##  [Hopf Umlaufsatz](/hopf-umlaufsatz) 

  ##  [Hopf Umlaufsatz](https://puremath.quantumdictionary.io/hopf-umlaufsatz-0) 

  

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**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The Hopf Umlaufsatz states that for a regular simple closed curve in the plane the total signed rotation (turning) angle of the tangent vector as one traverses the curve once equals 2π times the curve's winding (rotation) index; equivalently, the rotation index measures how many times the tangent completes a full turn.