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.