Vector field explorer
Pick one of seven classic fields, read its divergence and curl at any point, and ride a streamline through the flow.
F = (−y, x, 0)
Result
Vortex (rigid rotation) · probe (1.5, 0, 0)
The whole space rotates like a turntable. The curl is (0, 0, 2a) — exactly twice the angular velocity, everywhere — and the divergence is zero: it spins without spreading.
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position (1.5, 0, 0)
How is this calculated?
Using the tool
Pick a field, read its exact divergence and curl at any probe point, and then let go: the time slider rides a streamline from your probe through the flow, with the field vector live at every step. Seven canonical fields — sources, vortices, shears, drains, and the inverse square — each with the fact it exists to teach.
The math
A vector field assigns an arrow to every point in space, and two numbers capture what the arrows do locally. The divergence asks: if you sealed a tiny box around this point, is more flow leaving than entering? Positive means a source, negative a sink, zero means the flow just passes through. The curl asks: if you dropped a tiny paddle wheel here, would it spin? Its direction is the spin axis, its magnitude twice the local angular speed. The two are independent — a shear flow spins without spreading, a source spreads without spinning, a drain does both — and the inverse-square field does neither anywhere except at the singular point, where all of its divergence hides: that is exactly the fact Gauss's law turns into a tool. The streamline is the path a speck of dust would follow, its tangent everywhere equal to the field.
New to the math? Learn the concept: Higher dimensions, made visual