---
title: "EC.AR.5: Multivariable and Vector Integrals (Georgia Mathematics standard)"
code: EC.AR.5
cite_as: Georgia Standards of Excellence, Mathematics, EC.AR.5
subject: math
grades:
  - high-school
language: en
canonical_url: https://georgiahomeroom.org/standards/code/EC.AR.5
md_url: https://georgiahomeroom.org/standards/code/EC.AR.5.md
alternate_language_url: https://georgiahomeroom.org/es/standards/code/EC.AR.5.md
source_url: https://case.georgiastandards.org/ims/case/v1p1/CFDocuments/e9dd7229-3558-4df2-85c6-57b8938f6180
source_last_changed: 2024-10-09
data_built: 2026-09-18
publisher: Georgia Homeroom, a free public-interest project of Georgia Civic Data. Not affiliated with or endorsed by the Georgia Department of Education or any school district.
up: https://georgiahomeroom.org/standards/math/high-school.md
index: https://georgiahomeroom.org/standards/index.md
---

# EC.AR.5: Multivariable and Vector Integrals

Interpret integrals of functions of two independent variables and of vector functions to solve contextual, mathematical problems and to explain engineering-based phenomena.

- Subject: Mathematics
- Appears in: High School, https://georgiahomeroom.org/standards/math/high-school.md

## Expectations

- **EC.AR.5.1**: Manipulate integrals by changing the order of integration, introducing variable substitutions, or changing to curvilinear coordinates.
- **EC.AR.5.2**: Evaluate and apply line integrals that are independent of path.
- **EC.AR.5.3**: Apply properties of integrals to calculate and represent area, volume, or mass.
- **EC.AR.5.4**: Use integrals of vectors to define and apply the gradient, divergence, and the curl.
- **EC.AR.5.5**: Interpret the theorems of Green, Stokes, and Gauss and apply them to the study of real-world phenomena.

## Georgia Milestones coverage

No Georgia Milestones blueprint cites this standard.

Source: Georgia Standards of Excellence, published by the Georgia Department of Education at case.georgiastandards.org.