Math Insight

Pages similar to: The components of the curl

• 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.
• 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 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.
• 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 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 divergence of a vector field
  Intuitive introduction to the divergence of a vector field. Interactive graphics illustrate basic concepts.
• 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.
• Vector field overview
  An overview introducing the basic concept of vector fields in two or three dimensions.
• Vector fields as fluid flow
  Interpretation of vector fields as velocity fields of fluids.
• Line integrals as circulation
  Definition of circulation as the line integral of a vector field around a closed curve.
• 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 curl of a gradient is zero
  Calculation showing that the curl of a gradient is zero.
• A conservative vector field has no circulation
  How a conservative, or path-independent, vector field will have no circulation around any closed curve.
• 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.

Back to: The components of the curl