Calculus×Computing
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Vector calculus

Fields in space and the operators that describe them — gradient, divergence, curl, Laplacian — plus the integral theorems of Green, Gauss and Stokes. The mathematics of fluids, electromagnetism and physically based graphics.

10 topics

A field assigns a value to every point of space: a temperature (scalar field) or a velocity (vector field). Three differential operators summarize how a field behaves locally — the gradient (where it increases), the divergence (how much it spreads out from a point) and the curl (how much it swirls) — and the Laplacian combines two of them. The theorems of Green, Gauss and Stokes say that what happens inside a region is determined by what flows across its boundary; finite-volume simulators are built directly on that idea.

Topics

Scalar fields

A function ϕ:ℝ3→ℝ\phi : \R^3 \to \R seen as a quantity spread through space: temperature, pressure, density, electric potential, distance to a surface.

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Vector fields

A vector attached to every point: the wind, the flow of water, a magnetic field, the force of gravity. Integral curves (streamlines) follow the arrows.

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Divergence

∇⋅F=∂xP+∂yQ+∂zR\nabla\cdot F = \partial_x P + \partial_y Q + \partial_z R: the net outflow per unit volume at a point. Positive at sources, negative at sinks, zero for incompressible flow.

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Curl

∇×F\nabla\times F: a vector measuring the local rotation of a field — its axis is the axis of spin, its length twice the angular speed. Fields that are gradients have zero curl.

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Laplacian

Δf=∇⋅∇f=∑i∂2f/∂xi2\Delta f = \nabla\cdot\nabla f = \sum_i \partial^2 f/\partial x_i^2: how much ff at a point differs from the average of its neighbours. It appears in the heat, wave, Laplace, Poisson and Schrödinger equations, and its discrete version smooths meshes and detects edges.

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Line integrals

Integrating a field along a curve: the work ∫CF⋅dr\int_C F\cdot\dd r done by a force along a path. For gradient fields it depends only on the endpoints — the definition of a conservative force.

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Surface integrals and flux

Integrating over a surface: its area, or the flux ∬SF⋅n dS\iint_S F\cdot n\,\dd S — how much of a field passes through it per unit time (water through a net, light through a window).

Advanced

Green's theorem

In the plane, the circulation around a closed curve equals the integral of the curl inside it. A computational gem: the area of a polygon from its vertices (the shoelace formula) is Green's theorem.

AdvancedTheorem

Divergence theorem (Gauss)

The total divergence inside a region equals the flux out through its boundary: what is produced inside must leave through the surface. It is the foundation of conservation laws and finite-volume methods.

AdvancedTheorem

Stokes' theorem

The circulation of a field around a closed curve equals the flux of its curl through any surface bounded by the curve. Green's theorem in 3D, and the integral form of Faraday's and Ampère's laws.

AdvancedTheorem

Where this area leads in computing

3D Computer graphics ★★★★★

⚛ Physics and simulation ★★★★★

ψ Quantum computing and physics ★★★★★

∿ Signals, media and vision ★★★★★

ℒ AI and machine learning ★★★★★

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