# Double Integrals

Canonical: https://manyspace.aporia.systems/topics/double-integrals-mth-multc-176
Updated: 2026-10-13

Mathematics · Multivariable Calculus · MTH-MULTC-176

- Difficulty: 3 / 5
- Importance: 5 / 5
- Study time: 3–6 h

iterated integrals, regions, change of order

## Why it matters

Extends calculus to functions of several variables; essential for electromagnetism, fluid mechanics, GR, optimization and ML.

## Where it is used

Electromagnetic fields, fluid flow, gradient descent in ML, economic surfaces, thermodynamics potentials, graphics rendering.

## How you know you have learned it

Complete 3 independent examples correctly + explain common failure modes.

## What you need first

- [Riemann Sums & Definite Integral](https://manyspace.aporia.systems/topics/riemann-sums-definite-integral-mth-calc1-138) (required)
- [Integration by Parts](https://manyspace.aporia.systems/topics/integration-by-parts-mth-calc2-140) (required)
- [Partial Derivatives](https://manyspace.aporia.systems/topics/partial-derivatives-mth-multc-171) (required)
- [Gaussian Elimination](https://manyspace.aporia.systems/topics/gaussian-elimination-mth-linalg-155) (helpful)

## What it opens

- [Triple Integrals](https://manyspace.aporia.systems/topics/triple-integrals-mth-multc-178)
- [Change of Variables](https://manyspace.aporia.systems/topics/change-of-variables-mth-multc-179)

## Resources

- Thomas' Calculus — Hass, Heil & Weir
- Calculus: Early Transcendentals — James Stewart
- Calculus: Early Transcendentals — James Stewart