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Scalar fields

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

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

Formulas

\phi : \R^3 \to \R, \qquad \nabla\phi = \Big(\frac{\partial\phi}{\partial x}, \frac{\partial\phi}{\partial y}, \frac{\partial\phi}{\partial z}\Big)

Where it shows up in computing

Signed distance fields and ray marching★★★★★frequentComputer graphics
An SDF is a scalar field whose value is the signed distance to a shape.
Weather and climate modelling★★★★★frequentPhysics and simulation
Temperature, pressure and humidity are scalar fields on a 3D grid of the atmosphere.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

3D Computer graphics

Vector fields→Divergence→Laplacian→Mesh processing (discrete differential geometry)★★★★★
Signed distance fields and ray marching★★★★★
Vector fields→Surface integrals and flux→The rendering equation★★★★★

⚛ Physics and simulation

Vector fields→Electromagnetism (Maxwell's equations)★★★★★
Vector fields→Fluid dynamics and CFD★★★★★
Vector fields→Fluid dynamics and CFD→Weather and climate modelling★★★★★
Vector fields→Divergence→Laplacian→Heat equation and diffusion★★★★★
Vector fields→Divergence→Laplacian→Wave equation★★★★★
Vector fields→Line integrals→Classical mechanics★★★★★
+2

ψ Quantum computing and physics

Vector fields→Divergence→Laplacian→Schrödinger equation★★★★★

∿ Signals, media and vision

Vector fields→Divergence→Laplacian→Image processing and computer vision★★★★★
Vector fields→Divergence→Laplacian→Image processing and computer vision→Media compression (JPEG, MP3, video)★★★★★

ℒ AI and machine learning

Vector fields→Divergence→Generative models★★★★★

What depends on it

Vector fields

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.