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Vector Fields: Mathematics Of Flow And Force

4 micro-lessons

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Real-World Vector Fields

From hurricanes to robot navigation

Weather map showing hurricane with arrows indicating wind velocity and direction at multiple points

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

Compass surrounded by visible magnetic field lines with iron filings on paper

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

Diagram showing robot path with vectors pointing away from obstacles and toward goal

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

Water swirling down a drain showing circular flow pattern with faster motion at edges

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

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Vector Fields

Understanding vectors, fields, and circulation

🌊 Hurricane Maps Show This

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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Apply Vector Field Concepts

Use curl, circulation, and field thinking

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Electric field from a point charge has vectors pointing radially outward: $\vec{E} = k\frac{\vec{r}}{|\vec{r}|^3}$ . If you calculate the curl of this field, you get curl = 0. Why doesn't the paddlewheel spin?

← previous3d Surface In Calculus next →Fourier Series Approximations

Apply Vector Field Concepts

Use curl, circulation, and field thinking

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