Vector Fields: Mathematics Of Flow And Force

4 micro-lessons

Preview · progress not saved — Chapter 1 is free

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examples

Real-World Vector Fields

From hurricanes to robot navigation

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Hurricane wind field: each arrow shows wind speed and direction at that location

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Magnetic field: invisible force has direction and magnitude at every point in space

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Robot navigation: repulsive fields around obstacles, attractive field toward goal

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Vortex: circular vector field with curl \neq 0, velocity increases with distance from center

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concept

Understanding vectors, fields, and circulation

Weather apps show wind at every location: speed + direction. That's a vector (magnitude + direction).

Now they show this for EVERY point on the map. That's a vector field — a vector assigned to every location.

The visualization above does the same: each arrow shows the force/velocity at that point.

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quiz

Use curl, circulation, and field thinking

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You calculate curl for a vector field and get curl = 0. You drop a tiny paddlewheel into this field. What happens?