Pages similar to: A path-dependent vector field with zero curl
- A conservative vector field has no circulation
How a conservative, or path-independent, vector field will have no circulation around any closed curve. - 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 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. - The gradient theorem for line integrals
A introduction to the gradient theorem for conservative or path-independent line integrals. - 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. - Finding a potential function for conservative vector fields
How to find a potential function for a given conservative, or path-independent, vector field. - Finding a potential function for three-dimensional conservative vector fields
How to find a potential function for a given three-dimensional conservative, or path-independent, vector field. - A simple example of using the gradient theorem
An example of using the gradient theorem to calculate the line integral of a conservative, or path-independent, vector field. - Testing if three-dimensional vector fields are conservative
Examples of testing whether or not three-dimensional vector fields are conservative (or path-independent). - The idea of the curl of a vector field
Intuitive introduction to the curl of a vector field. Interactive graphics illustrate basic concepts. - The components of the curl
Illustration of the meaning behind the components of the curl. - When Green's theorem applies
A discussion of situations where you are allowed to use Green's theorem. - Green's theorem with multiple boundary components
How Green's theorem applies even to regions with holes in them - The definition of curl from line integrals
How the curl of a vector field is defined by line integrals representing circulation. - An introduction to conservative vector fields
An introduction to the concept of path-independent or conservative vector fields, illustrated by interactive graphics. - 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 divergence
Counterexamples illustrating how the divergence of a vector field may differ from the intuitive appearance of the expansion of a vector field. - Divergence and curl notation
Different ways to denote divergence and curl.