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Want to understand the *index* of a vector field V? Take a little circle around where V vanishes. At some point θ (blue), look at the unit vector V/|V| (orange). As θ goes around by 2pi, the unit vector traverses its own circle in some direction.
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Notice in the above that the blue dot went around its circle once in the anticlockwise direction, while the orange dot went around its circle in the clockwise direction. As these are opposite, this has an index of -1.
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Here's one where the blue and orange dots go around their circles in the same direction. This type of zero has an index of +1.
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What we're doing is making a *map* from a 1-sphere (S^1, better known as a circle) to another 1-sphere. The maps from S^n -> S^n are classified by their *degree*, which is an integer. In 1 dimension, we call it simply the winding number.
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But, the idea works in any number of dimensions. Find an isolated zero. Take a little n-1 sphere of radius epsilon around that zero. At every point on that S^(n-1), normalize the vector field to get V/|V| (which is possible since we are away from the zero).
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Normalizing the vector means it lives on its *own* (n-1) sphere! So for a vector field in any dimension, around an isolated zero, you get a map from S^(n-1) -> S^(n-1), and you can measure its degree.
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Back to two dimensionsâhere's an anticlockwise vortex on top, and a clockwise vortex on bottom. Notice that they have the *same degree*. Both blue dots go around anticlockwise; so do both orange dots on the right. They just start in different places.
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Same thing with this "hedgehog" (source) and "anti-hedgehog" (sink).
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And just to round things out, these two "hyperbolic" vector fields (saddle singularities). [Of course with these two, it's easier to see that the indices will be the same: one vector field can just be rotated by 90 degrees to give the other.]
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The index of a zero of a vector field doesn't change when you stretch/squash, rotate, or reflect the vector field around its zero. It's a fundamental topological property!
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But you *can* take a zero that has an index of +1, and another zero that has an index of -1, and get them to cancel each other out! Nicely visualized here, which was the inspiration for this thread:
https://twitter.com/gravity_levity/status/1361115214549311489
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https://twitter.com/gravity_levity/status/1361115214549311489
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Homework: try to visualize what a zero of index +1 and -1 look like for 3-dimensional vector fields! I've got to head to bed, maybe somebody will have posted it below by the time I wake up ;)
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