Calculus×Computing
ES

Surface integrals and flux

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

What is it?

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).

Formulas

∬SF⋅n dS=∬DF(r(u,v))⋅(ru×rv) du dv\iint_S F\cdot n\,\dd S = \iint_D F\big(r(u,v)\big)\cdot\big(r_u\times r_v\big)\,\dd u\,\dd v

Where it shows up in computing

  • The rendering equation★★★★★frequentComputer graphics

    Radiometry (flux, irradiance, radiance) is built on surface and solid-angle integrals.

  • Fluid dynamics and CFD★★★★★frequentPhysics and simulation

    Finite-volume solvers update each cell by the fluxes across its faces.

Where is it used?

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

⚛ Physics and simulation

3D Computer graphics

What depends on it

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

↑ ↓ to navigate · ↵ · Esc