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In Mathematics , the winding number is a property of a Curve , which plays a leading role in Complex Analysis and Algebraic Topology . It is the fundamental case of Degree Of A Continuous Mapping . INTUITIVE DESCRIPTION Intuitively, the winding number of a Curve γ with respect to a point is the number of times γ goes around in a counter-clockwise direction (number of Turn s). In the image on the right, the winding number of the curve (C) about the inner point pictured (''z''0) is 3, since the curve makes three full revolutions around the point. The small loop on the left does not go around the point and so has no effect overall. Note that if the direction of the curve were reversed, the winding number would be −3 instead of 3. TURNING NUMBER One can also consider the winding number of the path with respect to the tangent of the path itself. As a path followed through time, this would be the winding number with respect to the origin of the velocity vector. In this case the example illustrated on the right has a winding number of 4 (or −4), because the small loop ''is'' counted. This is only defined for immersed paths (i.e., for differentiable paths with nowhere vanishing derivatives), and is the degree of the tangential Gauss Map . This is called the turning number. FORMAL DEFINITIONS There are distinct but related concepts of a winding number in Complex Analysis and in Topology . It may be defined as follows. Complex analysis If γ is a closed rectifiable curve in C, and ''z''0 is a point in C not on γ, then the ''winding number'' of γ with respect to ''z''0 (alternately called the ''index'' of γ with respect to ''z''0) is defined by the formula: : This is verifiable from applying the Cauchy Integral Formula — the Integral will be a multiple of 2πi, since each time γ goes about ''z''0, we have effectively calculated the integral again. The winding number is used in the Residue Theorem . Topology In Topology , the winding number is an alternate term for the Degree Of A Continuous Mapping . In Physics , winding numbers are frequently called Topological Quantum Number s. In both cases, the same concept applies. The above example of a curve winding around a point has a simple topological interpretation. The complement of a point in the plane is es of maps from a Topological Space to the circle is called the first Homotopy Group or Fundamental Group of that space. The fundamental group of the circle is the integers Z and the winding number of a complex curve is just its homotopy class. Maps from the 3-sphere to itself are also classified by an integer which is also called the winding number or sometimes Pontryagin Index . SEE ALSO EXTERNAL LINKS |
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