Math Insight

Pages similar to: Vector fields as fluid flow

• Vector field overview
  An overview introducing the basic concept of vector fields in two or three dimensions.
• 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.
• 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.
• 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.
• The idea behind Green's theorem
  Introduction to Green's theorem, based on the intuition of microscopic and macroscopic circulation of a vector field.
• An introduction to conservative vector fields
  An introduction to the concept of path-independent or conservative vector fields, illustrated by interactive graphics.
• 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.

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