Multiple integrals and change of variables

What is it?

Formulas

$$\iint_R f(x,y)\,\dd A = \int_a^b\!\!\int_{c}^{d} f(x,y)\,\dd y\,\dd x$$

$$\int_{F(U)} g(x)\,\dd x = \int_U g\big(F(u)\big)\,\big|\det J_F(u)\big|\,\dd u$$

Where it shows up in computing

• Monte Carlo methods★★★★★fundamentalScientific computing and algorithms

  Grid quadrature in $d$ dimensions needs $n^d$ points; Monte Carlo's $O(N^{-1/2})$ error does not depend on $d$.

• The rendering equation★★★★★frequentComputer graphics

  Lighting integrals run over the hemisphere of directions and over areas of light sources.

Where it shows up in AI

• Generative models★★★★★advancedAI and machine learning

  Likelihoods of continuous generative models are multiple integrals; flows use the change-of-variables formula.

Where is it used?

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

What depends on it

• Surface integrals and flux
• Green's theorem