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Ryan Collison

I write about cool science and share it here.

Synthesis: Vector Calculus as the Language of Physics

Synthesis: Vector Calculus as the Language of Physics

Gradient tells you which way a field climbs. Divergence tells you if it's spreading out or converging. Curl tells you if it's rotating. Together, these three operators let you write Maxwell's equations, Navier-Stokes, and almost every other law of classical physics.

Maxwell's Equations: Vector Calculus in Electromagnetism

Maxwell's Equations: Vector Calculus in Electromagnetism

Maxwell's equations are vector calculus distilled to four lines. They completely describe classical electromagnetism—how electric and magnetic fields are created, how they interact, and how they propagate as light. Here they are in differential form: ∇ · E = ρ/ε₀ (Gauss's law) ∇ · B = 0 (no magnetic monopoles)

The Divergence Theorem: From Surface Flux to Volume Sources

The Divergence Theorem: From Surface Flux to Volume Sources

The Divergence Theorem (also called Gauss's Theorem) connects a surface integral over a closed surface to a volume integral over the region it encloses. It says: the flux out of the boundary equals the total divergence inside. ∫∫_S F · dS = ∫∫∫_V (∇ · F) dV where S is a closed

Stokes' Theorem: Generalizing Green's Theorem to Surfaces

Stokes' Theorem: Generalizing Green's Theorem to Surfaces

Stokes' theorem connects what happens on a surface to what happens on its boundary — showing that total rotation inside a region is fully encoded in the flow around its rim. A deep unity between local and global.

Green's Theorem: Relating Line and Double Integrals

Green's Theorem: Relating Line and Double Integrals

Green's Theorem connects a line integral around a closed curve to a double integral over the region it encloses. It says: the circulation around the boundary equals the total curl inside. ∫_C P dx + Q dy = ∬_R (∂Q/∂x - ∂P/∂y) dA where C is a simple

The Gradient Curl and Divergence: Del Operations

The Gradient Curl and Divergence: Del Operations

One symbol, three operations, infinite applications. The del operator ∇ is the Swiss Army knife of vector calculus — and understanding it unlocks electromagnetism, fluid dynamics, and more.

Curl: How Much Does a Field Rotate?

Curl: How Much Does a Field Rotate?

Drop a tiny paddle wheel in a fluid and watch it spin. Curl measures exactly that rotation — the tendency of a field to swirl around a point. It's one of three things the del operator tells you about a field.