Winding number
Definition
editLet be a cycle in , and let be a point that does not intersect. Then
is called the *winding number* of around .
Motivation
editFirst, consider the case where consists of a single closed curve. Then is homologous in to an -fold (for some ) traversed circle around with . Now,
Thus, this integral counts how many times the curve winds around the point .
Task
editLet the closed integration path be defined as:
1. Plot the trace of the integration path.
2. Determine the winding number .
3. Determine the winding number .
4. Determine the winding number .
Additivity of the Integral
editFor a cycle with closed , due to the additivity of the integral, we have
Thus, the winding number also counts how many times the point is encircled.
Length of the Cycle
editFor a cycle with closed , the length of the cycle is defined additively over the lengths of the individual integration paths:
See Also
edit
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- Date: 12/17/2024