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Divergence and curl example
Divergence and curl notation
Different ways to denote divergence and curl.
The components of the curl
Illustration of the meaning behind the components of the curl.
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 idea of the curl of a vector field
Intuitive introduction to the curl 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 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.
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 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.
Stokes' theorem examples
Examples illustrating how to use Stokes' theorem.
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 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.
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.
Finding a potential function for three-dimensional conservative vector fields
How to find a potential function for a given three-dimensional conservative, or path-independent, vector field.
Introduction to changing variables in double integrals
Introduction to the concepts behind a change of variables in double integrals. The transformation is illustrated with interactive graphics.
Introduction to differentiability in higher dimensions
An introduction to the basic concept of the differentiability of a function of multiple variables. Discussion centers around the existence of a tangent plane to a function of two variables.
Finding a potential function for conservative vector fields
How to find a potential function for a given conservative, or path-independent, vector field.
Vectors in two- and three-dimensional Cartesian coordinates
A introduction to representing vectors using the standard Cartesian coordinate systems in the plane and in three-dimensional space.
Divergence and curl example