Pages similar to: The integrals of multivariable calculus
- Length, area, and volume factors
A summary of the expansion factors that arise from mappings in integrals of multivariable calculus.
- 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.
- Examples of changing the order of integration in double integrals
Examples illustrating how to change the order of integration (or reverse the order of integration) in double integrals.
- Double integrals as area
Explanation of how double integrals could be used to represent area.
- Double integrals where one integration order is easier
Illustration of a region where integrating a double integral in one order is must easier than integrating in the other order.
- Introduction to triple integrals
An introduction to the definition of triple integrals as well as their formulation as iterated integrals.
- Triple integral examples
Examples showing how to calculate triple integrals, including setting up the region of integration and changing the order of integration.
- 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.
- 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.
- Line integrals as circulation
Definition of circulation as the line integral of a vector field around a closed curve.
- 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.
- Using Green's theorem to find area
A trick to use Green's theorem to calculate the area of a region
- 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.