve measurement, whereas the property of being closed, or of intersecting itself once, would be topological. The important thing about topological properties is that they are preserved when the geometric figure is deformed continuously without adding or subtracting points, whereas non-topological properties change under such a deformation. For example, if you deform a circle, it will not stay a circle, but it will still remain a closed curve that does not intersect itself. Topology is that branch of mathematics which studies topological properties of geomet- rical figures; it's a kind of geometry, but at the opposite pole from Euclidean geometry, which emphasizes measurement ("congruent triangles" "right angles", "circles"). Topology is a large and active branch of mathematics today, one which is attracting attention from other disciplines, like theoretical physics and molecular biology. Most students have never heard of it, because topological properties don't enter very often into the first few years of mathematics. However, they do right here, and in fact it was just in the study of the possible values of a line integral around a closed curve that the central ideas of modern topology first entered into mathematics, in the middle of the 1800's. So let F be a continuously differentiable vector field in a multiply-connected - i.e., not simply-connected - region D of the xy-plane, and suppose curl F = 0. What values can jc F F. dr have? We begin by considering an earlier example (Section V2, Example 2) in greater detail,
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mult_conectd_reg - MIT OpenCourseWare http/

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