Math Insight

Pages similar to: The curl of a gradient is zero

• 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.
• An introduction to the directional derivative and the gradient
  Interactive graphics about a mountain range illustrate the concepts behind the directional derivative and the gradient of scalar-valued functions of two variables.
• Directional derivative and gradient examples
  Examples of calculating the directional derivative and the gradient.
• Derivation of the directional derivative and the gradient
  Derivation of the directional derivative and the gradient from the definition of differentiability of scalar-valued multivariable functions.
• 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 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 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.
• 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 gradient vector
  The gradient vector is the matrix of partial derivatives of a scalar valued function viewed as a vector.

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