Pages similar to: The idea behind Green's theorem
- 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.
- Line integrals as circulation
Definition of circulation as the line integral of a vector field around a closed curve.
- Using Green's theorem to find area
A trick to use Green's theorem to calculate the area of a region
- The definition of curl from line integrals
How the curl of a vector field is defined by line integrals representing circulation.
- 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 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.
- 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.
- Vector line integral examples
Example of calculating line integrals of vector fields.
- 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
- 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 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.
- Length, area, and volume factors
A summary of the expansion factors that arise from mappings in integrals of multivariable calculus.
- Vector field overview
An overview introducing the basic concept of vector fields in two or three dimensions.
- Divergence and curl notation
Different ways to denote divergence and curl.
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