Pages similar to: Line integrals as circulation
- 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. - The idea behind Green's theorem
Introduction to Green's theorem, based on the intuition of microscopic and macroscopic circulation of a vector field. - Introduction to a line integral of a scalar-valued function
Introduction with interactive graphics illustrating the line integral of a scalar-valued function and informally deriving the formula for calculating the integral from the parametrization of the curve. - Line integrals are independent of parametrization
Description of how the value of a line integral over a curve is independent of the parametrization of the curve. - Examples of scalar line integrals
Examples demonstrating how to calculate line integrals of scalar-valued functions. - 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 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. - 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. - 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. - Introduction to double integrals
The definition of a double integral is motivated through a hair density example. - Double integrals as iterated integrals
A description of how to convert double integrals into two single integrals. - Double integral examples
Examples of integrating double integrals over rectangles and triangles. - Double integrals as volume
Explanation of how double integrals could be used to represent volume.
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