### Pages similar to: The idea behind Stokes' theorem

- Proper orientation for Stokes' theorem

The importance of orientating the surface and its boundary correctly when using Stokes' theorem. - The idea behind Green's theorem

Introduction to Green's theorem, based on the intuition of microscopic and macroscopic circulation of a vector field. - Stokes' theorem examples

Examples illustrating how to use Stokes' theorem. - The definition of curl from line integrals

How the curl of a vector field is defined by line integrals representing circulation. - 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. - 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. - The components of the curl

Illustration of the meaning behind the components of the curl. - 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 idea of the curl of a vector field

Intuitive introduction to the curl of a vector field. Interactive graphics illustrate basic concepts. - 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. - Length, area, and volume factors

A summary of the expansion factors that arise from mappings in integrals of multivariable calculus. - Green's theorem with multiple boundary components

How Green's theorem applies even to regions with holes in them - Line integrals as circulation

Definition of circulation as the line integral of a vector field around a closed curve. - The fundamental theorems of vector calculus

A summary of the four fundamental theorems of vector calculus and how the link different integrals. - Introduction to a line integral of a vector field

The concepts behind the line integral of a vector field along a curve are illustrated by interactive graphics representing the work done on a magnetic particle. The graphics motivate the formula for the line integral. - 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. - Introduction to a surface integral of a vector field

How to define the integral of a vector field over a parametrized surface, illustrated by interactive graphics. - The idea behind the divergence theorem

Introduction to divergence theorem (also called Gauss's theorem), based on the intuition of expanding gas. - The gradient theorem for line integrals

A introduction to the gradient theorem for conservative or path-independent line integrals. - Introduction to a surface integral of a scalar-valued function

How to define the integral of a scalar-valued function over a parametrized surface.