# line integrals

(1.3 hours to learn)

## Summary

The line integral gives a notion of integrating a vector field along a curve. Common uses include computing work done by a force and finding potential functions corresponding to a gradient field.

## Context

This concept has the prerequisites:

- functions of several variables
- vector fields (Vector fields are something we compute line integrals of.)

## Goals

- Be able to compute line integrals of vector fields with respect to curves

- Be able to express the line integral in terms of the arc length parameterization, or in terms of an arbitrary parameterization

- Why does the line integral of a vector field depend on the orientation of a curve?

- Be aware that the line integrals are independent of the parameterization of a path, for a given orientation

## Core resources (read/watch one of the following)

## -Free-

→ MIT Open Courseware: Multivariable Caclulus (2010)

Video lectures for MIT's introductory multivariable calculus class.

## -Paid-

→ Multivariable Calculus

An introductory multivariable calculus textbook.

Location:
Section 15.2, "Line integrals," pages 1019-1028

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

Location:
Section 8.3, "Line integrals and Green's Theorem," not counting subsections, pages 348-351

Additional dependencies:

- differential forms
- exterior derivative
- pullback

Other notes:

- This gives a more rigorous treatment than the other resources.

## See also

- Green's Theorem is a powerful theorem relating line integrals to integrals of functions
- Surface integrals are the higher-dimensional analogue