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Subtleties about curl
The idea of the curl of a vector field
Intuitive introduction to the curl of a vector field. Interactive graphics illustrate basic concepts.
The components of the curl
Illustration of the meaning behind the components of the curl.
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.
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.
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.
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.
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.
Subtleties about curl