### Pages similar to: Green's theorem with multiple boundary components

- The idea behind Green's theorem

Introduction to Green's theorem, based on the intuition of microscopic and macroscopic circulation of a vector field. - Calculating the formula for circulation per unit area

A sketch of the proof for the formula for the component of the curl of a vector field. - A path-dependent vector field with zero curl

A counterexample showing how a vector field could have zero curl but still fail to be conservative or path-independent. - When Green's theorem applies

A discussion of situations where you are allowed to use Green's theorem. - The idea behind Stokes' theorem

Introduction to Stokes' theorem, based on the intuition of microscopic and macroscopic circulation of a vector field and illustrated by interactive graphics. - The fundamental theorems of vector calculus

A summary of the four fundamental theorems of vector calculus and how the link different integrals. - Subtleties about curl

Counterexamples illustrating how the curl of a vector field may differ from the intuitive appearance of a vector field's circulation. - The components of the curl

Illustration of the meaning behind the components of the curl. - Green's theorem examples

Examples of using Green's theorem to calculate line integrals - The idea of the curl of a vector field

Intuitive introduction to the curl of a vector field. Interactive graphics illustrate basic concepts. - The definition of curl from line integrals

How the curl of a vector field is defined by line integrals representing circulation. - Using Green's theorem to find area

A trick to use Green's theorem to calculate the area of a region - Other ways of writing Green's theorem

Small variations in the way you can write Green's theorem - Line integrals as circulation

Definition of circulation as the line integral of a vector field around a closed curve. - A conservative vector field has no circulation

How a conservative, or path-independent, vector field will have no circulation around any closed curve. - How to determine if a vector field is conservative

A discussion of the ways to determine whether or not a vector field is conservative or path-independent. - The integrals of multivariable calculus

A summary of the integrals of multivariable calculus, including calculation methods and their relationship to the fundamental theorems of vector calculus. - Proper orientation for Stokes' theorem

The importance of orientating the surface and its boundary correctly when using Stokes' theorem. - Stokes' theorem examples

Examples illustrating how to use Stokes' theorem. - An introduction to conservative vector fields

An introduction to the concept of path-independent or conservative vector fields, illustrated by interactive graphics.