Pages similar to: The fundamental theorems of vector calculus
- 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. - The idea behind Green's theorem
Introduction to Green's theorem, based on the intuition of microscopic and macroscopic circulation of a vector field. - When Green's theorem applies
A discussion of situations where you are allowed to use Green's theorem. - Other ways of writing Green's theorem
Small variations in the way you can write Green's theorem - Green's theorem with multiple boundary components
How Green's theorem applies even to regions with holes in them - Using Green's theorem to find area
A trick to use Green's theorem to calculate the area of a region - Green's theorem examples
Examples of using Green's theorem to calculate line integrals - 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. - A conservative vector field has no circulation
How a conservative, or path-independent, vector field will have no circulation around any closed curve. - 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 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. - 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. - The idea behind the divergence theorem
Introduction to divergence theorem (also called Gauss's theorem), based on the intuition of expanding gas. - Divergence theorem examples
Examples of using the divergence theorem.