Pages similar to: The definition of curl from line integrals
- 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 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 idea of the curl of a vector field
Intuitive introduction to the curl of a vector field. Interactive graphics illustrate basic concepts. - 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. - Line integrals as circulation
Definition of circulation as the line integral of a vector field around a closed curve. - 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. - 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. - Divergence and curl notation
Different ways to denote divergence and curl. - Divergence and curl example
An example problem of calculating the divergence and curl of a vector field. - The arc length of a parametrized curve
Introduction to the arc length of a parametrized curve. The arc length definition is illustrated with interactive graphics. - Introduction to a line integral of a scalar-valued function
Introduction with interactive graphics illustrating the line integral of a scalar-valued function and informally deriving the formula for calculating the integral from the parametrization of the curve. - Line integrals are independent of parametrization
Description of how the value of a line integral over a curve is independent of the parametrization of the curve. - Examples of scalar line integrals
Examples demonstrating how to calculate line integrals of scalar-valued functions. - 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. - Alternate notation for vector line integrals
An alternative notation for the line integral of a vector field is introduced. - Vector line integral examples
Example of calculating line integrals of vector fields. - When Green's theorem applies
A discussion of situations where you are allowed to use Green's theorem.