Math Insight

Pages similar to: Stokes' theorem examples

• 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.
• The idea behind Green's theorem
  Introduction to Green's theorem, based on the intuition of microscopic and macroscopic circulation of a vector field.
• 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.
• Length, area, and volume factors
  A summary of the expansion factors that arise from mappings in integrals of multivariable calculus.
• 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.
• 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.
• Line integrals as circulation
  Definition of circulation as the line integral of a vector field around a closed curve.
• Vector line integral examples
  Example of calculating line integrals of vector fields.

Back to: Stokes' theorem examples